Symmetric Functions
and
Hall Polynomials


Second Edition


I. G. MACDONALD


Queen Mary and Westfield College
University of London




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Macdonald, 1. G. (Ian Grant)
Symmetric functions and Hall polynomials / I. G. Macdonald. - 2nd ed.
(Oxford mathematical monographs)
Includes bibliographical references and index.
1. Abelian groups. 2. Finite groups. 3. Hall polynomials.
4. Symmetric functions. 1. Title. 11. Series.
QA180.M33 1995 512'.2-dc20 94-27392 CIP
ISBN 0 19 853489 2 h/b
o 19 850450 0 p/b
Typeset by
Technical Typesetting Ireland
Printed in Great Britain by
Bookcraft (Bath) Ltd., Midsomer Norton, Avon

Preface to the second edition
The first edition of this book was translated into Russian by A. Zelevinsky
in 1984, and for the Russian version both the translator and the author
furnished additional material, both text and examples. Thus the original
purpose of this second edition was to make this additional material
accessible to Western readers. However, in the intelVening years other
developments in this area of mathematics have occurred, some of which I
have attempted to take account of: the result, I am afraid, is a much longer
book than its predecessor. Much of this extra bulk is due to two new
chapters (VI and VII) about which I shall say something below.
For readers acquainted with the first edition, it may be of use to indicate
briefly the main additions and new features of this second edition. The text
of Chapter I remains largely unchanged, except for a discussion of transi-
tion matrices involving the power-sums in 6, and of the internal (or inner)
product of symmetric functions in 7. On the other hand, there are more
examples at the ends of the various sections than there were before. To
the appendix on polynomial functors I have added an account of the
related theory of polynomial representations of the general linear groups
(always in characteristic zero), partly for its own sake and partly with the
aim of rendering the account of zonal polynomials in Chapter VII self-
contained. I have also included, as Appendix B to Chapter I, an account,
following Specht's thesis, of the characters of wreath products G,..., S n (G
any finite group), along the same lines as the account of the characters of
the symmetric groups in Chapter I, 7: this may serve the reader as a sort
of preparation for the more difficult Chapter IV' on the characters of the
fini te general linear groups.
In Chapter II, one new feature is that the formula for the Hall
polynomial (or, more precisely, for the polynomial gs(t) (4.1)) is now made
completely explicit in (4.11). The chapter is also enhanced by the appendix,
written by A. Zelevinsky for the Russian edition.
The main addition to Chapter III is a section (8) on Schur's Q-
functions, which are the case t = -1 of the Hall-Littlewood symmetric
functions. In this context I have stopped short of Schur's theory of the
projective representations of the symmetric groups, for which he intro-
duced these symmetric functions, since (a) there are now several recent
accounts of this theory available, among them the monograph of P.
Hoffman and J. F. Humphreys in this series, and (b) this book is already
long enough.

VI
PREFACE TO THE SECOND EDITION
Chapters IV and V are unchanged, and require no comment.
Chapter VI is new, and contains an extended account of a family of
symmetric functions PA(x; q, t), indexed as usual by partitions A, and
depending rationally on two parameters q and t. These symmetric func-
tions include as particular cases many of those encountered earlier in the
book: for example, when q = 0 they are the Hall-Littlewood functions of
Chapter III, and when q = t they are the Schur functions of Chapter I.
They also include, as a limiting case, Jack's symmetric functions depending
on a parameter a. Many of the properties of the Schur functions general-
ize to these two-parameter symmetric functions, but the proofs (at present)
are usually more elaborate.
Finally, Chapter VII (which was originally intended as an appendix to
Chapter VI, but outgrew that format) is devoted to a study of the zonal
polynomials, long familiar to statisticians. From one point of view, they are
a special case of Jack's symmetric functions (the parameter a being equal
to 2), but their combinatorial and group-theoretic connections make them
worthy of study in their own right. This chapter can be read independently
of Chapter VI.
London, 1995
I. G. M.

Preface to the first edition
This monograph is the belated fulfilment of an undertaking made some
years ago to publish a self-contained account of Hall polynomials and
related topics.
These polynomials were defined by Philip Hall in the 1950s, originally as
follows. If M is a finite abelian p-group, it is a direct sum of cyclic
subgroups, of orders pAl, pA2,..., pAr say, where we may suppose that
A1  A2  ...  Ar. The sequence of exponnts A = (At,..., Ar) is a parti-
tion, called the type of M, which describes M up to isomorphism. If now J.L
and v are partitions, let g;v(p) denote the number of subgroups N of M
such that N has type J.L and M/N has type v. Hall showed that g;v(p) is
a polynomial function of p, with integer coefficients, and was able to
determine its degree and leading coefficient. These polynomials are the
Hall polynomials.
More generally, in place of finite abelian p-groups we may consider
modules of finite length over a discrete valuation ring 0 with finite residue
field: in place of g;v(p) we have g;v(q) where q is the number of
elements in the residue field.
Next, Hall used these polynomials to construct an algebra which reflects
the lattice structure of the finite o-modules. Let H(q) be a free Z-module
with basis (uA) indexed by the set of all partitions A, and define a
multiplication in H(q) by using the g;v(q) as structure constants, i.e.
uJLUV = E g;v(q )UA.
A
It is not difficult to show (see Chapter II for the details) that H(q) is a
commutative, associative ring with identity, which is freely generated
(as Z-algebra) by the generators u(Y) corresponding to the elementary 0-
modules.
Symmetric functions now come into the picture in the following way.
The ring of symmetric polynomials in n independent variables is a poly-
nomial ring Z[e1,..., en] generated by the elementary symmetric functions
et,..., ell' By passing to the limit with respect to n, we obtain a ring
A = Z[et, e2,...] of symmetric functions in infinitely many variables. We
might therefore map H(q) isomorphically onto A by sending each genera-
tor u(Y) to the elementary symmetric function ere However, it turns out
that a better choice is to define a homomorphism «/1: H(q)  A  Q by
/(U(1r») = q-r(r-l)/2er for each r  1. In this way we obtain a family of

Vlll
PREFACE TO THE FIRST EDITION
symmetric functions t/1(u»), indexed by partitions. These symmetric func-
tions are essentially the Hall-Littlewood functions, which are the subject
of Chapter III. Thus the combinatorial lattice properties of finite 0-
modules are reflected in the multiplication of Hall-Littlewood functions.
The formalism of symmetric functions therefore underlies Hall's theory,
and Chapter I is an account of this formalism-the various types of
symmetric functions, especially the Schur functions (S-functions), and the
relations between them. The character theory of the symmetric groups, as
originally developed by Frobenius, enters naturally in this context. In an
appendix we show how the S-functions arise 'in nature' as the traces of
polynomial functors on the category of finite-dimensional vector spaces
over a field of characteristic O.
In the past few years, the combinatorial substructure, based on the 'jeu
de taquin', which underlies the formalism of S-functions and in particular
the Littlewood-Richardson rule (Chapter I, 9), has become much better
understood. I have not included an account of this, partly from a desire to
keep the size of this monograph within reasonable bounds, but also
because Schiitzenberger, the main architect of this theory, has recently
published a complete exposition [S7].
The properties of the Hall polynomials and the Hall algebra are devel-
oped in Chapter II, and of the Hall-Littlewood symmetric functions in
Chapter III. These are symmetric functions involving a parameter t, which
reduce to S-functions when t == 0 and to monomial symmetric functions
when t = 1. Many of their properties generalize known properties of
S - functions.
Finally, Chapters IV and V apply the formalism developed in the
previous chapters. Chapter IV is an account of J. A. Green's work [Gl1]
on the characters of the general linear groups over a finite field, and we
have sought to bring out, as in the case of the character'theory of the
symmetric groups, the role played by symmetric functions. Chapter V is
also about general linear groups, but this time over a non-archimedean
local field rather than a finite field, and instead of computing characters
we compute spherical functions. In both these contexts Hall's theory plays
a decisive part.
Queen Mary College,
London 1979
I. G. M.

Contents
I. SYMMETRIC FUNCTIONS
1. Partitions
2. The ring of symmetric functions
3. Schur functions
4. Orthogonality
5. Skew Schur functions
6. Transition matrices
7. The characters of the symmetric groups
8. Plethysm
9. The Littlewood-Richardson rule
Appendix A: Polynomial functors and polynomial representations
Appendix B: Characters of wreath products
1
17
40
62
69
99
112
135
142
149
169
II. HALL POLYNOMIALS
1. Finite o-modules
2. The Hall algebra
3. The LR-sequence of a submodule
4. The Hall polynomial
Appendix (by A. Zelevinsky): Another proof of Hall's theorem
179
182
184
187
199
III. HALL- LITTLEWOOD SYMMETRIC
FUNCTIONS
1. The symmetric polynomials R).
2. Hall-Littlewood functions
3. The Hall algebra again
4. Orthogonality
5. Skew Hall-Littlewood functions
6. Transition matrices
7. Green's polynomials
8. Schur's Q-functions
204
208
215
222
226
238
246
250
IV. THE CHARACTERS OF GLn OVER
A FINITE FIELD
1. The groups Land M
2. Conjugacy classes
3. Induction from parabolic subgroups
4. The characteristic map
5. Construction of the characters
269
270
273
276
280

x CONTENTS
6. The irreducible characters 284
Appendix: proof of (5.1) 291
V. THE HECKE RING OF GLn OVER
A LOCAL FIELD
1. Local fields
2. The Hecke ring H(G, K)
3. Spherical functions
4. Hecke series and zeta functions for GLn(F)
5. Hecke series and zeta functions for GSP2n(F)
292
293
298
300
302
VI. SYMMETRIC FUNCTIONS WITH
TWO PARAMETERS
1. Introduction
2. Orthogonality
3. The operators D
4. The symmetric functions PA(x; q, t)
5. Duality
6. Pieri formulas
7. The skew functions PAl,.,,' QA/,."
8. Integral forms
9. Another scalar product
10. Jack's symmetric functions
305
309
315
321
327
331
343
352
368
376
VII. ZONAL POLYNOMIALS
1. Gelfand pairs and zonal spherical functions
2. The Gelfand pair (S2n, Hn)
3. The Gelfand pair (GL,,(R), D(n»
4. Integral formulas
5. The complex case
6. The quaternionic case
388
401
414
424
440
446
BIBLIOGRAPHY
NOTATION
INDEX
457
467
473

I
SYMMETRIC FUNCTIONS
1. Partitions
Many of the objects we shall consider in this book will turn out to be
parametrized by partitions. The purpose of this section is to lay down some
notation and terminology which will be used throughout, and t6 collect
together some elementary results on orderings of partitions which will be
used later.
Partitions
A partition is any (finite or infinite) sequence
(1.1)
A = (A], A2, · · · , Ar' · · . )
of non-negative integers in decreasing order:
Al  A2  ...  A r  ...
and containing only finitely many non-zero terms. We shall find it conve-
nient not to distinguish between two such sequences which differ only by a
string of zeros at the end. Thus, for example, we regard (2, 1), (2, 1,0),
(2, 1,0,0, . . .) as the same partition.
The non-zero A; in (1.1) are called the parts of A. The number of parts
is the length of A, denoted by leA); and the sum of the parts is the weight
of A, denoted by I AI:
/ A/ = A] + A2 + . .. .
If I AI = n we say that A is a partition of n. The set of all partitions of n is
denoted by 9'n, and the set of all partitions by 9'. In particular, 9'0 consists
of a single element, the unique partition of zero, which we denote by O.
Sometimes it is convenient to use a notation which indicates the number
of times each integer occurs as a part:
A = (1 m12m2 . .. rm, . . . )
means that exactly m; of the parts of A are equal to i. The number
(1.2)
m; = miCA) = Card{j: Aj = i}
is called the multiplicity of i in A.
 \\rull

2
I SYMMETRIC FUNCTIONS
Diagrams
The diagram of a partition A may be formally defined as the set of points
(i,j) E Z2 such that 1 j  Ai. In drawing such diagrams we shall adopt
the convention, as with matrices, that the first coordinate i (the row index)
increases as one goes downwards, and the second coordinate j (the column
index) increases as one goes from left to right.t For example, the diagram
of the partition (5441) is
. . . . .
. . . .
. . . .
.
consisting of 5 points or nodes in the top row, 4 in the second row, 4 in the
third row, and 1 in the fourth row. More often it is convenient to replace
the nodes by squares, in which case the diagram is
We shall usually denote the diagram of a partition A by the same symbol
A.
The conjugate of a partition A is the partition A' whose diagram is the
transpose of the diagram A, i.e. the diagram obtained by reflection in the
main diagonaL Hence A'i is the number of nodes in the ith column of A, or
equivalently
(1.3)
A'i = Card{j: Aj  i}.
In particular, A'} = I( A) and Al = I( A'). Obviously A" = A.
For example, the conjugate of (5441) is (43331).
From (1.2) and (1.3) we have
(1.4)
mi(A) = Xi - A'i+1.
t Some authors (especially Francophones) prefer the convention of coordinate geometry (in
which the first coordinate increases from left to right and the second coordinate from bottom
to top) and define the diagram of A to be the set of (i, j) E Z2 such that 1  i  Aj. Readers
who prefer this convention should read this book upside down in a mirror.

1. PARTITIONS
3
For each partition A we define
(1.5)
n(A)= E (i-1)Ai'
i 1
so that n(A.) is the sum of the numbers obtained by attaching a zero to
each node in the top row of the diagram of A., a 1 to each node in the
second row, and so on. Adding up the numbers in each column, we see
that
(1.6)
n( A) = E (Xi).
i 1 2
Another notation for partitions which is occasionally useful is the
following, due to Frobenius. Suppose that the main diagonal of the
diagram of A consists of r nodes (i, i) (1  i  r). Let ai = Ai - i be
the number of nodes in the ith row of A to the right of (i, i), for 1  i  r,
and let {3i = Xi - i be the number of nodes in the ith column of A below
(i, i), for 1 <: i  r. We have at > a2 > ... > aT  0 and /3t > {32 > ... >
{3,  0, and we denote the partition A by
A=(at,...,a,1 (3t,...,/3,)=(al (3).
Clearly the conjugate of (a I ) is (  I a).
For example, if A = (5441) we have a = (421) and f3 = (310).
(1.7) Let A be a partition and let m  AI' n > Xl. Then the m + n numbers
Ai+n-i (lin),
n - 1 + j -  (1 j  m)
are a permutation of {O, 1,2,..., m + n - I}.
Proof. The diagram of A is contained in the diagram of (mn), which is an
n X m rectangle. Number the successive segments of the boundary line
between A and its complement in (mn) (marked thickly in the picture) with
the numbers 0, 1, . . . , m + n - 1, starting at the bottom. The numbers
attached to the vertical segments are Ai + n - i (1 <: i  n), and by trans-
position those attached to the horizontal segments are
(m + n - 1) - ( + m - j) = n - 1 + j - >.i
(1jm).

4
I SYMMETRIC FUNCTIONS
1iI1!11 II rI II ;;
fi*1j
:.....;;;
..:=:.:_:..-:_.t-: .
II m::::: II fI
,..::::..:::'
iiri;
II II II II
II
A=-=(5421), m=8, n=6
Let
n
fA, n (t) = E t Ai + n - i.
i-I
Then (1.7) is equivalent to the identity
(1.7') !A,n(t) + tm+n-1fA',m(t-1) = (1 - tm+n)/(I- t).
Skew diagrams and tableaux
If A, J-L are partitions, we shall write A::> J-L to mean that the diagram of A
contains the diagram of J-L, i.e. that Ai  J-Li for all i  1. The set-theoretic
difference () = A - J-L is called a skew diagram. For example, if A = (5441)
and J-L = (432), the skew diagram A - J.L is the shaded region in the picture
below:
II
tlllt'
.:1iir:'.
II II
II
A path in a skew diagram 8 is a sequence Xo, Xl'...' xm of squares in ()
such that Xi-l and xi have a common side, for 1  i  m. A subset cp of 0
is said to be connected if any two squares in cp can be connected by a path
in cpo The maximal connected subsets of 0 are themselves skew diagrams,
called the connected components of 8. In the example above, there are
three connected components.
The conjugate of a skew diagram 0 = A - J-L is 0' = A' - J-L'. Let 0i =
Ai - J-Lj, OJ = A - Ji;, and
181 = E Oi = 1'\1-1 J-LI.

1. PARTITIONS
5
A skew diagram 9 is a horizontal m-strip (resp. a vertical m-strip) if
I () I = m and 9;  1 (resp. 9;  1) for each i  1. In other words, a horizontal
(resp. vertical) strip has at most one square in each column (resp. row).
If 9 = A - J-L, a necessary and sufficient condition for () to be a horizon-
tal strip is that the sequences A and J-L are interlaced, in the sense that
Al  J-Ll  A2  J-L2  ... ·
A skew diagram () is a border strip (also called a skew hook by some
authors, and a ribbon by others) if () is connected and contains no 2 X 2
block of squares, so that successive rows (or columns) of () overlap by
exactly one square. The length of a border strip () is the total number 191 of
squares it contains, and its height is defined to be one less than the
number of rows it occupies. If we think of a border strip () as a set of
nodes rather than squares, then by joining contiguous nodes by horizontal
or vertical line segments of unit length we obtain a sort of staircase, and
the height of () is the number of vertical line segments or 'risers' in the
staircase ·
If A = (at,..., aT 1131'...' (3,) and J-L = (a2'...' a, 1 {32,...' (3,), then
A - J-L is a border strip, called the border (or rim) of A.
A (column-strict) tableau T is a sequence of partitions
J-L=A(O)CA(1)C ... c,\(r) = A
such that each skew diagram (}(i) = ,\(i) - A(i-l) (1  i  r) is a horizontal
strip. Graphically, T may be described by numbering each square of the
skew diagram (}(i) with the number i, for 1  i  r, and we shall often
think of a tableau as a numbered skew diagram in this way. The numbers
inserted in A - J-L must increase strictly down each column (which explains
the adjective 'column-strict') and weakly from left to right along each row.
The skew diagram A - J-L is called the shape of the tableau T, and the
sequence (I (}(1)I,.. ., 10(r)l) is the weight of T.
We might also define row-strict tableaux by requiring strict increase
along the rows and weak increase down the columns, but we shall have no
use for them; and throughout this book a tableau (unqualified) will mean a
column-strict tableau, as defined above.
A standard tableau is a tableau T which contains each number 1,2,..., r
exactly once, so that its weight is (1, 1, . . . , 1).
Addition and multiplication of partitions
Let A, J-L be partitions. We define A + J-L to be the sum of the sequences A
and J-L:
(A + J-L) i = Ai + J-Li.

6
I SYMMETRIC FUNCTIONS
Also we define A U J-L to be the partition whose parts are those of A and J-L,
arranged in descending order. For example, if A = (321) and J.L = (22), then
A + J-L = (541) and A U J.L = (32221).
Next, we define AJ-L to be the componentwise product of the sequences
A, J-L:
(AJ-L)i = Ai J-Li.
Also we define A X J-L to be the partition whose parts are min(Ai, J-Lj) for all
i  I( A) and j  I( J-L), arranged in descending order.
The operations + and U are dual to each other, and so are the two
multiplications:
(1.8)
(A U J-L)' = A' + J-L' ,
(A X J-L)' = A' J-L' ·
Proof The diagram of A U J.L is obtained by taking the rows of the
diagrams of A and J-L and reassembling them in order of decreasing length.
Hence the length of the kth column of A U J.L is the sum of the lengths of
the kth columns of A and of J-L, i.e. (A U J-LYk = Xk + J.Lk.
Next, the length of the kth column of AX J-L is equal to the number of
pairs (i, j) such that Ai  k and J.Lj  k, hence it is equal to Xk Jik.
Consequently (A X J-LYk = Xk J.Lk. I
Orderings
Let Ln denote the reverse lexicographic ordering on the set 9'71 of partitions
of n: that is to say, Ln is the subset of 9'71 X94'n consisting of all (A, J.L) such
that either A = J-L or the first non-vanishing difference Ai - J-Li is positive.
Ln is a total ordering. For example, when n = 5, Ls arranges 9'5 in the
sequence
(5),(41),(32),(312),(221),(213),(15).
Another total ordering on 9'71 is En' the set of all (A, J.L) such that either
A = J-L or else the first non-vanishing difference AT - J.LI is negative, where
Ai = An + 1- i. The orderings Ln' I.:n are distinct as soon as n  6. For
example, if A = (313) and J-L = (23) we have (A, J-L) E L6 and ( J-L, A) E I:6.
(1.9) Let A, J-L E9'n. Then
(A, J-L) EI:n  (J-L', A') ELn.
Proof Suppose that (A, J.L) E 1.:71 and A:;= J-L. Then for some integer i  1
we have Ai < J.Li' and Aj = J-Lj for j > i. If we put k = Ai and consider the
diagrams of A and J-L, we see immediately that  = JJ1 for 1 j  k,
and that Xk+ 1 < ,lk+ l' so that (J-L', A') E LII. The converse is proved
similarly. I

1. PARTITIONS
7
An ordering which is more important than either Ln or En is the natural
(partial) ordering Nn on 9'n (also called the dominance partial ordering by
some authors), which is defined as follows:
(A, J.L) E Nn .. At + . ., + Ai  J.Lt + ,.. + J.Li for all i  1.
As soon as n  6, Nn is not a total ordering. For example, the partitions
(313) and (23) are incomparable with respect to N6,
We shall write A  J.L in place of (A, J.L) E Nn.
(1.10) Let A, J.L EIPn, Then
A  J.L  (A, J.L) E Ln nEn.
Proof. Suppose that A  J.L, Then either At > J.Ll' in which case (A, J.L) E Ln,
or else  = J.Lt, In that case either A2 > J.L2' in which case again (A, J.L) E Ln,
or else A2 = J.L2' Continuing in this way, we see that (A, J.L) E Ln.
Also, for each i  1, we have
Ai+l + Ai+2 + ,.. = n - (AI + ,.. +Ai)
 n - ( J.L I + '" + J.Li)
= J.Li + t + J.Li + 2 + ".
Hence the same reasoning as before shows that (A, J.L) E En.
Remark. It is not true in general that Nn = Ln n En. For example, when
n = 12 and A = (632), J.L = (5212) we have (A, J.L) E L12 n E12, but (A, J.L) 
N12.
(1.11) Let A, J.L EIPn, Then
A  J.L  J.L'  A' ·
Proof. Clearly it is enough to prove one implication. Suppose then that
p.,'  A', Then for some i  1 we have
Xl + · .. + >-i  iiI + ... + P1
(1 j  i-I)
and
(1)
A'l + ... + Xi > J.il + · .. + Ji;
from which it follows that Xi > Ji;.
Let I = A';, m = Ji;. From (1) it follows that
(2)
Xi+l + Al+2 + .., < Ji;+1 + Jii+2 + ...

8
I SYMMETRIC Fl]NCTIONS
Now A'; + I + Xi + 2 + . .. is equal to the number of nodes in the diagram of A
which lie to the right of the ith column, and therefore
I
A'; + t + Xi + 2 + ... = E (Aj - i).
j==1
Likewise
m
+t+P:;+2+'" = E(lLj-i).
j-I
Hence from (2) we have
(3)
m 1 m
E (lLj-i» E (Aj-i) E (Aj-i)
j-I 1'=1 j=1
in which the right-hand inequality holds because 1 > m and Aj  i for
1 j <:.1. From (3) we have
ILt + ... + ILm > At + ... + Am
and therefore A 1:- IL.
Raising operators
In this subsection we shall work not with partitions but with integer vectors
a = (at,..., an) E zn. The symmetric group Sn acts on zn by permuting the
coordinates, and the set
Pn = {b E zn: bI  b2  ...  bn}
is a fundamental domain for this action, i.e. the Sn-orbit of each a E zn
meets Pn in exactly one point, which we denote by a+. Thus a+ is
obtained by rearranging a1,..., an in descending order of magnitude.
For a, b E zn we define a  b as before to mean
at + ... +ai  bI + ... + bi
(1 in).
(1.12) Let a E zn. Then
a E Pn  a  wa for all w E S n .
Proof. Suppose that a E Pn' i.e. a]  ...  an. If wa = b, then (bt,..., bn) is
a permutation of (at,..., an)' and therefore
al + ... +a;  bt +... +b;
(1in)
so that a  b.

1. PARTITIONS
9
Conversely, if a  wa for all W E Sn we have in particular
(a1,...,an)  (at,...,ai-l,ai+l,ai,ai+2,...,an)
for 1 <: i  n - 1, from which it follows that
at + ... +ai-1 +aial + ... +ai-t +ai+t,
i.e. a;  ai+ 1. Hence a E Pn.
Let S = (n - 1, n - 2,.. 0,1,0) E Pno
(1.13) Let a E Pn. Then for each w E Sn we have (a + 8 - wS)+ a.
Proof. Since 8 E Pn we have S  w8 by (1.12), hence
a + S - w8  a. '
Let b = (a + 8 - w8)+. Then again by (1.12) we have
b  a + 8 - w8.
Hence b  a.
For each pair of integers i, j such that 1  i <j  n define Rij: zn  zn
by
Rij(a) = (at,...,ai + 1,...,aj -1,...,an).
Any product R = OJ < j RiY is called a raising operator. The order of the
terms in the product is immaterial, since they commute with each other.
(1.14) Let a E zn and let R be a raising operator. Then
Ra  a.
For we may assume that R = R;j' in which case the result is obvious.
Conversely:
(1.15) Let a, b E zn be such that a  b and a) + ... +an = b1 + ... +bn.
Then there exists a raising operator R such that b = Ra.
Proof. We may take
n-l
R = n Rkk+l
k....l
where
k
rk = E (bi - ai)  o.
;=1

10
I SYMMETRIC FUNCTIONS
(1.16) If A, J.L are partitions of n such that A > IL, and if A, IL are adjacent for
the natural ordering (so that A  v  JL implies either v = A or Jl = IL), then
A = Rij IL for some i <j.
Proof. Suppose first that Al > ILl' and let i  2 be the least integer for
which Al + ... + Ai = ILl + .. · + J.Li. Then we have lLi > Ai  Ai+ 1  JLi+ l' so
that JLi > lLi + 1. Consequently v = R Ii JL is a partition, and one sees immedi-
ately that A  v. Hence A = v = Rli IL.
If now Al = ILl' then for some j > 1 we have At = ILk for k <j and
Aj > ILj. The preceding argument may now be applied to the partitions
(Aj, Aj+ 1'...) and (JLj,"JLj+ 1'...). I
Remark. This proposition leads directly to an alternative proof of (1.11);
for it shows that it is enough to prove (1.11) in the case that A = Rij IL, in
which case it is obvious.
Examples
1. Let A be a partition. The hook-length of A at x = (i,j) E A is defined to be
hex) = h(i,j) = Ai + Ai - i - j + 1.
From (1.7'), with A and A' interchanged, and m = AI' we have
AI n Al+n-1
E t Aj + Al - j + E t At - 1 + j - A1' = E t j
j-l j-=1 j-O
or, putting J.Li = Ai + n - i (1  i  n),
(1)
Al n ILl
E th(l,j)+ E t#LJ-#LJ= E ti.
j-l j-2 j-I
By writing down this identity for the partition (Ai' Ai + l' . . . ) and then summing over
i = 1,2,..., leA) we obtain
(2)
ILl
E th(x) + E t#LI-lLj = E E ti.
xeA i<j i;.1 j-l
From (2) it follows that
(3)
ILl
n n (1 - ti)
n (1 - t h(x») = i;. 1 j - 1
xE A n (1 - t IL;-#Lj)
i<j

1. PARTITIONS
11
and in particular, by dividing both sides of (3) by (1 - t)IAI and then setting t = 1,
that
(4)
n I-Li!
n h(x) = i;.1 .
xe A - - n (lLi - ILj)
i<j
2. The sum of the hook-lengths of A is
E hex) = n(A) + n(A') + tAl.
XE A
3. For each x = (i,j) E A, the content of x is defined to be c(x) = j - i. We have
E c(x) = n(A') - n(A).
xEA
If n is any integer '> leA), the numbers n + c(x) for x in the ith row of A are
n - i + 1,. . . , n - i + Ai' and therefore
n n 'PJ..,+n-;(t)
(1 - tn +c(x) =
XE A i> 1 'Pn-i(t)
where 'P,(t) == (1 - tX1 - t2)... (1 - t').
4. If A = (AI"'.' An) == (at,..., a, I (31'...' f3,) in Frobenius notation, then
n r
E ti(l - t- A;) = E (t /3j+ 1 - t- Qj).
i-I j-l
5. For any partition A,
E (h(x)2 - C(x)2) = IAI2.
xeA
6. Let A be a partition and let T, S be positive integers. Then Ai - Ai+,  s for all
i  I(A) if and only if >.1- Xj+s  r for all j  leA').
7. The set .9'n of partitions of n is a lattice with respect to the natural ordering: in
other words, each pair A, IL of partitions of n has a least upper bound 0' = sup( A, IL)
and a greatest lower bound T = inf( A, IL). (Show that l' defined by
, ( , ')
E Ti = min E Ai' E lLi
i-I i-I i-I
for all r '> 1 is indeed a partition; this establishes the existence of inf( A, IL). Then
define u= sup(A, p.) by u' = inf(A', IL'). The example A = (313), IL = (23), u= (321)
shows that it is not always true that
t uj = max( t Ai' t lLi) .)
i-I i-I i-I
6

12
I SYMMETRIC FUNCTIONS
8. Let P be an integer  2.
(a) Let A, J.L be partitions of length  m such that A::> J.L, and such that A - J.L is a
border strip of length p. Let 8m = (m - 1, m - 2, . . . , 1,0) and let g = A + 8m,
71 = J.L + 8m. Show that 71 is obtained from  by subtracting p from some part i of
 and rearranging in descending order . (Consider the diagrams of  and 71.)
(b) With the same notation, suppose that  has m, parts i congruent to r modulo
p, for each r = 0, 1, . . . ,p - 1. These i may be written in the form plr) + r
(1  k  m,), where 1(') > 4') > ... > ,)  o. Let A) = 1r) - mr + k, so that
,
).,(r) = (A'),..., ).,») is a partition. The collection A* = (A(O), A(1),..., A(p-l») is called
,
the p-quotient of the partition A. The effect of changing m  l( A) is to permute the
A(r) cyclically, so that A* should perhaps be thought of as a 'necklace' of partitions.
The m numbers ps + r, where 0  s  m, - 1 and 0  r p - 1, are all distinct.
Let us arrange them in descending order, say i1 > ... > im' and define a partition
A by Ai == ii - m + i (1  i  m). This partition A is called the p-core (or p-residue)
of ).,. Both X and ).,* (up to cyclic permutation) are independent of m, provided
that m  l( A).
If A = X (i.e. if A* is empty), the partition A is called a p-core. For example, the
only 2-cores are the 'staircase' partitions 8m = (m - 1, m - 2, . . . , 1).
Following G. D. James, we may conveniently visualize this construction in terms
of an abacus. The runners of the abacus are the half-lines x  0, y = r in the plane
R2, where r == 0,1,2,..., p - 1, and A is represented by the set of beads at the
points with coordinates (,), r) in the notation used above. The removal of a
border strip of length p from A is recorded on the abacus by moving some bead
one unit to the left on its runner, and hence the passage from A to its p-core
corresponds to moving all the beads on the abacus as far left as they will go.
This arithmetical construction of the p-quotient and p-core is an analogue for
partitions of the division algorithm for integers (to which it reduces if the partition
has only one part).
(c) The p-core of a partition A may be obtained graphically as follows. Remove a
border strip of length p from the diagram of A in such a way that what remains is
the diagram of a partition, and continue removing border strips of length p in this
way as long as possible. What remains at the end of this process is the p-core A of
A, and it is independent of the sequence of border strips removed. For by (a) above,
the removal of a border strip of length p from A corresponds to subtracting p
from some part of  and then rearranging the resulting sequence in descending
order; the only restriction is that the resulting set of numbers should be all distinct
and non-negative.
(d) The p-quotient of A can also be read off from the diagram of A, as follows. For
5, t = 0, 1,..., P - 1 let
R s = {( i , j) E A: Ai - i == s (mod p ) } ,
Ct = ((i,j) E A:j - Aj == t(JTIodp)},

1. PARTITIONS
13
so that Rs consists of the rows of A whose right-hand node has content (Example
3) congruent to s modulo p, and likewise for C,. If now (i,j) e Rs n C" the
hook-length at (i, j) is
h(i,j) = Ai +  -i -j + 1 ==s - t + l(modp)
and therefore p divides h(i, j) if and only if t == s + l(mod p).
On the other hand, if i = plr) + r as in (b) above, the hook lengths of A in the
ith row are the elements of the sequence (1,2,..., i) after deletion of i-
gi + l' · · · , i - gm. Hence those divisible by p are the elements of the sequence
(p, 2 p, . . . ,plr» after deletion of p( lr) - gl 1)' . . . , p( €r) - :». They are
therefore p times the hook lengths in the kth row of A(r), and in particular there
are A) of them.
It follows that each A(r) is embedded in ,\ as Rs () Cs+], where s = r - m(mod p),
and that the hook lengths in A(r) are those, of the corresponding nodes in
Rs n Cs+ l' divided by p. In particular, if m is a multiple of p (which we may
assume without loss of generality) then ,\(r) == ,\ n Rr n Cr+ 1 for each r (where
Cp == Co).
(e) From (c) and (d) it follows that the p-core (resp. p-quotient) of the conjugate
partition A' is the conjugate of the p-core (resp. p-quotient) of A.
(f) For any two partitions A, J.L we shall write
A ,..., p J.L
to mean that A == [L, i.e. that A and J.L have the same p-core. As above, let
 = A + 8m, 11 == J.L + 8m' where m  max(l( A), l( J.L». Then it follows from (a) and
(b) that A"'" p J.L if and only if 11 == w(mod p) for some permutation w e Sm. Also,
from (e) above it follows that A,..., p J.L if and only if A' ,..., p J.L'.
(g) From the definitions in (b) it follows that a partition A is uniquely determined
by its p-core A and its p-quotient A.. ince I AI == I AI + p I A *1, the generating
function for partitions with a given p-core A is
E tl1 = tIAlp(tPY'
jL-).
where P(t) == nn 1(1 - tn)-1 is the partition generating function. Hence the
generating function for p-cores is
EtlAI =P(t)/P(tPY'
(1 - tnpY'
=n 1 n ·
nl -t
In particular, when p == 2 we obtain the identity
(*)
1 - t 2n
E tm(m-l)/2 = n
m  1 n  1 1 - t 2n - 1 ·

14
I SYMMETRIC FUNCTIONS
We shall leave it to the interested reader to write down the corresponding identity
for p> 2. It turns out to be a specialization of the 'denominator formula' for the
affine Lie algebra of type A 1 [Kl]." Thus in particular (*) is a specialization of
Jacobi's triple product identity.
9. (a) A partition is strict if all its parts are distinct. If J.L = ( J.Lt,. . ., J.Lr) is a strict
partition of length r (so that J.Ll > J.L2 > ... > J.Lr > 0), the double of J.L is the
partition .,\ = ( J.L1'. · · , J.Lr I J.Ll - 1,. · ., J.Lr - 1) in Frobenius notation, and the dia-
gram of ,\ is called the double diagram D( J.L) of J.L. The part of D( J.L) that lies
above the main diagonal is called the shifted diagram S( J.L) of J.L; it is obtained from
the usual diagram of J.L by moving the i th row (i - 1) squares to the right, for each
i > 1. Thus D( J.L) consists of S( J.L) dovetailed into its reflection in the diagonal.
Let m  1(.,\) == J.Lh and let ==.,\ + 8m where 8m = (m - 1, m - 2,...,1,0) as in
Example 8. The first r parts of  are J.Ll + m,..., J.Lr + m, and since the partition
('\,+ 1'''.' Am) is the conjugate of (J.L1 - r,..., J.Lr - 1), it follows from (1.7) that  is
obtained from the sequence (J.Ll + m, J.L2 + m,..., J.L, + m, m - 1, m - 2,...,1,0) by
deleting the numbers m - J.L" . . . , m - J.Lt. Hence  satisfies the following condi-
tion:
( *) an integer j between ° and 2m occurs in  if and only if 2m - j does not.
Conversely, if  satisfies this condition, then A is the double of a strict partition.
(b) Let p be an integer  2, and consider the p-quotient and p-core (Example 8) of
.A. Without loss of generality we may assume that m is a multiple of p, so that
2m == (n + 1) p with n odd. As in Example 8, suppose that for each r = 0, 1, . . . , p - 1
the gi congruent to r modulo p are plr) + r (1  k  m ,), where gJr) > ... >
g)  o. Since p') + r  2m = (n + 1) p, it follows that l')  n if r * 0, and that
,
10)  n + 1.
Suppose first that r =#= 0, and let s = p - r. From above, for each k = 1,2,..., m,
the number 2m - (pr) + r) = p(n - l'» + s does not occur in . Hence the
numbers n - ,) (1  k  mr) and lS) (1  k  ms) fill the interval [0, n] of Z,
from which it follows first that m, + ms = n + 1, and second (by (1.7» that the
components .A(') and ,\(s) of the p-quotient of .A are conjugate partitions. In
particular, if p is even, .,\(p/2) is self-conjugate.
Next, if r = ° we have 2m - pO) = p(n + 1 - g1°» and therefore the sequence
g(O) = (gfO),..., gO;) satisfies the condition (*), so that 2mo = n + 1 and .,\(0) is the
double of a strict partition.
Finally, it fonows from the definition of the p-core in Example 8(b) and the fact
that m, + ms = n + 1 when r + s == ° (mod p) that the sequence f (loc. cit.)
satisfies the condition (*), and hence the p-core A is the double of a strict
partition.
(c) Let J.L as before be a strict partition of length r. If (i, j) e J.L and k  i, an
(i, j)k-bar of J.L consists of the squares (a, b) e J.L such that a = i or k and b  j,
and is defined only when the diagram obtained by the removal of these squares has
no two rows of equal length; so that (0 if k > i there is an (i, j)k-bar only when
j = 1, in which case it consists of the ith and kth rows of J.L; and (ii) when k = i
there is an (i,j)k-bar only when j - 1 is not equal to any of J.Li+ 1'"'' J.Lr The length
of a bar is the number of squares it contains.

1. PARTITIONS
15
Show that for each i  1 the lengths of the (i, j)k-bars of JL are the hook lengths
in the double diagram D( JL) at the squares (i, k) with k> i, that is to say at the
squares in the i th row of the shifted diagram S( JL).
Let II be the strict partition whose diagram is obtained from that of JL by
removing a bar of length p > 1 and then rearranging the rows in descending order
of length. Thus II is obtained from JL in case 0) by deleting JL; and JLk' where
lLi + JLk == p, and in case (ii) by replacing JL; by JL; - p and rearranging. Show that
the double diagram of II is obtained from that of IL by removing two border strips
of length p, one of which lies in rows i, i + 1,. . . , and the other in columns
i, i + 1,... .
(d) A strict partition is a p-bar core if it contains no bars of length p. By starting
with a strict partition JL, removing a bar of length p, rearranging the rows if
necessary, then repeating the process as often as necessary, we shall end up with a
strict partition JL called the p-bar core of JL. It follows from above that the double
of J.L is the p-core (Example 8) of the double of JL, and that JL is independent of
the sequence of moves described above to reach it.
10. For any partition A, let
h(A) == n h(x)
xe,\
denote the product of the hook-lengths of A (Example 1). With the notation of
Example 8, we have
h(A) == pIA.lh(A*)h'(A),
where h(A*) == nr.:l h(,\(r», and h'(A) is the product of the hook-lengths h(x)
that are not multiples of p. (Use formula (4) of Example 1.)
If, moreover, p is prime, then
h'(A) == OAh(A) (modp)
where OA = :i: 1. Hence in particular, when p is prime,' ,\ is a p-core if and only if
h(A) is prime to p.
11. Let A be a partition. The content polynomial of A is the polynomial
c,\(X) = n (X + c(x»
xe,\
(see 3, Example 4) where X is an indeterminate.
(a) Let m  1(,\) and let i = Ai + m - i (1  i  m) as in Example 8. Then
cA(X + m) =- fi X + i
c,\(X+m-l) i-I X+m-;.
(b) Let p be a prime number. If (J is a border strip of length p, the contents c(x),
x E (J are p consecutive integers, hence are congruent modulo p to 0, 1, . . . , p - 1
in some order. If (} == A - JL, it follows that
c,\(X) = c,.,,(X)(XP - X) (mod p).

16
I SYMMETRIC FUNCTIONS
Hence, for any partition '\,
c,\(X) == c.\(X)(Xp _X)IA.I (modp)
where (Example 8) A and ,\* are respectively the p-core and p-quotient of '\.
(c) From (a) and (b) it follows that if p is prime and 1,\1 = I J.LI, then
,\  p J.L  CA (X) == cJL(X) (mod p).
12. Let 9'n denote the set of partitions of n, and N + the set of positive integers.
For each r  1 let
a(r, n) = Card{('\, i) E9'n X N+: '\; == r},
b(r, n) = Card{('\, i) E9'n X N+: m;('\)  r}.
Show that
a(r,n) =b(r,n) =p(n -r) +p(n - 2r) +...
where p(m) is the number of partitions of m.
Deduce that
n (n ;mm) = n (n mj(A)!).
,\ e9',. i> 1 ,\ E9'n i > 1
Let h(r, n) == Card{('\, x): 1,\1 = n, x e ,\ and h(x) == r), where (Example 1) h(x) is
the hook-length of ,\ at x. Show that
h (r , n) == ra (r , n) .
13. A matrix of non-negative real numbers is said to be doubly stochastic if its row
and column sums are all equal to 1.
Let '\, J.L be partitions of n. Show that ,\  J.L if and only if there exists a doubly
stochastic n X n matrix M such that M,\ = J.L (where '\, J.L are regarded as column
vectors of length n). (If ,\  J.L we may by (1.16) assume that ,\ = R;j J.L. Now define
M == (m,s) by
m.. = m.. == (,\. -,\. - 1)/(,\. - ,\.)
II. }} ,'} '} ,
m.. == m.. == 1/(,\. - ,\.)
I}}I I J
and m,s == 8,s othelWise. Then M is doubly stochastic and M A = J.L.)
14. Let A be a partition. If s == (i, j), with i, j  1, is any square in the first
quadrant, we define the hook-length of ,\ at s to be h(s) == A; + ,\i - i - j + 1.
When sEA this agrees with the previous definition, and when sEA it is negative.
For each r E Z let u,( A) denote the number of squares s in the. first quadrant such
that h(s) = r. Show that u_,(A) = u,(A) + r for all r e Z.
If u,(,\) = u,( J.L) for all r E Z, does it follow that A = J.L or ,\ = J.L'?

2. THE RING OF SYMMETRIC FUNCTIONS
17
15. (a) Let ,\ be a partition, thought of as an infinite sequence, and let u be the
sequence (t - i)i> 1. Show that ,\ + u and _(,\' + u) are complementary subse-
quences of Z + t.
(b) Let g,\(t)=(1-t}!:i>1t'\j-i, which is a polynomial in t and t-1. Show that
g).(t) = gA,(t-1).
16. Let '\, J..L, II, 'TT' be partitions such that ,\  II and J..L  'TT'. Show that A + J..L 
p + '1T', ,\ U J..L  II U '1T', '\J..L  II'TT', and ,\ X J..L  II X 'IT.
17. Let ,\ =: ('\1' . . · , '\n) be a partition of length  n with '\1  n, so that ,\ c (nn).
A A /lit. A
The complement of ,\ in (nn) is the partition A == (A},..., An) defined by Ai == n -
A
An +}- i' so that the diagram of ,\ is obtained by giving the complement of the
diagram of ,\ in (nn) a half-turn.
A A
Suppose now that ,\ == (a I 13) in Frobenius notation. Show that A == ( 13 I a),
A
where a (resp. J3) is the complement in [0, n -,I] of the sequence a (resp. J3).
18. For '\, J..L partitions of n, let 'TT'n be the probability that ,\  J..L. Does 'TT'n -+ 0 as
n ----+ oo?
Notes and references
The idea of representing a partition by its diagram goes back to Ferrers
and Sylvester, and the diagram of a partition is called by some authors the
Ferrers diagram or graph, and by others the Young diagram. Tableaux and
raising operators were introduced by Alfred Young in his series of papers
on quantitative substitutional analysis [Y2].
Example 8. The notion of the p-core of a partition was introduced by
Nakayama [Nl], and the p-quotient by Robinson [R6] and Littlewood
[LI0].
Example 9. The notions of a bar and of the p-bar core are due to A. O.
Morris [MI4]. See also Morris and Yaseen [MI6] and Humphreys [Hl2].
2. The ring of symmetric functions
Consider the ring Z[xl,..., xn] of polynomials in n independent variables
Xl'.'" xn with rational integer coefficients. The symmetric group Sn acts
on this ring by permuting the variables, and a polynomial is symmetric if it
is invariant under this action. The symmetric polynomials form a subring
An = Z[ XI' . . . , X n ] S n .
An is a graded ring: we have
A = in Ak
n W n
k__O
where A consists of the homogeneous symmetric polynomials of degree
k, together with the zero polynomial.

18
I SYMMETRIC FUNCTIONS
For each a = (at,..., an) E Nn we denote by xa the monomial
Xo =XOI Xa"
t ... n ·
Let A be any partition of length  n. The polynomial
(2.1)
m,\(xt,...,Xn) = Lxa
summed over all distinct permutations a of A = (AI'...' An), is clearly
symmetric, and the m,\ (as A runs through all partitions of length  n)
form a Z-basis of An. Hence the mA such that I(A) <: nand IAI = k form a
Z-basis of A; in particular, as soon as n;;' k, the m,\ such that IAI == k
form a Z-basis of A.
In the theory of symmetric functions, the number of variables is usually
irrelevant, provided only that it is large enough, and it is often more
convenient to work with symmetric functions in infinitely many variables.
To make this idea precise, let m ;;, n and consider the homomorphism
Z[ Xl' · · · , X m]  Z[ x t , .. · , X n ]
which sends each of x n + l' . . . , X m to zero and the other Xi to themselves.
On restriction to Am this gives a homomorphism
Pm,n: Am  An
whose effect on the basis (m,\) is easily described; it sends mA(xt,..., xm)
to m,\(xt,..., xn) if I(A)  n, and to 0 if I(A) > n. It follows that Pm,n is
surjective. On restriction to A we have homomorphisms
ok · Ak  Ak
rm, n · m n
for all k  0 and m  n, which are always surjective, and are bijective for
m  n  k.
We now form the inverse limit
Ak = lim Ak
+- n
n
of the Z-modules A relative to the homomorphisms P,n: an element of
Ak is by definition a sequence f= (fn)n> 0' where each In = [n(xt,...,xn) is
a homogeneous symmetric polynomial of degree k in x],..., xn' and
fm(xt,. .., Xn, 0,. · .,0) = [n(xI,. . ., xn) whenever m  n. Since p,n is an
isomorphism for m  n ;;, k, it follows that the projection
Pk. Ak  Ak
n.11. 11.n,

2. THE RING OF SYMMETRIC FUNCTIONS
19
which sends f to fn' is an isomorphism for all n  k, and hence that Ak
has a Z-basis consisting of the monomial symmetric functions m A (for all
partitions A of k) defined by
P: (m A) = m A (x 1 , · · · , X n )
for all n  k. Hence Ak is a free Z-module of rank p(k), the number of
partitions of k.
Now let
A = EB Ak,
k;;.O
so that A is the free Z-module generated by the mA for all partitions A.
We have surjective homomorphisms
Pn = EB P:: A --+ An
k>O
for each n  0, and Pn is an isomorphism in degrees k  n.
It is clear that A has a structure of a graded ring such that the Pn are
ring homomorphisms. The graded ring A thus defined is called the ring of
symmetric functionst in countably many independent variables Xl' x2,... .
Remarks. 1. A is not the inverse limit (in the category of rings) of the
A
rings An relative to the homomorphisms Pm.n' This inverse limit, A say,
contains for exaple the infinite product n_1(1 + x;), which does not
belong to A, since the elements of A are by definition finite sums of
monomial symmetric functions mAt However, A is the inverse limit of the
An in the category of graded rings.
2. We could use any commutative ring A in place of Z as coefficient ring;
in place of A we should obtain AA ::: A  A. .
Elementary symmetric functions
For each integer r  0 the rth elementary symmetric function e, is the sum
of all products of r distinct variables Xi' so that eo = 1 and
e =
,
X; x; ...x; =m(t')
12,
;1<;2< ... <;,
E
for r  1. The generating function for the e, is
(2.2)
E(t) = E e,t' = n (1 +x;t)
r> 0 ;;> 1
t The elements of A (unlike those of A,.) are no longer polynomials: they are formal infinite
sums of monomials. We have therefore reverted to the older terminology of 'symmetric
functions' .

20
I SYMMETRIC FUNCTIONS
(t being another variable), as one sees by multiplying out the product on
the right. (If the number of variab'es is finite, say n, then e, (i.e. Pn(e,)) is
zero for all r> n, and (2.2) then takes the form
n n
E e,t'= n (1 +x;t),
r-O i-I
both sides now being elements of An[t]. Similar remarks will apply to many
subsequent formulas, and we shall usually leave it to the reader to make
the necessary (and obvious) adjustments.)
For each partition A = (A1, A2,...) define
e). = e Al e A2 · .. ·
(2.3) Let A be a partition, A' its conjugate. Then
e)., = mA + E aAJJ.mJJ.
p.
where the aAp' are non-negative integers, and the sum is over partitions J.L < A
in the natural ordering.
Proof. When we multiply out the product e A' = e ,\'1 e ,\'2 . .. , we shall obtain a
sum of monomials, each of which is of the form
(X;IXi2"')(XhXh...)... =xQ,
say, where i1 < i2 < ... < iA'I' jt <j2 < ... <j ,\'2' and so on. If we now enter
the numbers it, i2,..., iA'1 in order down the first column of the diagram of
A, then the numbers jt, j2"'" j A'2 in order down the second column, and so
on, it is clear that for each r  1 all the symbols  r so entered in the
diagram of A must occur in the top r rows. Hence a] + ... + a, 
A] + ... + A, for each r  1, i.e. we have a  A. By (1.12) it follows that
eA, = E a'\IJ.mIJ.
p.<A
with aAJJ.  0 for each J.L  A, and the argument above also shows that the
monomial x). occurs exactly once, so that aAA = 1. I
(2.4) We have
A = Z[ e 1 , e 2 , .. · ]
and the e, are algebraically independent over Z.

2. THE RING OF SYMMETRIC FUNCTIONS
21
Proof. The mA form a Z..basis of A, and (2.3) shows that the eA form
another Z-basis: in other words, every element of A is uniquely expressible
as a polynomial in the e,. ,I
Remark. When there are only finitely many variables Xt,..., xn' (2.4)
states that An = Z[el,..., en], and that e1,..., en are algebraically indepen-
dent. This is the usual statement of the 'fundamental theorem on symmet-
ric functions'.
Complete symmetric functions
For each r  0 the rth complete symmetric function h, is the sum of all
monomials of total degree r in the variables Xl' X2'''.' so that
h,= E mA.
IAI-r
In particular, ho = 1 and hI = et. It is convenient to define h, and e, to be
zero for r < O.
The generating function for the h, is
(2.5)
H(t) = E h,t' = n (1-xit)-1.
r;>O ;;>1
To see this, obselVe that
(1 -xit)-l = E x;tk,
k;>O
and multiply these geometric series together.
From (2.2) and (2.5) we have
(2.6)
H(t)E( -t) = 1
or, equivalently,
(2.6')
n
E (-1)' e,hn_, = 0
r-O
for all n  1.
Since the e, are algebraically independent (2.4), we may define a
homomorphism of graded rings
w: A -+ A
by
w(e,) = h,

22
I SYMMETRIC FUNCTIONS
for all r  O. The symmetry of the relations (2.6') as between the e's and
the h's shows that
(2.7) w is an involution, i.e. w2 is the identity map. I
It follows that w is an automorphism of A, and hence from (2.4) that
(2.8) We have
A = Z[ hI' h 2 , · · · ]
and the h, are algebraically independent over Z.
Remark. If the number of variables is finite, say n (so that e, = 0 for r> n)
the mapping w: An -+ An is defined by w(e,) == h, for 1  r  n, and is still
an involution by reason of (2.6'); we have An == Z[h1,..., hn] with hI'...' hn
algebraically independent, but hn+ l' hn+2,... are non-zero polynomials in
hI' · · · , h n (or in e I' · · · , en).
As in the case of the e's, we define
hA = h).lhA2...
for any partition A == (AI' A2,...). By (2.8), the h). form a Z-basis of A. We
now have three Z-bases, all indexed by partitions: the mA, the e)., and the
hA, the last two of which correspond under the involution w. If we define
fA == w(m).)
for each partition A, the fA form a fourth Z-basis of A. (The fA are the
'forgotten' symmetric functions: they have no particularly simple direct
description.)
The relations (2.6') lead to a determinant identity which we shall make
use of later. Let N be a positive integer and consider the matrices of
N + 1 rows and columns
H== (hi-j)O<i,j<N'
E = (( _l)H ei-j )O<i,j<N
with the convention mentioned earlier that h, = e, == 0 for r < O. Both H
and E are lower triangular, with l's down the diagonal, so that det H =
det E = 1; moreover the relations (2.6') show that they are inverses of each
other. It follows that each minor of H is equal to the complementary
cofactor of E', the transpose of E.
Let A, JL be two partitions of length  p, such that A' and IL' have
length  q, where p + q = N + 1. Consider the minor of H with row
indices Ai + P - i (1  i p) and column indices JLi + P - i (1  i p). By

2. THE RING OF SYMMETRIC FUNCTIONS
23
(1.7) the complementary cofactor of E' has row indices p - 1 + j - A}
(1 j  q) and column indices p - 1 + j - J.L} (1 j < q). Hence we have
( ) - ( -1)1"1+11 d ( _l)A"-P1-;+j )
det hA1-Jl.j-i+j 1<i,j<p - et eAj-Jl.j_;+j 1<;,j<q.
The minus signs cancel out, and therefore we have (Aitken [AI])
(2.9)
det(h" - II- -i+J.)1 .. = det(eA, -II! -;+J.)1 .. ·
I j <',J<p , j "',J<q
In particular, taking J.L = 0:
(2.9') det(hA,_i+j) = det(e,\',_;+j).
power sums
For each r  1 the rth power sum is
Pr = Ex; = mer).
The generating function for the Pr is
P(t) = E Prtr-1 = E E x[tr-1
r-.1 ;-.1 r1
= E Xj
;1 I-xit
d 1
= E - log
i1 dt I-xit
so that
dId
(2.10) P(t) = -d log n (l-xit)- = -d log H(t) =H'(t)/H(t).
t i 1 t
Likewise we have
(2.10')
d
P(-t)= dt logE(t}=E'(t}/E(t).
From (2.10) and (2.10') we obtain
(2.11)
n
nhn = E Prhn-r,
r-1
(2.11')
n
nen= E (-1)r-1Pren_r
r-1

24
I SYMMETRIC FUNCTIONS
for n  1, and these equations enable us to express the h's and the e's in
terms of the p's, and vice versa. The equations (2.11') are due to Isaac
Newton, and are known as Newt"on's formulas. From (2.11) it is clear that
hn E Q[PI'''.' Pn] and Pn E Z[hI,..., hn], and hence that
Q[Pl,...,Pn] = Q[hJ,...,hn].
Since the h, are algebraically independent over Z, and hence also over Q,
it follows that
(2.12) We have
AQ = A  Q = Q[ PI' P2'''.]
and the Pr are algebraically independent over Q.
Hence, if we define
PA = PAl PA2 · · ·
for each partition A = (AI' A2,...), then the PA form a Q-basis of AQ. But
they do not form a Z-basis of A: for example, h2 = t(p; + P2) does not
have integral coefficients when expressed in terms of the PA.
Since the involution w interchanges E(t) and H(t) it follows from (2.10)
and (2.10') that
W(Pn) = (_l)n-I Pn
for all n  1, and hence that for any partition A we have
(2.13)
w( PA) = 8,\ PA
where 8A = ( -1)IAI-/('\).
Finally, we shall express hn and en as linear combinations of the PA" For
any partition A, define
ZA = Oimi.mj!
i 1
where mj = mj(A) is the number of parts of A equal to i. Then we have
(2.14)
H(t) = EZ;lp,\tl'\l,
A
E(t) = E 8,\z;lpAtIAI,
A

2. THE RING OF SYMMETRIC FUNCTIONS
25
or equivalen tIy
(2.14')
hn = E Z;lpA,
/AI-n
en = E BAZ;lPA.
IAI-n
Proof. It is enough to prove the first of the identities (2.14), since the
second then follows by applying the involution wand using (2.13). From
(2.10) we have
H(t) = exp E p,t' /r
, 1
= n exp( p,t' /r)
, 1
ex>
= n E (p,t,)mr Irm, . m,!
r 1 m - 0
r
= Ez;IPAtIAI. I
,\
(2.15) Remark. In the language of A-rings [B5], [K13]) the ring A is the
'free A-ring in one variable' (or, more precisely, is the underlying ring).
Consequently all the formulas and identities in this Chapter can be
translated into this language. It is not our intention to write a text on the
theory of A-rings: we shall merely provide a brief dictionary.
If R is any A-ring and x any element of R, there exists a unique
A-homomorphism A -+ R under which e1( = hI = PI) is mapped to x.
Under this homomorphism
p,
U'(x) = (-1)' A'( -x)
At(x)
ut(x) = A-t( -x)
rfr'(x)
(rth exterior power)
(rth symmetric power)
e, is mapped to A'(x)
h,
E(t)
H(t)
(Adams operations)
and the involution lJ) corresponds in R to X"-+ -x. So, for example, (2.14')
becomes
an(x) = E z;II/1A(X)
IA/-n
valid for any element x of any A-ring (where of course t/1 A(X) =
t/JAt(x)fjJ A2(X)...).

26
I SYMMETRIC FUNCTIONS
Examples
1. (a) Let Xl = ... =Xn = 1, xn+l xn+2" ... ==0. Then E(t)==(1 +t)n and H(t)
= (1 - t)-n, so that
e, = ( ; ) ,
h,=(n+-l)
and Pr = n for all n  1. Also
mA=UA(l(:»)'
where
UA :=:
I( A)!
nmi(A)!
;> I
(b) More generally, let X be an indeterminate, and define a homomorphism
8X: AQ -+ Q[X] by 8X(Pr) :=: X for all r  1. Then we have
&x(e,) = ( ), &x(h,) = ( X + ; - 1 ) = ( _1)' ( -,X),
for all ,  1, and
8X(mA) = UA( X ).
I(A)
For these formulas are correct when X is replaced by any positive integer n, by (a)
above. Hence they are true identically.
2. Let Xi = l/n for 1  i  n, Xi = 0 for i > n, and then let n --+ 00. From Example 1
we have
e :=: lim 2. ( n) = 2-
r n  00 nr, r!
and likewise hr = 1/,!, so that E(t) a: H(t) = e'. We have PI = 1 and Pr = 0 for
r> 1; more generally, mA := 0 for all partitions A except A:= (1r)(r  0).
3. Let Xi = qi-l for 1 < i  n, and Xi = 0 for i > n, where q is an indeterminate.
Then
E{t) = Yi. (1 + qit) = t q,<,-I}/2[;] tr
,- 0 r- 0
where [;] denotes the 'q-binomial coefficient' or Gaussian polynomial
[n] :=: (l_qn)(l_qn-l)...(l_qn-r+l)
r (1-q)(1-q2)...(1-q')'
and
n-l 00 [ ]
. -1 n+r-l
H(t) == n (1 - q't) == E r tr.
,- 0 r- 0

2. THE RING OF SYMMETRIC FUNCTIONS
27
These identities are easily proved by induction on n. It follows that
e, = qr(,-1)/2 [ ; ] ,
h,=[n+-ll
h is the generating function for partitions A such that leA) <. rand leA') < n - 1,
ad e, is the generating function for such partitions with all parts distinct.
4. Let n -+ 00 in Example 3, i.e. let Xi == qi-l for all i ;> 1. Then
00
00
E(t) == n (1 + qit) == E qr(,-I)/2/, /cp,(q),
i- 0 ,- 0
00 00
H(t) == n (1- qil) -1 == E t' /cp,(q),
i-O ,-0 '
where
lp, (q) == (1 - q ) (1 - q 2) . . . (1 - q') .
Hence in this case
e, == q,(,-1)/2 / lp,(q),
h, = 1/ cp,(q)
and p, = (1 - q')-I.
5. Since the h, are algebraically independent we may specialize them in any way,
and forget about the original variables Xi: in other words, we may take H(t) (or
E(t» to be any power series in I with constant term 1. (We have already done this
in Example 2 above, where H(/) == et.) Let a, b, q be variables and take
00 1 - bqit
H(t) == n
; _ 0 1 - aq it'
Then we have
, a - bqi - 1
h,'" n 1 qi '
i-I -
, aqi-l - b
e =n
, 1 qi
;-1 -
(see e.g. Andrews [A3], Chapter II.) Also p, == (a' - b') /(1 - q').
6. Take H(t) == n: _ 1(1 - tn) -1, so that hn == pen), the number of partitions of n.
Then E( -I) = n:_1(1 - tn), and so by Euler's pentagonal number theorem en = 0
unless n is a pentagonal number, i.e. of the form !m(3m + 1) for some m e Z; and
en == ( _l)m(m + 1)/2 if n == tm(3m + 1).
From (2.10) we obtain p, == u(r), the sum of the divisors of r. Hence (2.11) gives
in this case
(1)
1 n
pen) == - E u(r)p(n -r).
n ,-1

28
I SYMMETRIC FUNCTIONS
7. Take H(t) = Il:_1(1-tn)-n, so that hn =P2(n), the number of plane partitions
of n (5, Example 13). From (2.10) we obtain Pr = u2(r), the sum of the squares of
the divisors of r. Hence by (2.11) .
(2)
1 n
P2(n) = - E u2(r) P2(n - r).
n r= 1
It is perhaps only fair to warn the reader that the obvious generalization of (1)
and (2) to m-dimensional partitions (m > 2) is false.
8. By solving the equations (2.6') for en we obtain
en = det(ht-i+j)l<i,j<n
and dually
hn = det(el-i+j)t <i,j<n.
Likewise from (2.11) we obtain the determinant formulas
e1 1 0
2e2 el 1
Pn =
nen en-l en-2
PI 1 0
P2 PI 2
n! en =
Pn-l Pn-2
Pn Pn-l
and dually
hI 1 0
( )n-l 2h2 hI 1
- 1 Pn =
nhn hn-I hn-2
PI -1 0
P2 PI -2
n! hn =
Pn-l Pn-2
Pn Pn-]
o
o
el
o
o
n-1
Pl
o
o
hI
o
o
-n+ 1
P1

2. THE RING OF SYMMETRIC FUNCTIONS
29
9. (a) Let G be any subgroup of the symmetric group Sn. The cycle indicator of G
is the symmetric function
1
c(G) = IGI :nG( p)pp
p
where nG( p) is the number of elements in G of cycle-type p, and the sum is over
all partitions p of n. In particular,
c(Sn) = E z;lpp = hn
Ipl-n
by (2.14'), and for the alternating group An we have
c(An) = hn + en
(b) If G is a subgroup of Sn and H a subgroup of Sm, then G X H is a subgroup of
S n X S m C S n + m' and we have
c(G X H) = c(G)c(H).
(c) Let G be a subgroup of Sn and let I be the set of all sequences a = (a],..., an)
of n Positive integers. For each such sequence a, define xa =xa ... xa . The group
1 "
G acts on I by permuting the terms of these sequences, and the function a....... xa
is constant on each G-orbit. Show that
(1)
c(G) = EXa
a
where a runs through a set of representatives of the orbits of G in I (Polya's
theorem). (Let
X=IGI-1 E Xa
(g, a)
summed over all (g, a) E G X I such that ga = a, and show that X is equal to
either side of (1).)
10. From Examples 8 and 9 it follows that the number of elements of cycle-type p
in Sn is equal to the coefficient of Pp in the determinant
P1 -1 0
P2 PI -2
d" = n! hn =
Pn-1 Pn-2
Pn Pn-l
o
o
-n + 1
PI
Let I be a prime number . We may use this formula to count the number of
conjugacy classes in Sn in which the number of elements is prime to I, by reducing
the determinant d,. modulo I. Suppose that n = ao + n11, where 0  ao  1 - 1.

30
I SYMMETRIC. FUNCTIONS
Then since the multiples of 1 above the diagonal in d" become zero on reduction,
it follows that
d" == di1 dao (mod. 1)
Now it is clear from the original definition of d" == n!c(S,,) that
d, = p - p, (mod. 1)
and therefore we have
(1)
d" == (p{ - p,)"1 . dao (mod. 1).
Hence if n = ao + all + a2/2 + ..., with 0  a;  1- 1 for all i  0, it follows from
(1) that
n (/i '1_1)41
d" == dao PI - p, (mod. I).
i> 1
Consequently, if ,.",(S,,) denotes the number of conjugacy classes in Sn of order
prime to 1, we have
,.",(S,,) = IL,(Sa ) n (ai + 1)
o . 1
.>
= p(ao) n (ai + 1)
;> 1
where p(ao) is the number of partitions of ao. In particular, if 1=2, we see that
IL2(Sn) is always a power of 2, because each ai is then either 0 or 1: namely
""2(Sn) == 2r if [n/2] is a sum of r distinct powers of 2.
11. Let
00 i tn
I(t)= E,
,,-0 n.
00 g tn
g(t)== E 
,,-0 n!
be formal power series (with coefficients in a commutative Q-algebra) such that
g(O) = O. We may substitute g(t) for t in f(t), and obtain say
00 H tn
H(t) == f(g(t» == E --7-.
o n.
Clearly each coefficient H" is of the form
n
H" == E IkB",k(g)
k-1
where the B", k are polynomials in the coefficients of g, called the partial Bell
polynomials. Since each H" is linear in the coefficients of I, in order to compute
the polynomials B",k we may take fk = ak, so that I(t) = eat. Writing
00
H(t) == E hntn
,,-0

2. THE RING OF SYMMETRIC FUNCTIONS
31
as usual, we have H(t) == exp(ag(t)) and therefore by (2.10)
d 00 ag tn - 1
P(t) = - log H(t) = ag'(t) = E (n )'
dt 1 n - 1 !
so that Pn = agn/(n - 1)1 for all n  1. Hence by (2.14')
n!
Hn ==n!hn = E -PA
IAI-n ZA
and consequently
n!
Bn,k = E -PA == EC.\gA
,\ ZA. A
where the sum is over partitions A of n such that l(A) == k, and
cA = n!jflr;!(iO'i
i> 1
gA,==gA1gAZ... ,
if ,\ = (1'12'2 . . .). These coefficients cA are integers, because cA is the number of
decompositions of a set of n elements into disjoint subsets containing A], A2, . . .
elements. Hence each Bn,k is a polynomial in "the gn with integer coefficients.
Particular cases:
(a) if get) = 10g(1 + t), then Bn,k = s(n, k) are the Stirling numbers of the first kind;
(-l)n-ks(n, k) is the number of elements of Sn which are products of k disjoint
cycles. We have
 tn a (a )
'-' s(n,k),ak==(I"+t) = E n t'.,
n,k>O n. n>O
from which it follows that
n
E s(n,k)ak==a(a -1)...(a -n + 1)
k-O
and hence that s(n, k) is the (n - k)th elementary symmetric, function of
- 1, - 2, . . ., - n + 1.
(b) if get) = e' - 1, so that gn = 1 for all n  1, then Bn,k = S(n, k) are the Stirling
numbers of the second kind; S(n, k) is the number of decompositions of a set of n
elements into k disjoint subsets, and is also the (n - k )th complete symmetric
function of 1,2,.. . , k.
12. Deduce from Example 11 that if f and g are n times differentiable functions
of a real variable, and if fk, gk, (f 0 g)k denote the kth derivatives of f, g, and
fog, then
n
(f 0 g ) n == E B n, k (g 1 , g 2 , .. · ) (fk 0 g ) .
k-l

32
I SYMMETRIC FUNCTIONS
13. If H(t) = (1 - tT) /(1 - t )T, we have
hn = ( n.+ r - 1 ) _ ( n - 1 )
,-1 ,-1
and by (2.10) we find that Pn =, if n  0 (mod r), whereas Pn = 0 if n = 0 (mod r).
Hence from (2.14')
E Z;I,I(A) = ( n +, - 1) _ ( n - 1)
A ,-1 ,-1
(,  2)
where the sum on the left is over partitions A of n none of whose parts is divisible
by r.
In particular (r = 2)
E Z; 12/(..\) = 2
A
summed over all partitions of n into odd parts.
14. Suppose that Pn = anll/n! for n  1. Then
h - ( )n-l/,
n - a a + n n .,
( )n-1
en =a a -n In!.
(Let t = xe-x and use Lagrange's reversion formula to show that P(t) =
aex/(1-x).)
15. Show that
" -1 _" _ 1 _ 1.3.5... (2n - 1)
zp - zu - ,
p u 2.4.6.. .2n
where the first sum. is over all partitions p of 2n with all parts even, and the second
sum is over all partitions u of 2n with all parts odd.
16. Suppose that en = Pn for each n  1. Show that
an
hn = (n + I)! '
e =
n
(_1)n anBn
n!
for some a, where Bn is the nth Bernoulli number.
17. If hn = n for each n  1, the sequence (en)n  1 is periodic with period 3, and
the sequence (Pn)n> 1 is periodic with period 6.
18. (Muirhead's inequalities.) For each partition A of n, the A-mean of x =
(X1' X1"'" xn) is defined to be
1
MA(x) = - E w(xA).
n!
weSn
In particular, M(n)(x) is the arithmetic mean of Xl,".' x, and M(lll)(X) is their
geometric mean.

2. THE RING OF SYMMETRIC FUNCTIONS
33
Let '\, J.L be partitions of n. Then the following statements are equivalent;
(0 ,\  JL; (ii) MA(x)  MJL(x) for all x = (Xl'''.' Xn) E R .
(To show that (n implies (iO, we may assume by (1.16) that A = Rij J.L, in which case
it is enough to show that
X ; X j + X ; X j  X JL; X !-"J + x !-"i x !-'-1
') } '?'} } ,.
To show that (ii) implies (D, set Xl = ... = x, = X and xr+ 1 = ... = xn = 1, where
X is large, and deduce that Al + ... + Ar  J..Ll + .. . + J.Lr.)
19. Let pr) = Em).., summed over all partitions A of n of length r (so that
p1) = Pn). Show that
Epr)tn-rur = E(u - t)H(t)
n t r
and that
( _1)'
E pr)tn-r = ,E(r)( -t)H(t)
n'>r r.
where E(r)(t) is the rth derivative of E(t) with respect to t. Deduce that
pr) = E (_l)o-r(; )eohb
a+b-n
and that (if n  r) pr) is equal to the determinant of the matrix (aij)o < ;, j  n - r'
where a;o = (r ; i )er+i and aij = ei-j+ 1 if j ;;;. 1.
20. For any partition A, let uA = I(A)!/fli> 1 miCA)!, as in Example l(a). Show that
( _ 1)/(,\)- 1 _
Pn = n E 1(J..) UAhA
1..\1= n
8A
= -n E 1(J..) uAeA
IAI=n
and that
en = E B,\u,\h,\.
IAI-n
(Let H+(t) = En> 1 hntn, and pick out the coefficients of tn in log(l + H+(t» and
(1 + H+(t»-l, expanded in powers of H+(t).)
21. Let xn = 1/n2 for each n  1. Then
(1)
2 ( t 2 ) sin 'IT t
E( -t ) = TI 1- 2 =
n>l n 'TTt

34
I SYMMETRIC .FUNCTIONS
so that en = 'TT2n /(2n + I)! for each n  O. We have
. 1
p, = i..J 2; = (2,)
",>1 n
where ( is Riemann's zeta function. Deduce from (1) that
d
-2t2P(t2) = t-Iog E( -t2) == 11"t cot 11"t-l
dt '
and hence that
l (2,) == ( _1),-12 2r-l17" 2rB2r/(2,)!,
where B2r is the 2,th Bernoulli number.
22. (a) From Jacobi's triple product identity
E tn2un == rI (1 - t2n)(1 + t2n-1U)(1 + t2n-1U-1)
neZ n> 1
by setting u = - 1 we obtain
1- tn
E (_1)ntn2= rI
n e Z n,> 1 1 + tn ·
Deduce that if
(1)
1 + tn
E(t) = rI 1 n
n,>1 -t
then h n = 2 or 0 according as n  1 is a square or not.
(b) Deduce from (1) and (2.10') that
Pn = 2( _1),,-1 a 'en),
where a 'en) is the sum of the divisors d  1 of n such that n/d is odd.
(c) Let N,.(n) denote the number of representations of n as a sum of , squares,
that is to say the number of integer vectors (Xh..., xr) E zr such that xl + ... +
x; = n. Deduce from (b) above and Example 8 that
a '(1) 1/2, 0
(2,)" a '(2) u '(1) 2/2,
Nr(n) =
n!
a'(n -1) u ' (n - 2)
a 'en) a' (n - 1)
o
o
(n -1)/2,
u '(1)
23. If G is a finite group and d is a positive integer, let W d( G) denote the number
of solutions of Xd == 1 in G. Show that
(1)
1 ( 1).
E ,. wd(Sn)tn = exp E - tr .
n,> 0 n. rid '

2. THE RING OF SYMMETRIC FUNCTIONS
35
(If x e Sn has cycle-type A, then xd = 1 if and only if all the parts of the partition A
divide d. Hence
(2)
Wd(Sn)/n! = Ez;1
summed over such partitions ,\ of n. Let 'Pd= AQ --. Z be the homomorphism
defined by 'Pd(Pr) 11I:I 1 or 0 according as r does or does not divide d. Then the
right-hand side of (2) is by (2.14') equal to 'Pd(hn) and hence the generating
function (1) is ((Jd(H(t».)
24. Another involution on the ring A may be defined as follows. Let
(1)
u =tH(t) =t+htt2+h2t3+....
Then t can be expressed as a power series in u, say
(2)
t = u + hi u2 + h u3 .+ ... ,
with coefficients h E N' for each r;> 1. The formulas (1) and (2) show that the
ring homomorphism .p: A -+ A defined by .p(h,) = h for each r  1 is an involu-
tion on A.
For each / E A, let f* == .p(/). Thus for example h = hiJhi2 . .. for each
partition A = (At, A2, . . .), and the hi form a Z-basis of A.
(a) To calculate h: explicitly, we may argue as follows. From (2) we have
dt = E (n + l)h:un du
n>O
and therefore (n + l)h: is the residue of the differential
dt/un+1 = dt/tn+IH(t)n+l,
hence is equal to the coefficient of tn in the expansion of H(t)-n-l in powers of t.
Writing H(t) = 1 + H+ (t) as in Example 20, it follows that
(n + l)h: = E (-1)/().){ n + I(A») U).hA
IAI-n n
with u A as in Example 20.
(b) Show likewise that
*  -I( )'()')
Pn = I..J z). -n PA
IAI-n
and that
(n-l)
(n - l)e = - E u).e)..
I,\I-n l( A)
25. Let f(x), g(x) E Z[[x]] be formal power series with constant term 1. Let
t = xf(x) and express g(x) as a power series in t:
g(x) =H(t) = E hntn.
11;>0

36
I SYMMETRIC .FUNCTIONS
Let C{Jn(f,g) denote the coefficient of xn in (f+xf')glfn+l, where f' IS the
derivative of f. Then we have
hn = !Pn(f, g),
en = ( - 1) n C{Jn ( f, g - 1 )
and (with the notation of Example 24)
h: = 'Pn(fg, g-1),
e: = ( - 1) n 'Pn (fg , g ) .
Moreover, if t/ln(f, g) is the coefficient of xn -1 in g I Ifn g, we have
Pn = t/ln(f, g),
P: = I/1n (fg , g -1 ) .
(a) Take f(x) = (1 + x)-a and g(x) = (1 + x) 13. Then
f3 (na+f3)
'Pn(f,g) = na+{3 n =un(a,{3),
say, and therefore
hn=un(a,f3),
en = un(1- a, f3),
h = un( a - f3, - f3),
e = Un (1 - a + {3, - f3 ) ,
and
Pn = : (nna),
*= f3 (na-n{3)
Pn .
{3-a n
In particular, un(2,1) is the nth Catalan number Cn = (n + 1)-1( 2:), and
un(2,2) = Cn+ I' un(2, -1) = - Cn-I' Hence when Pn = a( 2:) with a = t, 1, - t
we have respectively hn = Cn' Cn+ l' - Cn _], and h: = - 81n, (-1)n(n + 1),
n - 1 ( 3n ) .
n-1
(b) Take f(x) = e-ax, g(x) = e (3x. Then
C{Jn(f,g) = f3(na + f3)n-l In! = vn(a, {3),
say, and hence
hn=vn(a,{3),
en = vn( - a, {3),
h: = vn(a - f3, - {3),
e=vn(f3-a,-f3),
and
f3(na)n-1
Pn = (n - I)! '
( )n-1
-f3 na-n{3
P:=
(n - 1)!

2. THE RING OF SYMMETRIC FUNCTIONS
37
26. Let k be a finite field with q elements, and let V be a k-vector space of
dimension n. Let S = S(V) be the symmetric algebra of V over k (so that if
Xl'... ,xn is any k-basis of V, then S = k[Xt,..., xn]). Let
Iv(t) = n (t + v),
veV
a polynomial in S[t].
(a) Show that
[(at + bu) = af(t) + bl(u)
for all a, b E k and indeterminates t, u. (If a E k, a  0 we have
I(at) = n (at+v)=aq" n (t+a-1v)
veV veV
= af(t).
N ext, let
I(t + u) - f(t) - I(u) = E trg,(u)
,;>0
with g,(u) E S[u]; since f(t + v) = I(t) and f(v) = 0 for all v E V, it follows that
g,(v) = 0 for all v E V. Since g, has degree < qn, we conclude that gr = 0 for each
r, and hence that I(t + u) = I(t) + f(u).)
(b) Deduce from (a) that [v(t) is of the form
" n-l
Iv(t) = tq + at(V)tq + ... +an(V)t,
where each a,(V) E S, and in particular an(V) is the product of the non-zero
vectors in V.
Show that
(1)
E at(L) =0
L<V
where the sum is over all lines (i.e. one-dimensional subspaces) L in V. (Since each
v =1= 0 in V lies in a unique line L = kv, it follows that
Iv(t) := t n t-1IL(t) = t n (tq-t + at(L))
L<V L<V
and therefore the sum (1) is equal to the coefficient of tq" -1 in f v(t), which is
clearly zero.)
(c) Let U be a vector subspace of V. The mapping v  1 u(v) of V into S is
k-linear, by (a) above, and its kernel is U. Hence its image 1 u(V) is isomorphic to
the quotient of V by U, and we shall denote it by VIV. Each element of VIV is a
product of the form nu e u(v + u) for some v E V, i.e. it is the product in S of the
elements of a coset of V in V.

38
I SYMMETRIC FUNCTIONS
Show that
!v(t) = !v/u(!u(t»
and that if T is a vector subspace of U then
V/U=(V/T)/(U/T).
(d) Show that for 1  r  n - 1 we have
(2)
a,(V) = E a,(V /L)
L<V
summed as before over all lines L in V. (From (c) above we have !v(t) = !v / L(!L(t»,
from which it follows that
n-,
a,(V) = a, (V /L) + a,-l(V /L)al(L)q .
Since the number of lines L in V is 1 + q + . . . + qn - t == 1 (mod. q), it follows that
(3)
"-,
a,(V) = E a,(V/L) + E a,_1(V/L)a1(L)Q
L L
By induction on n = dim V we may assume that
(4)
a,_l(V/L) = Ea,_t(V/M)
M
summed over all two-dimensional subspaces M in V that contain L. The second
term on the right-hand side of (3) is therefore equal to
"-,
E a,_l(V/M)( E a1(L)f
M L<M
which is zero by (1) above.)
(e) Deduce from (d) that
a,(V) = E a,(V /V)
u
summed over all subspaces V of V of dimension n - r, where a,(V /V) is the
product of the vectors v E V such that v  U.
27. As in Example 26, let k be a finite field with q elements, let x),..., xn be
independent indeterminates over k, and let k[x] = k[xt,..., xn]. Let V be the
k-vector space spanned by Xl"..' xn' and let
(1)
f ( ) - q" q" - 1
V t - t + alt + ... +ant
be the monic polynomial whose roots are the elements of V. Let k[a] =
k[a),..., an] c k[x].

2. THE RING OF SYMMETRIC FUNCTIONS
39
(a) The coefficients al'.'" an in (1) are algebraically independent over k. (Let
bl,..., bn be independent indeterminates over k, and let W be the set of roots of
the polynomial
( ) - q" b q"-l b
g t - t + It + ... + nt
in a splitting field. The roots are all distinct, since g '(t) = bn :;: 0, and since
g(at + f3u) = ag(t) + f3g(u) for a, 13 E k it follows that W is a k-vector space of
dimension n. Choose a basis (Yl'.'.' Yn) of Wand let 8: k[x] -+ k[y] be the
k-algebra homomorphism that sends Xi to Yi (1  i  n). Then 8(ai) = bi (1  i  n);
since the bi are algebraically independent over k, so are the aj.)
(b) Let G = GL(V)  GLn(k) be the group of automorphisms of the vector space
V. Then G acts on the algebra k[ x]; let k[ x]G denote the sub algebra of G-
invariants. Since G permutes the roots of the polynomial [vet), it fixes each of the
coefficients ai' so that k[a] c k[x]G. In fact (see (d) below) k[a] = k[x]G.
(c) k[x] is a free k[a]-module with basis (XO)oeE' where
E = {a = (at,..., an): 0  a;  qn - qi-l - I}.
(Let v;. denote the subspace of V spanned by Xl'...' X r' for 0  r  n - 1 (so that
Vo = 0). The polynomial gr(t) = [v (t)/[v,(t) is monic of degree qn - qr, has coeffi-
cients in the ring k[a, Xl"." xr], and has xr+ 1 as a root.
Now let h E k[x]. Use the polynomial gn-l to reduce the degree of h in xn
below qn - qn -t. Then use gn _ 2 to reduce the degree of h in xn -1 below
qn - qn - 2, and so on. In the end we shall obtain say
(2)
h = E hoxo
creE
with coefficients ho E k[a] c k[x ]G.
Hence the xO, a E E, span k[x] as k[x]G-module. They therefore also span the
field k(x) = k(Xl'.." xn) as vector space over k(x)G, since every element of k(x)
can be written in the form u/v with u E k[x] and v E k[x]G. But by Galois theory
the dimension of k(x) over k(x)G is
n
IGI = n (qn - qi-l) = lEI;
i- 1
hence (xQ)o e E is a basis of k(x) over k(x)G, i.e. the expression (2) for h E k[x] is
unique.)
(d) Suppose now that h e k[x]G. Then in (2) we must have ho = ° if a:;: (0,... ,0),
and h = ho,...,o E k[a]. Hence k[x]G = k[a] (Dickson's theorem).
Notes and references
Example 11. For more information on Bell polymomials, Stirling numbers
etc., see for example L. Comtet's book [C3].
Example 13. This example is due to A. O. Morris [M15].
Example 27. The proof of Dickson's theorem given here I learnt from R.
Steinberg.

40
I SYMMETRIC FUNCTIONS
3. Schur functions
Suppose to begin with that the number of variables is finite, say Xl"'" Xn.
Let XQ = XII... X:" be a monomial, and consider the polynomial aa
obtained by antisymmetrizing xa: that is to say,
aa=aa(xt"",xn)= E s(w).w(xa)
weSn
where s(w) is the sign ( :t 1) of the permutation w. This polynomial aa is
skew-symmetric, i.e. we have
w(aa) = s(w)aa
for any w E S,.; in particular, therefore, aa vanishes unless al,..., an are
all distinct. Hence we may as well assume that al > a2 > ... > an  0, and
therefore we may write a = A + 8, where A is a partition of length  n,
and S = (n - 1, n - 2, . . . , 1,0). Then
aa=aA+S= E s(w).w(xA+S)
w
which can be written as a determinant:
- d t( Aj+n -j)
aA+8 - e Xi l<i,j<n'
This determinant is divisible in Z[xl,..., Xn] by each of the differences
Xi - Xj (1  i <j  n), and hence by their product, which is the Vander..
monde determinant
TI (xi-xj)=det(xr-j)=as'
1 <. i <j <. n
So aA+ S is divisible by as in Z[xt,..., xn], and the quotient
(3.1)
SA = SA (X I , .. . , X n) = a A + s/ as
is symmetric, i.e. is in An. It is called the Schur function in the variables
Xl"'.' Xn, corresponding to the partition A (where I(A)  n), and is homo-
geneous of degree I AI.
Notice that the definition (3.1) makes sense for any integer vector
A E Zn such that A + 8 has no negative parts. If the numbers Ai + n - i
(1  i  n) are not all distinct, then SA = O. If they are all distinct, then we
have A + 8 = w( J.L + 8) for some w E Sn and some partition J.L, and SA =
s(w)sl-£'
The polynomials aA+ s, where A runs through all partitions of length
 n, form a basis of the Z-module An of skew-symmetric polynomials in
Xl"'" xn' Multiplication by as is an isomorphism of An onto An (i.e. An is
the free An..module generated by as), and therefore

3. SCHUR FUNCTIONS
41
(3.2) The Schur functions SA(X1)"., xn), where leA)  n, form a Z-basis
of An' I
Now let us consider the effect of increasing the number of variables. If
lea)  n, it is clear that aa(xt,.", xn' 0) = aa(xt,,,,, xn). Hence
Pn + 1, n ( SA (X 1 , · · · , X n + 1 )) = SA (X 1 , . · · , X n )
in the notation of 2. It follows that for each partition A the polynomials
s (Xt,. .., xn), as n -+ 00, define a unique element SA E A, homogeneous of
degree I AI. From (3.2) we have immediately:
(3.3) The SA form a Z-basis of A, and for each k  0 the SA such that I AI = k
form a Z-basis of Ak. I
From (2.4) and (2.8) it follows that each Schur function SA can be
expressed as a polynomial in the elementary symmetric functions e" and
as a polynomial in the complete symmetric functions hr' The formulas are:
(3.4)
SA = det(h,\/-i+j)l <i,j<n
where n  leA), and
(3.5)
SA = det( e A',- i + j ) 1< i, j < m
where m  leA').
By (2.9'), it is enough to prove one of these formulas, say (3.4). We shall
work with n variables Xl'.." xn' For 1  k  n let ek) denote the elemen-
tary symmetric functions of Xl'.." Xk-l, Xk+ 1"'.' xn (omitting xk), and let
M denote the n X n matrix
M- ( _1)n-i (k) ) .
- en-i l<i,k<n'
The formula (3.4) will be a consequence of
(3.6) For any ex = (al,..., an) E Nn, let
Aa = (xt'), Ha = (haj-n+j)
(n X n matrices). Then Aa = HaM.
Proof. Let
n-l
E(k)(t) = L: ek)tr = n (1 +xjt).
r-O i+k
Then
H(t)E(k)( -t) = (1-Xkt)-1.

42
I SYMMETRIC FUNCTIONS
By picking out the coefficient of t aj on either side, we obtain
n
"h (_l)n-j (k) _ at
!-J OJ-n+j. en-j -Xk
j-l
and hence HaM =Aa. I
Now take determinants in (3.6): we obtain
ao = det(Aa) = det(Ha)det(M)
for any a E Nn, and in particular det M == as, since det(Hs) = 1. Hence
(3.7)
Da = as det(Ha)
or equivalently
(3.7')
aa=as E B(w)ha-ws
wes,.
for any a E Nn. Taking a = A + 8 in (3.7), we obtain (3.4), or equivalently
from (3.7')
(3.4')
SA = E e(w)hA+8-w8.
weS,.
From (3.4) and (3.5) it follows that
(3.8)
W(SA) = SA'
for all partitions A.
Also from (3.4) and (3.5) we obtain, in particular,
(3.9)
S(n) = hn,
S(1") = en ·
Finally, the formula (3.4) or (3.4') which expresses SA as a polynomial in
the h's can also be expressed in terms of raising operators (1):
(3.4" )
SA = n (1 - Rij)h).
i<j
where, for any raising operator R, RhA means hRAot
t It should be remarked that if R, R' are raising operators, RR'h). = hRR'A is not necessarily
equal to R(R' h)). For it may well happen that R' A has a negative component, but RR' A does
not, in which case R'h). = 0 but RR'hA + O. See [03] for a discussion of this point.

3. SCHUR FUNCTIONS
43
Proof. In the ring Z[X1:t 1,..., x! 1] we have
E e(w)x.\+8-w8 =xA+8a_s =XA+8 n (xiI -xiI)
we S 1<)
,.
= n (I-XiX}!) .X.\
i<j
= n (I - R;j)xA
i<j
where R{xA) =XRA for any raising operator R. If we now apply the
Z-linear map : Z[X1:t 1,..., xn:t l] -+ An defined by cp(xa) = ha for all a E
Zn, we see that
E e(w)h.\+8-wS= n(1-R;j)hA
we S i <j
n
and therefore (3.4") follows from (3.4'). I
(3.10) Remark. In view of (2.15) we may use (3.4) or (3.5) to define 'Schur
operations' in any A-ring R. If J.L is any partition and x is any element of
R, we define
S#£(x) = det(ultl-i+j(X»l<i,j<n
, . .
= det( A Itl-' +J(x» ..
1 < ',j < m
where n  I( JL) and m  I( p.,'). We have
SJI.( -x) = (-l)I,uIS,u'(x)
and in particular
s(n)(x) = un(x), S(l")(X) = An(x).
For example, the results of Examples 1-3 below evaluate S A(l +
q + ... +qn-l), S.\«1 - q)-l) and SA«a - b)/(l - q », where a, b, q are
elements of rank 1 in a A-ring R such that 1 - q is a unit in R.
Since each f e A is an integral linear combination of the sit' say
f = E a,.,.s,u ,
it follows that f determines a 'natural operation'
F = E a,uS""
on the category of A-rings. F is natural in the sense that it commutes with
all A-homomorphisms (because it is a polynomial in the A'). Conversely,
any natural operation F arises in this way, from f = F(el).

44
I SYMMETRIC FUNCTIONS
Examples
1. Take Xi = qi-l (1 <; i <; n) as in 2, Example 3. If A is any partition of length
 n, we have
aA+6 == det(q(i-1XAj+n-j»)1<i,j.;;n
which is a Vandermonde determinant in the variables qAj+n-j (1 j n), so that
aA+6 = n (qAj+n-j - qA;+n-i)
i<j
= qn(A)+n(n -lXn - 2)/6 n (1 _ qAi-Aj-i+j)
i<j
which by use of  1, Example 1 is equal to
qn(A)+ n(n -lXn - 2)/6 n 'PA.+n _ i(q)
i 1 I
fI (1 - qh(x»)
xE A
where h(x) is the hook-length at x E A, and 'Pr(q} = (1- q).. .(1- q'). Hence (1,
Example 3)
1_qn+c(x)
sA = aA+ 6/a6 = qn(A) fI 1 h(x)
XE A - q
where c(x) is the content (1, Example 3) of x E A.
For any partition A define
[n] = 1 - qn-c(x)
A }l 1 - qh(x)
(which when A = (r) agrees with the notation [] for the q-binomial coefficients
introduced in 2, Example 3). Then we have
[ n ]
n-l _ n(A)
s).<1,q,...,q ) -q A'.
[ :] is a polynomial in q, of degree
n
d== E (n-c(x)-h(x))= E(n+1-2i)A'i
xEA i-I
by using U, Examples 2 and 3. If aj is the coefficient of qi in [:] for 0  i  d,
then clearly a i = ad _ j' We shall show in 8, Example 4 that [:] is unimodal (or
'spindle-shaped'), i.e. that ao <; al <; ...  a[d/2].
Finally, we can express [:] as a determinant in the q-binomial coefficients [],
by using (3.5).

3. SCHUR FUNCTIONS
45
2. Let n -+ 00 in Example 1, so that H(t) = n_o(1 - qit)-l. From Example 1 we
have
n h -1 1
SA = qn().) (1 - q (x» = qn().)H). (q)-
xe ,\
where H).(q) is the 'hook polynomial' nxE).(1 - qh(x».
3. More generally, let
 1 - bqi t
H(t) = n
i - 0 1 - aq it
as in 2, Example 5. Then
(*)
a - bqc(X)
s = qn().) n .
A 1 h(x).
xE'\ - q
For if we replace t by a-1t, the effect is to replace s,\ by a-I).ls).. Hence we may
assume that a = 1. Both sides of (*) are then polynomials in b, hence it is enough
to show that they are equal for infinitely many values of b. But when b = qn and
a = 1 we are back in the situation of Example 1, and (*) is therefore true for
b = qn.
4. Suppose Xi = 1 (1 <; i <; n), Xi = 0 for i> n. Then E(t) = (1 + t)n, and
SA = n n + c(x)
xe,\ hex)
by setting q = 1 in Example 1.
More generally, if E(t) = (1 + t)X, where X need not be a positive integer, then
SA = n X+c(x)
xE'\ hex)
for the same reason as in Example 3: both sides are polynomials in X which take
the same values at all positive integers.
These polynomials may be regarded as generalized binomial coefficients, and
they take integer values whenever X is an integer. For any partition A define
(X) _ n X-c(x)
A xE'\ hex)
(which is consistent with the usual notation for binomial coefficients). Then
( ) = det( ( Ai  + j ))
by (3.5). Also
( X) = (_nIAI(;).

46
I SYMMETRIC FUNCTIONS
5. As in 2, Example 2, take Xi = 1/n for 1 <; i <; n, Xi = 0 for i > n, and let n  00,
Then E(t) = H(t) == et, and from Example 4 we have
1 n+c(x)
SA = Iim !Ai fl h()
noo n xeA X
= fl h(X)-I.
xeA
6. Let p(n) denote the number of partitions of n. Then
det(p(i - j + 1))1 <i,j<n
is equal to :I: 1 or 0 according as n is or is not a pentagonal number. (Use 2)
Example 6 together with (3.4).)
7. Let m be a positive integer. Then
m m
fl Xi -Xj ameS
= - =S(m-l)c5
l<i<j<n Xi -Xj ac5
== det(h(m-1Xn-i)-i+j)1<;j<n
== det(hmi-j)l <i,j<n-] .
In particular,
fl (Xi +Xj) == det(h2i_j).
i<j
8. Consider the ring Qn == Q[ x]I. 1, . . . , X nf:. 1] of polynomials in X],..., X n and their
inverses. For each a E zn the monomial xa ==xi1... x:" generates a symmetric
function
ma== E xwa
weS"
and the mOl such that al  a2  ...  an form a basis of Q".
Define a linear mapping cp: Q"  An  Q by cp(ma) == ha (with the usual
convention that h a = 0 if any ai is negative).
(a) For all a, 13 E zn we have
cp(aOlaa) == det(ha.+ Q)ll" 'I' .
,., , "" ....I,J....n
For
Qa al3 =
E e(WIW2)XWla+W213
Wt,W2eS"
= E e(w) E xWt(a+wl3)
we S" WI eSn
== E e( w )ma+w13
weS"

3. SCHUR FUNCTIONS
47
SO that
cp(aaa13) == E s(w)ha+w13 = det(hat+13j).
WE SrI
(b) In particular, if ,\ is any partition of length <; n, we have
cp(sAa8a-8) == cp(a,\+sa_s) = det(h'\t-i+j) = s.\
by (3.4). Since the SA form a Z-basis of An' it follows that cp(fasa-a) = f for ail
f E An.
(c) Let a, 13 E Nn and let f3 = (f3n" .., (31) be the reverse of f3. Then s-p = a13- 8/a- 8'
and hence from (a), (b) we have
SaS" = cp(aa+8al3-8) = det(ha,+l3j-i+j)l.-i,j.-n'
a formula which expresses the product of two Schur functions (in a finite number
of variables) as a determinant in the hr.
9. Let a, b  0, then (a I b) is the Frobenius notation (1) for the partition
(a + 1,lb). From the determinant formula (3.4) we have
b
s(alb)=ha+1eb-ha+2eb-l + ...+( -1) ha+b+l
If a or b is negative, we define s(a I b) by this formula. It follows that (when a or b is
negative) s(alb) == 0 except when a + b = -1, in which case s(alb) = ( -1)b.
Now let A be any partition of length <; n. By multiplying the matrix
(h,\,-i+j)l<i,j.-n on the right by the matrix « -I)j-Ien+l-j-k)t.-j,k<n, we obtain
the matrix (S('\I-iln-k»l < i,k.- n. By taking determinants and using 1, Example 4
we arrive at the formula
S(a 113) = det(s(ad I3J»l.-i,j.-r
where (a I (3) = (at,..., ar I f3t,..., (3r).
10.
s.\(1 +x},1 +x2,...,1 +xn) = E d,\Jl.SJl.(Xl,...,Xn)
J.L
summed over all partitions J.£ c A, where
( (Ai + n - i ))
dAJ.L == det J.L' + n - j ...
J l<',j.-n
(Calculate aA+ s(1 + Xl' . . . , 1 + xn) and observe that a8(1 + x, . . . , I + xn) =
as(x1,. · ., xn).)

48
I SYMMETRIC FUNCTIONS
This formula can be used to calculate the Chern classes of the exterior square
A 2E and symmetric square S2E of a vector bundle E. If c(E) = flJ:.l(1 + x) is the
total Chern class of E, then .
c( A2E) = Il (1 +xi +Xj)
i<j
= 2-m(m -1)/2 Il (1 + 2xi + 1 + 2xj)
i<j
= 2-m(m-l)/2sa(1 + 2xl'.'" 1 + 2xm)
by Example 7, where 8 = (m - 1, m - 2,...,0), and therefore
c(A2E) = 2-m(m-l)/2 E d<'3JL211£Isl£(Xl,...,Xm).
IJ-c8
Likewise
c(S2E) = Il (1 + Xi + Xj)
i<j
= 2-m(m -1)/2 "d 21vls (x X)
 sv vi'...' m
vCC
where B = (m, m - 1,. . . , 1).
11. Let IL = ( ILl"'" J.Ln) be a partition of length  n, and r a positive integer.
Then, the variables being Xl'...' X n' we have
(1)
n
ap.+ aPr = E aJL+ 8+r&q
q==l
where Bq is the sequence with 1 in the qth place and 0 elsewhere. We shall
rearrange the sequence IL + 8 + rBq in descending order. If it has two terms equal,
it will contribute nothing to (1). We may therefore assume that for some p <:. q we
have
J.Lp - 1 + n - p + 1 > J.Lq + n - q + r > ILp + n - p,
in which case aJ.l.+8+rcq = ( -1)Q-PaA+8, where A is the partition
A = ( ILl' . · · , ILp - l' J.Lq + P - q + r , ILp + 1,..., JLq - 1 + 1, ILq + 1 , . · . , J.Ln)'
and therefore (} = A - J.L is a border strip of length r. Recall (1) that the height
ht( (}) of a border strip 8 is one less than the number of rows it occupies. With this
terminology, the preceding discussion shows that
(2)
= ,,( _l)ht(A-)
S J.l.P , i..J SA
A
summed over all partitions A::> IL such that A - IL is a border strip of length r.

3. SCHUR FUNCTIONS
49
From (2) it follows that, for any partitions A, J.L, P such that I AI = I J.LI + I pi, the
coefficient of SA in sJ-tPp is
E ( - l)ht(S)
S
summed over all sequences of partitions S = (A(O), ,\(1),..., A(m») such that J.L =
A(O) C A(t) c ... c A(m) = A, with each A(i) - A(i-1) a border strip of length Pi' and
ht(S) = E ht( A(i) - A(i-1»).
i
12. Let 0': Z[X1'''.' xn]  An be the Z-linear mapping defined by u(xa) = sa for
all a E Nn. Then 0' is An-linear, i.e. u(fg) = fu(g) for fE An and g E
Z[Xl'.'.'Xn]. For u(xa) =ai1a(xa+8), where
a = E B(W)W
weSn
is the antisymmetrization operator. By linearity it follows that u(g) = ai1a(gx6)
for all g E Z[x1,..., xn], and the result follows from the fact that a is An-linear.
13. If a, b  0 we have
PI
P2
c} 0
p} C2
o
o
(a + b + 1)a! hI S(alb) =
Pa+b
Pa+b+I
Ca+b
PI
where (ct,...,Ca+b)=( -1, -2,..., -a,b,b-1,...,1).
(Use the relation s(alb) +s(a+llb-l) =ha+1eb which follows from the first formula
in Example 9, together with the determinant forulas of 2, Example 8, and
induction on b.)
1 + ux'
D I =E(u)H(t)=1+(t+u) E s(alblaub.
1-Ix; a,b;;.O
15. Let M be an n X n matrix with eigenvalues x],..., X n' Then for each integer
r  0 we have
14.
n-l
Mn+r= E (-l)Ps(rIP)(Xt,...,xn)Mn-p-l.
p-o
(If Mn"t.'= EapMn-p-l, we have xj+r= Eapxi-p-l for 1 i n; now solve
these equations for ao,.. . , a p -1.)
16. Let '\, J.L be partitions of length  n, and let
Pn(A, J.L) = det(p,\.+II.+2n-i-J')}I" 'I' '
I '-}  , ,j  n

50
I SYMMETRIC FUNCTIONS
with the understanding that Po = n. Then in An we have
S). = Pn( A, J.L) / Pn (0, J.L),
s).sp, = Pn (A, /-') / Pn (0,0).
(Observe that Pn(A, J.L) = aA+8alA-+8, by multiplication of determinants.)
17. (a) Let p be an integer  2 and let w == e21ri/ p. If A is any partition of length
<;p, we have
(*)
sA(l, w,..., wp-1) = Il
l<cj<k<.p
wAj+p-j - WAk+P-k
wp-j - wp-k
from which it follows that sA(I, W,..., wp-1) = :f: 1 if A -pO (1, Example 8(f)) and
is zero otherwise. More precisely, if A -pO we have sA(1, W,..., wp-1) = ap(A),
where up(A) is the sign sew) of the unique permutation wE Sp such that A + Op ==
w8p(mod. p), where 8p = (p - 1, p - 2, . . . , 1,0).
(b) Assume from now on that p is an odd prime. Then
p
E(t) = Il (1 + w,-lt) = 1 + tP
,-1
= (1 + t)p (mod. p)
and therefore
sA(l, W,..., wp-1) == s,\(l,..., 1) (mod. p)
(1)
=(:,)
(Example 4). Hence (:) ;;;; 0 (mod. p) unless A - pO
(c) Let q + P be another odd prime and let A = (q - 1)8p. Then
(2)
SA(Xl'...,Xp) = Il (x? -x1)/(x; -Xj)
i<j
from which it follows that
s,,(1,...,1) =qp(p-I)/2 = (;) (mod.p)
where (; ) is the Legendre symbol, equal to + 1 or -1 according as q is or is not
a square modulo p. From (1) and (3) we deduce that
(3)
(4)
s,,(1, W,..., wp-I) = (;).

3. SCHUR FUNCTIONS
51
(d) Let Gp (resp. Gq) denote the set of complex pth (resp. qth) roots of unity, and
choose S c Gp and T c Gq such that
(xP-l)/(x-l)= n (x-oo)(x-oo-1),
O'eS
(xq -1)/(x - 1) = n (x - l' )(x - 1'-1).
'TeT
From (2) and (4) we have
(5)
(;) = TI(aq-q)/(a-M
the product being taken over all two-element subsets {a, {3} of G p' where we may
assume that a-1{3 e S. We have '
( ex q - (3 q) / ( a - (3) == n (a - (31' ) ( a - (31' -1 )
'TeT
(6)
= TI a{3(I-UT)(oo-1-1'-1)
'TeT
where 00 = a -1{3 e S. Deduce from (5) and (6) that
(7)
(!) == n(l- 001')(00-1- 1'-1)
P 0',1'
(product over all 00 e S, l' E T). (Observe that n a{3 == 1, that each U E S arises as
a-p from p subsets {a, {3}, and that (1- oo1'Xoo-1 - 1'-1) is a real number.)
By interchanging q and p we have likewise
(7')
(P)== n(l-001')(1'-l_.oo-l)
q 0','"
and therefore
() = (_O(P-IXQ-I)/4(;)
(the law of quadratic reciprocity).
18. Let k be a finite field with q elements, let Xl'...' Xn be independent indeter-
minates over k, and let k[x] = k[Xb'..' xn]. Let a == (al"'.' an) E Nn, let qQ ==
(qaJ,..., qQ,. ),. and let Aa E k[x] be the polynomial obtained by antisymmetrizing
the monomial xqO, so that
(1)
0"
Aa =AQ(x1,...,xn) == det(x? J)l<i,i<n.
As in the text, we may assume that al > a2 > ... > an  0, so that ex == A + 8,
where ,\ is a partition of length  n.

52
I SYMMETRIC FUNCTIONS
(a) As in 2, Example 26, let V denote the k-vector space spanned by Xl'"'' xn in
k[x], and let G = GL(V) = GLn(k, acting on k[x]. Show that
(2)
gAa = (det g )Aa
for g E G, and deduce that A a is divisible in k[ x] by each non-zero v E V. (The
definition (1) shows that Aa is divisible by each Xi' and G acts transitively on
V - {O).)
(b) Consider in particular the case A = 0, i.e. a = 8 = (n - 1, n - 2, . . . , 1,0). The
polynomial A 8 is homogeneous of degree
qn-l + qn-2 + ... + 1 = (qn - l)/(q -1),
and has leading term xqd. Let Vo denote the set ,of all non-zero vectors 'U = I. a iX i
in V for which the first nonvanishing coefficient a i is equal to 1. Then we have
Al) = Il V
UEVO
and each Aa is divisible in k[x] by As.
(c) Now define, for any partition A of length  n,
(3)
SA = SA ( Xl' . .. , X n) = A A + sl As.
From (2) above it follows that SA is G-invariant, and hence depends only on A and
V, not on the particular basis (Xl'''.' Xn) of V. Accordingly we shall write SA(V) in
place of SA(XI,..., xn). It is a homogeneous polynomial of degree E?_l(qA; - l)qn-i.
(d) If t is another indeterminate, we have from (b) above
Al) (t,xl'''.'Xn) =Al) (x1,...,xn) Il (t+ u)
n+J n
uEV
=Al) (Xl'.'" Xn)[V(t)
n
in the notation of 2, Example 25. By expanding the determinant Al) (t, Xl'.'" Xn)
n+1
along the top row, show that
[Vet) =tqn -Et(V)tqn-1 +... +( -l)"En(V)t,
where
E,(V) = S(IT)(V)
(1 rn).
The E,(V) are the analogues of the elementary symmetric functions, and in the
notation of 2, Example 26 we have
E,(V) = ( _1)' a,(V)
= (-I)'E a,(VIU)
u
summed over subspaces U of V of codimension r.

3. SCHUR FUNCTIONS
53
19. In continuation of Example 18, let
Hr(V) = S(r)(V)
(r  0)
with the usual convention that Hr(V) = 0 if r < O. (Likewise we define Er(V) to be
zero if r < 0 or r> n.) Let S(V) (= k[x]) be the symmetric algebra of V over k,
and let cp: S(V) -+ S(V) denote the Frobenius map u t-+ uq, which is a k-algebra
endomorphism of S(V), its image being k[Xf,..., x%]. Since we shall later en-
counter negative powers of cp, it is convenient to introduce
A -r
S(V) = U S(V)q ,
rO
where S(V)q-' = k[Xf-',..., x:-']. On S(V), cp is an automorphism.
(a) Let E(V), H(V) denote the (infinite) matrices
R(V) = ( lpi+lHj_;(V) )i,jEZ'
E(V) = ( _1)j-i lpjEj_;(V) );,jEZ'
Both are upper triangular, with 1's on the diagonal. Show that
E(V) =H(V)-l.
(We have to show that
( )k-j k( ) i+l( )_
i..J -1 lp Ek-j lp Hj-i - O;k
j
for all i, k. This is clear if i  k. If i < k, we may argue as follows: since [vex) = 0
it follows from Example 18 that
lpn(xi) _Etlpn-l(x;) +... +( -1)nEnx;=O
and hence that
(1) lpn+r-l(xi) -lpr-l(E1)cpn+r-2(xi) +... +( _1)nlpr-l(En)lpr-l(x) =0
for all r  0 and 1  i  n. On the other hand, by expanding the determinant
A(r)+ c5 down the first column, it is clear that Hr = Hr(V) is of the form
(2)
n
Hr = E Uilpn+r-l (x;)
i= 1
with coefficients ui E k(x) independent of r. From (1) and (2) it follows that
(3) Hr - lpr-l(El )Hr-1 + .. . + ( _1)" lpr-l (En )Hr-n = 0
for each ,  O. Putting r = k - i and operating on (3) with lpi+ 1, we obtain the
desired relation.)

54
I SYMMETRIC FUNCTIONS
(b) Let A be a partition of length  n. Then
S,\(V):;: det( q;1-jH,\/_i+j(V))
= det( q;j-1EAi_i+j(V))
in strict analogy with (3.4) and (3.5). (Let a = (at,..., an) E Nn. From equation (2)
above we have
n
q;1-i(HCI;_n+i) = E q;Cli(Xk)q;l-i(Uk)
i-1
which shows that te matrix (q;1-iHCli_n+i)j,j is the product of the matrices
(q;CI;Xk)i,k and (q;l-JUk)k,j-all three matrices having n rows and n columns. On
taking determinants it follows that
(4)
det( q;t-iHai_n+i) =AaB
where B = det(lp1-iuk). In particular, taking a =  (so that ai - n + j = j - i) the
left-hand side of (4) becomes equal to 1, so that A8B = 1 and therefore
det( q;1-iHCli_n+J) =Aa/As
for all a E Nn. Taking a = ,\ + , we obtain the first formula. The second Gnvolv..
ing the E's) is then deduced from it and the result of (a) above, exactly as in the
text.)
20. Let R be any commutative ring and let a = (an)n e z be any (doubly infinite)
sequence of elements of R. For each r E Z we define 'T' a to be the sequence whose
nth term is an+r. Let
(X I a)' = (x+at)...(x+ar)
for each r  o.
Now let x = (x l' . . . , xn) be a sequence of independent indeterminates over R,
and for each a = (at,..., an) E Nn define
(1)
Aa(x I a) = det«xj I a)ClJ)t<i,J<n.
In particular, when a = 8 = (n - 1, n - 2,...,1,0), since (Xi I a)n-J is a monic
polynomial in xi of degree n - j, it follows that
(2)
A8(x I a) = det(xi-i) = ac5(x)
is the Vandermonde determinant, independent of the sequence a. Since Aa(x I a)
is a skew-symmetric polynomial in Xl'''.' xn' it is therefore divisible by Ac5(x I a) in
R[xt,..., xn]. As in the text, we may assume that at > a2 > ... > an  0, i.e. that
a = ,\ + 8 where ,\ is a partition of length  n. It follows therefore that
(3)
s,\(X I a) =AA+c5(X I a)/A8(x I a)

3. SCHUR FUNCTIONS
55
is a symmetric (but not homogeneous) polynomial in Xl'.'" xn with coefficients in
R. Moreover it is clear from the definitions that
A,\+s(x I a) = a,\+8(x) + lower terms,
and hence that
SA(X I a) == s).(x) + lower terms.
Hence the s,\(x I a) form an R-basis of the ring An.R.
When ,\ == (r) we shall write
h,(x I a) == s(r)(x I a)
(r  0)
with the usual convention that h,(x I a) = 0 if r < 0; and when ,\ == (1') (0  r  n)
we shall write
e,(x , a) == S(l')(X I a)
with the convention that e,(x I a) == 0 if r < 0 or r> n.
(a) Let t be another indeterminate and let
n
I(t) == f1 (t -Xi).
i-I
Show that
(4)
n
I(t) a& E ( _1)' e,(x I a)(t I a)n-r.
,-0
(From (2) above it follows that
/(t) ==At.,,+ l(t, Xl,..., Xn I a)/As,,(xt "..., Xn I a);
now expand the determinant Ac5 along the top row.)
,,+1
(b) Let E(x I a), H(x I a) be the (infinite) matrices
H(x I a) == (hj_i(x I Ti+ la») 'tje Z'
E(x I a) == ( _l)j-i ej_i(x I Tja) )i,je z.
Both are upper triangular, with l's on the diagonal. Show that
E(xla) =H(xla)-1.
(We have to show that
E ( _l)k-j ek_j(x I 'Tka)hj_,(x I 'Ti+ la) == 8ik
j

56
I SYMMETRIC. FUNCTIONS
for all i, k. This is clear if i  k, so we may assume i < k. Since f(x;) = 0 it follows
from (4) above that
n
E ( _l)r er(x I a)(x; I a)n-r = 0
,- 0
and hence, replacing a by Ts-1a and multiplying by (Xi I a)s-l, that
(5)
n
E ( _l)r er(x I TS-1a)(xj \ a)n-r+s-1 = 0
,-0
for all s > 0 and 1  i  n. Now it is clear, by expanding the determinant
A(m)+ s(x \ a) down the first column, that hm(x I a) is of the form
(6)
n
hm(x I a) = E (Xi I a)m+n-l Ui(X)
i-1
with coefficients Ui(X) rational functions of Xt'..., xn independent of m. (In fact,
ui(x) = 1If'(x).)
From (5) and (6) it follows that
n
E ( _1)' e,(x I TS-1a)hs_r(x I a) = 0
,-0
for each s > o. Putting s = k - i and replacing a by Ti+ la, we obtain the desired
relation.)
(c) Let A be a partition of length  n. Then
s,,(x I a) = det( h,\;_i+j(X I T1-ja»),
= det( eAi-i+j(x I Tj-1a»),
again in strict analogy with (3.4) and (3.5). (Let a = (al,.." an) E Nn. From (6)
above we have
n
h al- n + j (x I T 1 - j a) = E (x kiT 1 - j a) aj + j - I Uk (X)
k=l
n
= E (Xk \ a)a;(Xk I Tl-ja)j-l Uk(X)
k-l
which shows that the matrix Ha=(hat_n+j(X\Tl-ja»i,j is the product of the
matrices «Xk \ a)a/)i,k and B = «Xk I 'T1-ja)i-tuk(X»k,j. On taking determinants it
follows that
det(Ha) =Aa deteR).
In particular, when a = 8 the matrix Ht, is unitriangular and hence has determi-
nant equal to 1. It follows that At, deteR) = 1 and hence that det(Ha) =Aa/A8 for I

3. SCHUR FUNCTIONS
57
all a E Nn. Taking a = A + , we obtain the first formula. The second formula,
involving the e's, is then deduced from it and the result of (b) above, exactly as in
the text.)
The results of this Example, and their proofs, should be compared with those of
Example 19. It should also be remarked that when a is the zero sequence (an = 0
for all n) then SA (x I a) is the Schur function SA(X).
21. Let R be a commutative ring and let S = R[hrs: r  1, S E Z], where the hrs are
independent indeterminates over R. Also, for convenience, define hos = 1 and
h == 0 for r < 0 and all s E Z. Let cp: S -+ S be the automorphism defined by
;(h,s)==hr,S+I. Thus hrs = cpS(hr), where hr=hrO' and we shall use this notation
henceforth.
For any partition ,\ we define
(1)
... - d ( - j + Ih )
SA - et cp Aj-i+j  <i,j<n
where n  l( ,\). These 'Schur functions' include as special cases the symmetric
functions of the last two Examples: in Example 19 we take R = k and specialize hrs
to lpsHr(V), and in Example 20 we specialize hrs to hr(x I TSa).
From (1) it follows that hr = s(r). We define
...
er = S(1')
for all r  0, and set e r = 0 for r < o.
(a) Let E, H be the (infinite) matrices
H= (cpi+lhj_i)i,jEZ'
( j-i' )
E = -1) cpJej_i i,jE Z.
Both are upper triangular with 1's on the diagonal. Show that
E = H-1 .
(We have to show that
(2)
E ( _1)j-i cpi(ej_i )cpj+ 1 (hk- j) = ;k
j
for all i, k. This is obvious if i  k, so assume i < k and let r = k - i. Then (2) is
equivalent to
r
E ( _1)s cp-S(er_s) cp-s+ l(hs) = 0
s.o
which follows from er=det(cpl-jht-i+i)l<i,j<r by expanding the determinant
along the top row.)
(b) Deduce from (a) that
... d (j-l )
SA = e t cp e Ai - i + j .

58
I SYMMETRIC FUNCTIONS
(c) Let w: S --. S be the R-algebra homomorphism that maps cpsh, to cp-se, for aU
T, s. Then w is an involution and wi).  SA' for all partitions A.
(d) Let A == (ah. .., a, I /31'...' /3,) in Frobenius notation (1). Show that with SA as
defined in (1) we have
SA = det(s(Ool QO»)l .. ·
''''J <ltl<'
(Copy the proof in Example 9 above.)
22. Let Xh...' xn, u.,..., un be independent indeterminates over Z, and let let) ==
(t-X1)...(t-xn). For each rEZ let
n
H, == E u;xi+,-t If'(x;)
i-1
and for each sequence a == (alf..., a,) e Z', where r  n, let Ma denote the n X n
matrix
M =: (u ..xj+n -j)
a I}'
where Uij == ui if 1 <:j <: r, uij == 1 if j > r, and aj = 0 if j > r. Then let
Sa = aB(x)-l det(Ma).
Show that
Sa == det(Ho/-i+j)l <;,j<'.
(Multiply the matrix MOl on the left by the matrix whose (i,j) element is
xJ-1If'(xj) if i < T, and is 8;) if i > T.)
23. The ring An of symmetric polynomials in n variables Xl"..' X n is the image
of A under the homomorphism Pn of 2, which maps the formal power series
E(t):: E,>oe,t' to the polynomial ni_1(l +x;t) of degree n. More generally, we
may specialize E(t) to a rational function of I, say
(*)
m / n
Ex/y(t) == n (1 +x;t) n (1 +yjt).
i-1 j-I
(In the language of A-rings, this amounts to considering the difference x - y, where
x has rank m and y has rank n.)
Let x(m) == (Xl'...' xm), yen) == (YI'.'" Yn) and let SA(X(m) Iy(n» (or just SA(X/Y))
denote the image of the Schur function s). under this specialization. From (*) we
have
e,(x(m) Iy(n» == E (-I)j e;(x )hj(y)
i+j-,
and the formula (3.5) shows that SA(X(m) Iy(n» is a polynomial in the x's and the
y's, symmetric in each set of variables separately.

3. SCHUR FUNCTIONS
59
These polynomials have the following properties:
(1) (homogeneity) SA(X(m) /y(n» is homogeneous of degree I AI.
(2) (restriction) The result of setting xm == 0 (resp. Yn == 0) In SA(X(m) /y(n» is
s (X(rn -1) /y(n» (resp. S,\(X(m) /y(" -1».
(3) (cancellation) The result of setting Xm == Yn in SA(X(m) lye"» is SA(X(m -1) /y(n-I»
(if m, n  1).
(4) (factorization) If the partition A satisfies Am  n  Am+ l' SO that A can be
written in the form A =- «nm) + a) U ' , where a (resp. ) is a partition of length
 m (resp.  n), then
SA(X(m) /y(n» == ( -1)IIJIR(x(m), y(n»sa(x(m»slJ(y(n»
where R(x(m), yen» is the product of the mn factors Xi - yj (1  i  m, 1 j  n).
It is clear from the definition that the SA satisfy (1), (2), and (3), and it may be
shown directly that they satisfy (4) (see the notes and references at the end of this
section). We shall not stop to give a direct proof here, because this property (4) will
be a consequence of the results of this Example and the next.
Moreover, these four properties characterize the SA(X/y). More precisely, let
Am.n( = Am  An) denote the ring Z[Xt,..., xm, Yh..., Yn]SmXS" of polynomials in
the x's and y's that are symmetric in each set of variables separately, and suppose
that we are given polynomials s:(x(m) /y(n» e Am,n for each partition A and each
pair of non-negative integers m, n, satisfying the conitions (1*)-(4*) obtained
from (1)-(4) by replacing SA by s: throughout. Then s:(x(m) /y(n» == s)..(x(m) /y(n»
for all A, m, n.
(a) First of all, when n :II: 0 it follows from (4*) that
(5)
s!(x(m) /0) == SA(X(m»
for partitions A of length  m, and likewise when n == 0 that
(5')
s:,(0/y(n» == ( -1)IAlsA(y(n»
for partitions A of length  n,
(b) Next, let Ami" denote the subring of Am,n consisting of the polynomials I in
which the result of setting xm .. Yn == t is independent of t. It follows from (3*) that
s*(x(m)/y(n» e A
A min'
Let rm,n be the set of lattice points (i,j) e Z2 such that i  1, j> 1 and either
i < m or j  n. We shall show that
(6)
the s* (x(m)/y(n» such that A c r snan A
A m,n r mln'
This is true when m == 0 or n == 0, by (5) and (5') above. Assume then that
m,n> 1 and that (6) is true for m -1,n -1. Let Ie Amln and let to = Ilxm-y,,-O,
so that 10 E Am _ 1 I" _ I and therefore is of the form
10 -= E aAs:(X(m-1)/y(m-1»
A

60
I SYMMETRIC FUNCTIONS
summed over partitions A c r m _ I, n _ I' with coefficients a)" E Z. Let
(7)
g = f - E a,\s: (x(m) /y(n».
,\
Then g E Am/n, so that glxm-YII = glxm-YII-O = O. Consequently g is divisible in
Am,,. by Xm - Yn and hence (by symmetry) by R(x(m), y(n»: say g = Rh where
h E Am,,." By writing h in the form
h = E ba, {3 sa (x(m»Sfj (y(n»
a,{3
summed over partitions a,  such that l(a)  m and I( /3)  n, it follows from (4*)
that g == Rh is a linear combination of the s;(x(m), y(n» such that (nm) c J.L c rm,n.
In view of (7) above, this establishes (6).
(c) Now let A be any partition. In order to show that s:(x(m) /y(n» = s),,(x(m) /y(n»)
we may, by virtue of (2) and (2*), assume that m and n are large, and in particular
that m;> I AI. Since s,\(x(m) /y(n» E Am / n by (3), it follows from (6) above that we
may write
(8)
s,\(x(m)/y(n» = E cs:(x(m)/y(n»
IJ-
where (by (1) and (1 *» the sum is over partitions J.L such that I J.LI = I AI and hence
I( J.L)  m (since I AI  m). If we now set YI = ... = Yn = 0 in (8), we obtain
s,\(x(m» = E cs(x(m»
IJ-
by virtue of (2), (2*), and (5). Hence cp. = A and finally s,\ = s:.
24. (a) Let x = (Xl'''.' xm), Y = (Yl'''.' Yn). If A is any partition let
fA (x, y) = TI (Xi - Yj)
(i,j)e'\
with the understanding that X; = 0 if i > m, and Yj = 0 if j > n. Also let
(x) = n (1-xi1Xj), (y) = n (l_y;-lYj).
14;'<)4;m t<'<l<n
Then we have
(1)
s,\(x/y) == E w(f),,(x, y)/ (x) (y ».
weSmxsn
In this formula, Sm permutes the x's and Sn permutes the y's.
(Let s:(x/y) denote the right-hand side of (1). We have
(2) s:(x/y) ==a8m(x)-ta8"(y)-t E e(w).w(x8my8I1f),,(x,y»
weSmxSn
from which it is clear that s: E Am,n' in the notation of Example 23, and it is
enough to verify that s: has the properties (1*)-(4*), loco cit. Of these, all but the

3. SCHUR FUNCTIONS
61
cancellation property (3*) are obviously satisfied. As to (3*), let cp(u) =
s:(x/y)l.m-Y"-U' which is a polynomial in u of degree say d. It will be enough to
shoW that cp(u) = cp(O), i.e. that d = O. If w E Sm X Sn' let w-1(xm) = Xi and
W-1(Yn) = Yj. If (i, j) E A, then xm - Yn = w(x; - Yj) is a factor of wf(x, y), which
therefore vanishes when xm = Yn = u. Hence cp(u) is a sum over those w E Sm X Sn
such that (i, j)  A, that is to say i >  and j > Ai. For such a permutation w, the
degree in u of w(x8my8"!,\(x,y»lxm_y"_u is
m - i + n - j + Ai + Ai  m + n - 2;
on the other hand, the degree in u of a8m(x)a8"(y)lx",-y"-u is (m -1) + (n -1) =
rn + n - 2. It now follows from (2) that d = 0, as required.)
(b) When n = 0, we have !,\(x, y) =x\ and the formula (1) reduces to the
definition (3.1) of s,\(x). Next, when m = 0, it follows from the definition (Example
23) that s,\(x/y) becomes ( -1)1,\lws,\(y). On the other hand, !A(X, y) becomes
( _1)1,\ly'\', so that the formula (1) in the case m = 0 reduces to (3.8). Finally, as we
have already remarked, the factorization property (4) of Example 23 is an immedi..
ate consequence of (1), and so is the fact that s,\(x/y) = 0 unless A C fm,n.
(c) The s>..(x/y) such that A cfm,n form a Z..basis of the ring Am/n defined in
Example 23(b). (It follows from Example 23 that the s,\(x/y) such that A c rm,n
span Am/n, and it remains to be shown that they are linearly independent. This is
clearly true if m = 0 or n = 0, by (3.2) and (3.8). Hence we may assume m  1 and
n  1 and the result true when either m is replaced by m - 1, or n by n - 1.
Suppose then that
E a,\s,\(x/y) = 0
A
where the sum is over partitions A C r m, n. By setting X m = 0 (resp. Y n = 0) we see
that a,\ = 0 for A c rm-1,n (resp. A C rm,n-l). Since rm,n is the union of fm-l,n
and r m, n -], it follows that a,\ = 0 for all A E r m, n' as required.)
Notes and references
Schur functions, despite their name, were first considered by Jacobi [J3], as
quotients of skew..symmetric polynomials by the polynomial as, just as we
have introduced them. Their relevance to the representation theory of the
symmetric groups and the general linear groups, which we shall describe
later, was discovered by Schur [S4] much later. The identity (3.4) which
expresses SA in terms of the h's is due originally to Jacobi (loc. cit.), and is
often called the Jacobi-Trudi identity.
The results of Examples 1-4 may be found in Littlewood [L9], Chapter
VII, which gives other results of the same sort. The formula in Example 8
for the product of two Schur functions as a determinant in the h's is
essentially due to Jacobi (loc. cit.), though rediscovered since. The result of
Example 9 is due to Giambelli [G8]. Example 10 is due to A. Lascoux [L1].
The proof of the law of quadratic reciprocity sketched in Example 17 is

62
I SYMMETRIC FUNCTIONS
essentially Eisenstein's proof (see Serre [S13], Chapter I). The presentation
here is due to V. G. Kac [K1].
For Examples 18-21 see [M8J For Examples 23 and 24, and the history
of the Sergeev-Pragacz formula (i.e., formula (1) of Example 24), see [P3],
also [P2] and [S27]. Other proofs of this formula, in the context of Schubert
polynomials, are due to A. Lascoux (see for example [M7]). The factoriza-
tion property (4) in Example 23 (which is a special case of the
Sergeev-Pragacz formula) is due to Berele and Regev [B2].
4. Orthogonality
Let x -= (XI' X2,...) and Y == (Yl' Y2'...) be two finite or infinite sequences
of independent variables. We shall denote the symmetric functions of the
x's by SA(X), PA(X), etc., and the symmetric functions of the y's by
SA(Y)' PA(Y)' etc.
We shall give three series expansions for the product
n (1-x;y;)-1.
; tj
The first of these is
(4.1)
n (l-xiYj)-1 == EZ;lp.\(X)p.\(y)
;,j A
summed over all partitions A.
This follows from (2.14), applied to the set of variables x;Yj.
Next we have
(4.2) n (l-x;Yj)-1 == Eh.\(x)m.\(y) == Em.\(x)h.\(y)
;,j ,\ A
summed over all partitions A.
Proof. We have n(l-x;Yj)-1 = nH(Yj)
;,j j
00
= n E h,(x)y!
j r-O
== Eha(x)ya
a
== Eh.\(x)m,\(y)
,\

4. ORTHOGONALITY
63
where a runs through all sequences (at, a2,...) of non-negative integers
such that E ai < 00, and A runs through all partitions. I
The third identity is
(4.3)
n (1-X;Yj)-1 = E s.\(x)s.\(Y)
i,j ,\
summed over all partitions A.
Proof. This is a consequence of (4.2) and (3.7'). Let x = (Xl'...' xn),
y == (YI'...' Yn) be two finite sets of variables, and as usual let S = (n - 1,
n - 2,. ..,0). Then from (4.2) we have
n
(4.4) ac5(x)a8(y) n (1-xiYj)-1 =a8(x) E ha(x)s(w)ya+w8
i,j-l a,w
summed over a E Nn and w E Sn,
=ac5(X) E s(w)h_ws(x)yl3
tW
= E a13(x)y13

by (3.7'). Since aw = s(w )a, it follows that this last sum is equal to
E a'Y(x)a'Y(Y) summed over i'] > i'2 > ... > i'n  0, i.e. to
E a,\+s(x)a.\+s(y),
,\
summed over partitions ,\ of length  n. This proves (4.3) in the case of n
variables Xi and n variables Yi; now let n -+ 00 as usual. I
We now define a scalar product on A, i.e. a Z-valued bilinear form
(u,v), by requiring that the bases (h.\) and (m,\) should be dual to each
other:
(4.5)
(h.\, m J.I.) = 8.\J.I.
for all partitions A, j.L, where 8.\J.I. is the Kronecker delta.
(4.6) For each n  0, let (u.\), (v.\) be Q-bases of A'Q, indexed by the
partitions of n. Then the following conditions are equivalent:
(a) (u.\, vJ.l.) = 8.\J.I. for all A, J.L;
(b) L,\ u,\(x)v.\(Y) = ni,j(1 - XiYj)-I.

64
I SYMMETRIC FUNCTIONS
Proof. Let
UA = E aAphp'
p
Vp.= Ebp.umu.
u
Then
(UA, Vp.) = E aApbp.p
p
so that (a) is equivalent to
(a')
E aApbp.p = 8AJ.L'
P
Also (b) is equivalent to the identity
E UA(X)VA(Y) = E hp(x)mp(Y)
A p
by (4.2), hence is equivalent to
(b')
E aApbAu = 8pu'
A
Since (a') and (b') are equivalent, so are (a) and (b).
From (4.6) and (4.1) it follows that
(4.7)
(PA'Pp.) = 8Ap.ZA
so that the PA form an orthogonal basis of Ao' Likewise from (4.6) and
(4.3) we have
( 4.8) ( SA' S p.) = 8A J.L
so that the SA form an orthonormal basis of A, and the SA such that IAI =n
form an orthonormal basis of An. Any other orthonormal basis of An must
therefore be obtained from the basis (SA) by transformation by an orthogo-
nal integer matrix. The only such matrices are signed permutation matri-
ces, and therefore (4.8) characterizes the SA' up to order and sign.
Also from (4.7) or (4.8) we see that
(4.9) The bilinear form < u, v) is symmetric and positive definite.
(4.10) The involution CJJ is an isometry, i.e. (CJJu, CJJv) = (u, v).
Proof. From (2.13) we have CJJ(PA) = -:1::.PA' hence by (4.7)
(CJJ(PA)' CJJ(Pp.» = (PA'PJ.L)
which proves (4.10), since the PA form a Q-basis of 'Ao (2.12).

4. ORTHOGONALITY
65
Finally, from (4.10) and (4.5) we have
< eA, IlL) = BAIL
where Ill- = w(mIL), i.e. (eA) and (fA) are dual bases of A.
Remarks. 1. By applying the involution w to the symmetric functions of
the x variables we obtain from (4.1), (4.2), and (4.3) three series expan-
sions for the product Ili,j(l + XiYi)' namely
(4.1')
n (1 +xiYj) = E 8AZ;lpA(X)PA(Y)'
i,j A
(4.2')
n (1 +xiYj) = E mA(x)eA(y) = E eA(x)mA(y),
i,j A A
(4.3')
n (1 +XiYi) = E SA(X)SA'(Y)'
i,j A
the last by virtue of (3.8).
2. If x, yare elements of a A-ring R, we have
Ut() = E Z;1A(X)A(y)tIAI
A
= E SA(x)SA(y)tIAI
A
from (4.1) and (4.3), and
At(.XY) = E 8A Z; 11/1 A(X) 1/1 A(y )tlAI
A
= E SA(X)SA'(y)tIAI
A
from (4.1') and (4.3').
Examples
1. If we take Yl = ... =Yn =1, Yn+1 =Yn+2 = ... = 0 in (4.3'), we obtain
E(t)n = E SA(X)SA'(Y)
A
=  (:)SA(X)tIAI
in the notation of 3, Example 4.

66
I SYMMETRIC FUNCTIONS
The coefficients of the powers of I on each side are polynomials in n (with
coefficients in A) which are equal for all positive integral values of n, and hence
identically equal. Consequently we have
E(t}x =  ( )s"tl"l
for all X. By replacing X, t by -X, -t we obtain
H(t)x= (X)s tlAI
i..J A' A ·
A
These identities generalize the binomial theorem.
2. Let Yi = qi-1 for 1 < i < n, and Yi = 0 for i > n. From (4.3') we obtain
Ii: E(qi-l) = E qn(".)[n ]SA
i-1 A A
in the notation of A3, Example 1. Likewise, from (4.3),
Ii: H(qi-l) = E qn(,,)[ n,] SA'
i- 1 ,\ A
In these formulas we may let n -+ 00 and obtain
q n( A')
.n (l+Xjql-l)= E H( )s,,(x),
1.j;.1 A A q
q n( ,\)
.n (l-Xjqi-lrl = E H ( )s,,(x),
',J  t A A q
where HA(q) == nx e ,\(1 - qh(x» is the hook-length polynomial corresponding to
the partition A.
3. Let Yt = ... = Yn = tin, Yn+ 1 == Yn+2 = ... = 0, and then let n -+ 00. We have
( )n
x.1
I;.Il + -;- --. I;.I exp(xit) = exp(e1t)
and
1 (n) 1 1
!Ai \ -+ n h(x)- =h(A)-
n 1\ xe'\
where h(A) is the product of the hook-lengths of A. Hence from (4.3') we obtain
SA
exp(e11) = E _tlAI
.\ h( A)
and therefore
n!
ei'= E h(>')s"
I,\I-n

4. ORTHOGONALITY
67
or equivalently
(ei, SA) = n !/h( A).
4. From (2.14') and (4.7) we have
(hn,PA) = 1
for all partitions A of n. Dually,
(en,PA) = eA.
5.
m n
n n(xi+Yj)= ESA(X)S.\,(Y)
;-1 j-1 .\
summed over all partitions A = (AI'...' Am) such that Al  n (i.e. A c (nm», where
A' = (m - Xn,..., m - Xl). (Replace Yi by Yi-1 in (4.3'), and clear of fractions.)
Hence from A3, Example 10 we have
m n
n n(1+xi+Yj)= EdAp.sp.(x)SA:'(Y)
i-1 j-I A,P.
summed over pairs of partitions A, J.L such that J.L cAe (nm). (This formula gives
the Chern classes of a tensor product E  F of vector bundles, since if c(E) =
n(1 + x;) and c(F) = n(1 + Yj) are the total Chern classes of E and F, we have
c(E  F) = n(1 + Xi + Yi).)
6. Let 4 = det«1-x;Yj)-1)t < i,j< n (Cauchy's determinant). Then
n
4 =a8(x)a8(Y) n (l-X;Yj)-l.
;,j-l
For if we multiply each element of the ith row of the matrix «1 -x;Yj)-t) by
nj.l(l - x;Yj)' we shall obtain a matrix D whose (i, j) element is
II
n(1-xiy,)= E x7-k.(-1)n-kek(Y)
r.. j k - 1
in the notation of (3.6). This shows that D = A6(x)M(y), so that det(D) =
as(x )aB (y). On the other hand, it is clear from the definition of D that det(D) =
A. n(1 - xiYj).
Since also
4 = det(l +xiYj+xfyf+ ...)
= E det(x;oJyjoJ)
a
= E ao(x)yQ
a

68
I SYMMETRIC FUNCTIONS
the summation being over all a = (aI'.. ., an) E Nn, it follows that
11 = E aA+(x)a).+(y)
It
summed over all partitions A of length  n. Hence we have another proof of (4.3),
7. Likewise the identity (4.3') can be proved directly, without recourse to duality.
Consider the Vandermonde determinant as(x, y) in 2n variables
Xt,...,xn'YI,...,Yn; on the one hand, this is equal to a8(x)a8(y)n(X;-Yj); on the
other hand, expanding the determinant by Laplace's rule, we see that it is equal to
(1)
E ( - 1) e( JJ-) a JL ( X ) a {L ( Y ) ,
JL
summed over J.L E Nn such that 2n - 1  J.Ll > J.L2 > ... > J.Ln  0, where iL is the
strictly decreasing sequence consisting of the integers in [0, 2n - 1] not equal to any
of the J.Li' and e( J.L) = E (2n - i - J.L;). By writing J.L = A +  and using (1.7), we see
that (1) is equal to
( - 1) n (y 1 . .. Y n ) 2n - I E a). + 8 (x) a)., + 8 ( - Y - I )
A
summed over all partitions A such that I( A)  nand I( A')  n. If we now replace
each Y; by y;-l, we obtain (4.3').
8. Let M be a module over a commutative ring A, and let cp: M X M -+ A be an
A -bilinear form on M. The standard extension of cp to the symmetric algebra S(M)
is the bilinear form defined on each sn(M) by
n
<I>(u, v) = E n cp(u;, VW(i»
weSn i-I
where u = Ut ... un' V = VI ... vn and the ui' vj are elements of M. In other words,
<I>(u,v) is the permanent of the n X n matrix (cp(u;,vj».
In particular, let A = Q and let M be the Q..vector space with basis PI' P2,.'"
so that S(M)=Q[Pt,P2,...]=AQ. Define CP(Pr,Ps)=rrs for all r,s 1. Then
the scalar product (4.5) on AQ is the standard extension of cpo
9. Let C(x, Y) = n (1 - X;Yj)-1. Then for all ! E A we have
i, j
(C(x,y),!(x» = fey)
where the scalar product is taken in the x variables. (By linearity, it is enough to
prove this when f = SA' and then it follows from (4.3) and (4.8).)
In other words, C(x, y) is a 'reproducing kernel' for the scalar product.
10. Let pr) = EmA summed over partitions A of n of length r, as in 2, Example
19. Show that
(r)_ " (_l)b+r-l( b )
Pn - i..J r _ 1 S(a,lb).
a+b-n

5. SKEW SCHUR FUNCTIONS
69
(Under the specialization hr...... X for all r  1, the Jacobi-Trudi formula (3.4)
showS that SA t-+ 0 if A2 > 1, that is to say if ,\ is not a hook; and if ,\ = (a, 1b) it is
easily shown that SA t-+ X(X - 1)b. Hence from the identity «4.2), (4.3»
E mA(x)hA(y) = E s,\(x)s,\(y)
we obtain
E mA(x)X1(A) = E s(a,tb)(x)X(X _1)b
a,b>O
from which the result follows.)
11. (a) Let ,\ be a partition of n. Show that when mA is expressed as a polynomial
in PI'''.' Pn' the coefficient of Pn is non-zero. (The coefficient in question is
n-1(Pn, mA), which is also the coefficient of hA in n-1Pn expressed as a polyno-
mial in the h's, and is given explicitly in fi2, Example 20.)
(b) For each integer r  llet Ur be a monomial symmetric function of degree r (i.e.
u, = mA for some partition A of r). Show that the u, are algebraically independent
over Q and that AQ = Q[Ut, U2'...]. (From (a) above we have u, = c,p, +
a polynomial in PI'.'" Pr-t' where c, * O. This shows by induction on r that
Q[Ut,..., u,] = Q[Pt,..., Pr] for each r  1. Hence for each m  1 the monomials
of degree m in the U variables span A Q' and are therefore linearly independent
over Q.)
Notes and references
The scalar product on A was apparently first introduced by Redfield [Rl]
and later popularized by P. Hall [H3]. Example 5 is due to A. Lascoux [Ll],
and Example 11 to D. G. Mead [MID].
5. Skew Schur functions
Any symmetric function ! E A is uniquely determined by its scalar prod-
ucts with the sA: namely
!= E(!,SA)S.\
,\
since the SA form an orthonormal basis of A (4.8).
Let A, JL be partitions, and define a symmetric function SA/JL by the
relations
(5.1)
(SA/JL'Sv) = (s,\,sJLsv)

70
I SYMMETRIC FUNCTIONS
for all partitions v. The SA/p, are called skew Schur functions. Equivalently,
if c;v are the integers defined by
(5.2)
Sp,Sv = E C;vSA'
A
then we have
(5.3)
SA/p, = E c;vSv.
v
In particular, it is clear that SA(O = SA' where 0 denotes the zero partition.
Also c;v=O unless IAI=IILI+lvl, so that SA/p, is homogeneous of degree
I AI-IILI, and is zero if I AI < I ILl. (We shall see shortly that SA/p, = 0 unless
A  IL.)
Now let x = (Xl' x2'...) and Y = (Yt, Y2'. ..) be two sets of variables.
Then
E SA/(X)S,\(Y) = E C;vSV(X)SA(Y)
A A,V
= E sv(X)Sp,(Y)Sv(Y)
v
by (5.2) and (5.3), and therefore
E SA/p,(X)SA (y) = Sp,(Y) E hv(x )mv(y)
A v
by (4.2) and (4.3). Now suppose that Y = (Yt, .. ., Yn), so that the sums
above are restricted to partitions A and v of length  n. Then the
previous equation can be rewritten in the form
E SA/p,(x)aA+ B(Y) = E hv(x)mv(Y )ap,+B(Y)
A v
= E ha(x) E B(W)y"+w(p,+B)
a weSn
summed over a E Nn. Hence SA/P,(X) is equal to the coefficient of yA+B in
this sum, i.e. we have
SA/p, = E s( W )hA+ B-w( p,+ B)
weSn
with the usual convention that ha = 0 if any component ai of a is
negative. This formula can also be written as a determinant
(5.4)
SA I" = det(hA -" -i+J.)1 ..
1 r- I r-J __ , ,J __ n
where n  I(A).
When IL = 0, (5.4) becomes (3.4).

5. SKEW SCHUR FUNCTIONS
71
From (5.4) and (2.9) we have also
(5.5)
SAJ"- = det(eA/._ ,,-' -i+}. )1 ..
,,.- I 1""') <',J<m
where m  leA'), and therefore
(5.6) w ( s AI IJ-) = SA' I IJ- ' ·
From (5.4) it follows that sA/IJ- = 0 unless Ai  J.L; for all i, i.e. unless
A:J #-t. For if A, < J.L, for some r, we have Ai  A, < 1L,  J.Lj for 1 <j  r  i
 n, and therefore Ai - ILj - i + j < 0 for this range of values of (i, j).
Consequently the matrix (h AJ _ IJ- _ i + j) has an (n - r + 1) X r block of zeros
in the bottom left-hand corner,} and therefore its determinant vanishes.
The same considerations show that if A:J J.L and J.L,  A,+ 1 for some
r < n, the matrix (hA,- J'/-i+j) is of the form (  ), where A has r rows
and columns, and B has n - r rows and columns, so that its determinant is
equal to det(A) det(B). Hence if the skew diagram A - J.L consists of two
disjoint pieces 8, 'P (each of which is a skew diagram), then we have
s)./JI. = S8 · s'P. To summarize:
(5.7) The skew Schur function s).11J- is zero unless A::> J.L, in which case it
depends only on the skew diagram A - IL. If 8; are the components (1) of
,\ - J.L, we have s).11J- = n S8J. I
If the number of variables Xi is finite, we can say more:
(5.8) We have sA/IJ.(xl,..., xn) = 0 unless 0  Xi - J4  n for all i  1.
Proof. Suppose that X, - J£, > n for some r  1. Since en+l = en+2 = ... =
0, it follows as above that the matrix (eAi-IJ-' -i+j) has a rectangular block of
zeros in the top right-hand comer, with ode vertex of the rectangle on the
main diagonal, hence its determinant vanishes. I
Now let X = (Xl' X2, .. .), Y = (Yl, Y2,. . .), z = (Zl' Z2,.. .) be three sets of
independent variables. Then by (5.2) we have
(a) E sA/IJ-(x )SA (z )slJ- (y) = E sJ.£(Y )slJ-(z). n (1 - XiZk)-l
A,  i,k
which by (4.3) is equal to
n (I-X;Zk)-l . n (l-Yjzk)-t
i,k j,k
and therefore also equal to
(b)
E SA (X, y)s,\(z)
A

72
I SYMMETRIC FUNCTIONS
where SA(X, y) denotes the Schur function corresponding to A in the set of
variables (Xl' x2'...' Yl' Y2'''.). From the equality of (a) and (b) we con..
clude that
SA(X,y) = E SA/(X)S(Y)
JL
(5.9)
= E c;vs(y)sv(x).
JL, v
More generally, we have
(5.10)
SA/(X,y) = E SA/V(X)Sv/(Y)
v
summed over partitions v such that A :> v:> J.L.
Proof. From (5.9) we have
E s A/ (x, Y ) S p. (z) = SA (X, y, z)
p.
= E SA/v(X)Sv(Y, z)
v
= E SA/V(X)Sv/p.(Y )S(z)
p., v
by (5.9) again; now equate the coefficients of s(z) at either end of this
chain of equalities. I
The formula (5.10) may clearly be generalized, as follows. Let x(1),.. ., x(1J)
be n sets of variables, and let A, J.L be partitions. Then
(5.11)
n
( (I) (n») -  n . «i»)
SA/IJ. X , · · · ,X - i...J Sv(l) /v(.-l) X
(v) i-I
summed over all sequences (v) = (v(O), v(l),..., v(fJ») of partitions, such that
v(O) = /L, v(n) = A, and v(O) C vel) C ... C V(n). I
We shall apply (5.11) in the case where each set of variables xci) consists
of a single variable Xi. For a single x, it follows from (5.8) that SA/r(X) = 0
unless A -/L is, a horizontal strip (1), in which case SA/(X) =XIA-. Hence
each of the products in the sum on the right-hand side of (5.11) is a
monomial Xfl... x:", where aj == Iv(i) - v(i-I)I, and hence we have
SA/P.(Xh. . ., xn) expressed as a sum of monomials xa, one for each tableau

5. SKEW SCHUR FUNCTIONS
73
(1) T of shape A - J.L. If the weight of T is a == (at,..., an), we shall write
xT for xa. Then:
(5.12) SA/J1. == EXT
T
summed over all tableaux T of shape A - J.L.
For each partition v such that I v I == I A - J.LI, let KA- p., v denote the
number of tableaux of shape A - I-L and weight v. From (5.12) we have
(5.13 )
SA/J1. == E KA- p., vmv
v
and therefore
(5.14)
KA-p.,v == (SA/J1.' hv> == (SA' sp.hv>
so that
(5.15)
SJ.LhV = E KA - p.t vSA.
,\
In particular, suppose that v == (r), a partition with only one non-zero part.
Then KA- p.,(r) is 1 or 0 according as A - I-L is or is not a horizontal r-strip,
and therefore from (5.15) we have
(5.16) (Pieri's formula)
Sp.hr == E SA
,\
summed over all partitions A such that A - I-L is a horizontal r-strip.
By applying the involution w to (5.16), we obtain
(5.17)
SILer == E SA
,\
summed over all partitions A such that A - I-L is a vertical r-strip.
Remarks. 1. It is easy to give a direct proof of (5.17). Consider (for a finite
set of variables Xl'...' Xn) the product
al-L+ 8 er == E e( w )XW( p.+ 8) E xa
we SrI a
= E al-L+a+8
a
where the sum is over all a E Nn such that each a; is 0 or 1, and L at == r.
For each such a, the sequence
I-L + a +  = ( J.Lt + at + n - 1, 1-L2 + a2 + n - 2,..., I-Ln + an)

74
I SYMMETRIC FUNCTIONS
is in descending order, so that we have only to reject those a for which
two consecutive terms are equal. We are then left with those a for which
A = I-L + a is a partition, i.e. such that A - I-L is a vertical r-strip. This
proves (5.17), hence also (5.16) by duality. We can now play back the rest
of the argument: (5.16) implies (5.15) by induction on the length of v,
hence (5.14), which in turn is merely a restatement of (5.13).
2. Proposition (5.12) is the origin of the application of Schur functions to
enumeration of plane partitions (see the examples at the end of this
section). For this reason, combinatorialists often prefer to take (5.12) as
the definition of Schur functions (see e.g. Stanley [823]). This approach has
the advantage of starting directly with a simple explicit definition, but it is
not clear a priori why one should be led to make such a definition in the
first place.
3. In any A-ring we can define operations S A/JI. by the formula (5.3):
SA/JI. = " cA S'II
L.J I-L'II
V
Then (5.9), for example, takes the form of an addition theorem:
SA(X+Y) = E SA/JI.(X)SIL(y)
p.
for any two elements x, y of a A-ring. Similarly for the other formulas in
this section.
4. The formula (5.4) shows that the skew Schur functions SA/JI.(X), where A
and J..L are partitions of length p, are the p xp minors of the matrix
Hx = (hi_j(x)), i.e. they are the entries in the pth exterior power !f(Hx).
The relation (5.10) is therefore equivalent to
!f(Hxty) = N'(Hx) /f(Hy)'
Thus it is a consequence of the functoriality of N, since Hx,y = HxHy.
Examples
1. Let ,\ - J..L be a horizontal strip. Then s./JI. = hv = hV1hv2 . .. where the integers
Vi are the lengths of the components of the strip. (Use (5.7).) Likewise, if A - JL is
a vertical strip, we have s A/JI. == eVJ eV2 ..., where again the Vi are the lengths of the
components of the strip.
2. (a) Let A be a partition of n. Then the number of standard tableaux of shape A
IS
KA,(lPl) = (s).,hi>
by (5.14). By 4, Example 3 it follows that the number of standard tableaux of
shape A is equal to nl/h(A), where h(A) is the product of the hook-lengths of A.

s. SKEW SCHUR FUNCTIONS
75
1bis result is true more generally if A is a skew diagram all of whose connected
components are right diagrams (i.e. diagrams of partitions).
(b) Let P be a positive intege! and let A be a partition, X its p-core (1, Example
8). A p-tableau of shape A - A is a sequence of partitions
A == IL (0) C IL (1) C ... C lL(m) == A
such that lL(i) - lL(i -1) is a border strip of length 1!, for 1  i  m. (Thus when
p == 1, a p-tableau is just a standard tableau (and A == 0); when p == 2, it is also
called a domino tableau.)
Let A* be the p-quotient of A, thought of as a skew diagram with components
A(i) (0 < i <p - 1), and let h(A.*) =- nh(A.(i» be the product of the hook..lengths of
A *. From ii, Example 8 it follows that the p-tableaux of shape A - A are in
one-one correspondence with the standard tableaux of shape A., and hence by (a)
above the number of p-tableaux of shape A - X is equal to m!jh(A*), where
m -IA.*I. This number is also equal to pmm!jhp(A), where hj'(A) is the product of
the hook-lengths of A that are divisible by P (1, Example 8td».
3. For each symmetric function f e A, let f.l. : A -. A be the adjoint of multiplica-
tion by f, i.e.
(f.l. u, v) == (u, tv )
for all u, v e A. Then f..... fJ. : A  End(A) is a ring homomorphism.
(a) Since (S: SA' sv) = (s.\, sp.s.,) = (s'\/IJ,' S.,) for all partitions A, IL, JI, it follows that
S: SA == s)./p.. Hence from (5.9) we have
SA (x , y) == E S 1J,.l. S). ( x) . S IJ, ( Y )
J.L
and therefore, for any f e A,
f(x,y) == E s:f(x). Sp.(Y).
J.L
(b) We have h).J. mp' = 0 unless IL == A. U JI for some partition II, and in that case
hfmp. -= m.,. For
(htmlJ"h.,) == (mp.,h).h.,) == (mp.,h).u.,)
which is zero unless IL == A. U II.
In particular, h; mp' == 0 if n is not a part of J..L, and h;- mp' == mv if n is a part of
JL, where II is the partition obtained by removing one part n from J..L. It follows that
for every f(xo, Xl' X2,...) e A, (h: fXXh..., xn) is the coefficient of x8 in f.
(c) Next consider pn.l. . If N;> n we have
(pnJ.hN,p.\) == (hN,PnP).) == (hN-n,p).)

76
I SYMMETRIC FUNCTIONS
for all partitions A of N - nt by 4t Example 4. Hence
Pn.1. hN = hN-n
and therefore
pn.l. = E h, a/ ahn+r
rO
acting on symmetric functions expressed as polynomials in the h's.
Dually
.1. ( )n-l"
Pn = -1 i.J e, a/ aen+r
r 0
acting on symmetric functions expressed as polynomials in the ets.
Further, we have (pn.l.PA'PJ1.) = (PA,PnPp,), which is zero if A + J-L U (n), and is
equal to z). if A == IL U (n). It follows that pnJ.PA =Z).Z;1pp, if n is a prt of A, and 
is the partition obtained by removing one part n from A. From the definition of z,\
it follows that Z).Z;1 = n. mn(A)t where mn(A) is the multiplicity of n as a part of
A, and therefore
p,:-=na/iJPn
acting on symmetric functions expressed as polynomials in the p's. In particular,
each pn.l. is a derivation of A.
Since each Ie A can be expressed as a polynomial CP(P1,P2'.") with rational
coefficients, it follows that
fJ. = cP ( a/a PI' 2 a/a P 2 , . . . )
is a linear differential operator with constant coefficients.
"(d) For each neZ, let ?Tn: A -+ A be the operator defined as follows: if n  1, 'TTn is
multiplication by Pn; if n  -1, then 'TTn = P n; and 'TTo is the identity. Then we
have
[ 'TT m , 'TTn] = n 8m + n ,0 'TT 0
for all m, neZ, so that the linear span of the 7Tn is a Heisenberg Lie algebra.
4. We have
E s). = TI (I-Xi)-l TI (I-X;Xj)-l,
.\ i i <j
where the sum on the left is over all partitions A.
It is enough to prove this for a finite set of variables Xl'...' Xn' Let <I>(Xt,..., xn)
denote LA SA (Xl' ..., Xn), which is now a sum over partitions A of length <: n. By
induction on n, it is enough to show that
n
<I>(Xt,...tXn'Y) = <I>(xt,...,xn)(l-y)-l TI (l-x;y)-l.
i-I

5. SKEW SCHUR FUNCTIONS
77
From (5.9) it follows that
m ( )  1 A-#-'I ( )
'AI Xl'...' Xn, Y =  Y S#-, XI'...' Xn
A, J£
where the sum on the right is over all pairs of partitions A::> J.L such that I( J.L)  n
and A - J.L is a horizontal strip. For each such pair A, J.L, define veIL by
J.Li - Vi = Ai+ I - J.Li+ 1 (i> 1), so that IA - J.LI = Al - J.L1 + I J.L - vi. Then A can be
reconstructed from J.L, v, and the integer Al - J.Lh and hence
(*) E yIA-#-'lsJ£(Xl,...,Xn) = E yl#-,-V1(1-y)-ls#-,(Xl'...'Xn),
A,J£ ,v
the sum on the right being over pairs of partitions J.L::> II such that I( IL)  nand
J.L - v is a horizontal strip. By (5.16), the right-hand side of (*) is equal to
E y'(l - y) -1 h'(X1 ,. . . , xn )SV(XI, . . ., xn)
v,r
summed over all partitions v of length  n, and all integers r> 0; and this last
sum is equal to (l-y)-l n7_t(1-x;y)-I<1>(x1,...,xn), as required.
5. (a) We have
E sJ£= n(l-x;)-ln<l-xixj)-I,
J£ cvcn i i <j
where the sum on the left is over all even partitions J.L (i.e. with all parts J.Li even).
Each partition A can be reduced to an even partition J.L by removing a vertical
strip, in exactly one way: we take J.Li = Ai if Ai is even, and J.Li = Ai - 1 if Ai is odd.
From this observation and (5.17) it follows that
( E Si£)( E e,)  ESA'
#Lcven r>O A
the sum on the right being over all partitions A. Since E e, = n(l + Xi), the result
now follows from Example 4.
(b) We have
E sv=n(l-xixj)-t.
v' even i <j
The proof is dual to that of (a): each partition A can be reduced to one with even
columns by removing a horizontal strip, in exactly one way. From this observation
and (5.16) it follows that
( E Sp)( E h,) = E SA'
JI' even , > 0 A
and since E,> 0 h, = n(l-x;)-I, the result again follows from Example 4.

78
I SYMMETRIC FUNCTIONS
The involution CIJ interchanges the identities in (a) and (b).
6. We have
E ( -1)n(,\)s,\ == n (I-Xi)-l n (1 +X;X})-I.
A i i<j
For if we replace each variable Xi by  . Xi in Example 5(b), we shall obtain
n (1 + XiXj) -1 == E ( _1)1111/2 sJI
;<j JI
summed over partitions v with all columns of even length. Each partition ,\ is
obtained uniquely from such a partition v by adjoining a horizontal strip, and
therefore
n (1 - Xi) -1 n (1 + XiXj) -1 = E E ( _1)'111/2 sl1h,.
; i <j r;> 0 JI
== E ( -1)/('\) SA
,\
where !(A) == Ei> l[!Xi] (and [x] is the greatest integer  x). Since
[ ;] E (;) (mod 2), it follows from (1.6) that to..);: n(,\) (mod 2).
7. The same argument as in Example 5(b) shows that
E tC('\)s,\ == n (1- tx;)-1 n (l-xixj)-l
A i i <j
where the sum is over all partitions A, and c( A) is the number of columns of odd
length in A. This includes the identities of Example 4 (when t == 1) and Example
5(b) (when t == 0).
8. By applying the involution CIJ to Example 7 we obtain
E tr(A)SA = n 1 + IX: n 1
,\ ; I-x; i<j 1 -XiXj
where the sum is over all partitions A, and r( A) is the number of rows of odd
length in A. When t == 1 this reduces to Example 4, and when t = 0 it reduces to
Example 5(a).
9. The products
n (I-XiX,),
. . }
, <}
n (I-x;) n (1-xixj), n (I-x;) n (I-xixj).
; ; <j i i <j
(i.e. the reciprocals of those of Examples 4, 5(a) and 5(b» can also be expanded as
series of Schur functions. The expansions may be derived from W eyl's identity for
the root..systems of types D,., Bn, Cn respectively. (If R is a root system with Weyl
group W, R+ a system of positive roots, p half the sum of the positive roots, then
Weyl's identity ([B8], p. 185) is
(*) E e(w)ewp== n (eo/2_e-o/2)
we W aeR+

s. SKEW SCHUR FUNCTIONS
79
where &(w) is the sign of we W, and the e's are formal exponentials.)
(a) When R is of type DfI' the identity (*) leads to
E ( _1)1"'1/2 S1r(Xt,..., xn) == n (1 - X;Xj)
 i<j
summed over all partitions 'T1' == (at - 1,. . . , ap - 11 at,. . . , ap) in Frobenius nota-
tion, where at < n - 1.
(b) When R is of type C,., we obtain from (*)
E ( -1)lpI/2sp(Xt,..., xn) == n (1 -xl) n (1 -X;Xj)
p i i <j
summed over all partitions p == (at + 1,..., ap + 11 at,..., ap), where at  n - 1.
(c) When R is of type B,., we obtain from (*)
E ( -lilQ'l+p(Q'»/2 SO'(Xl"." xn) == n (1- Xi) n (1 - X;Xj)
CT i i <j
summed over all self-conjugate partitions 0' == (at,..., ap I at,..., ap) such that
a} <n -1, where p(O') =p.
10. In the language of '\-rings, the identities of Examples 5(a), 5(b), and 9 give
series expansions (in terms of Schur operations) for 0',(0' 2(X», O't(,\2(x», '\t(oo 2(x»,
and At(,\2(X», namely
00,( 0' 2(x» == E s JA.(x )tl "I /2 ,
JA. even
oo,(,\2(X» == E SI'(x)tIVI/2,
1"even
'\t(002(X)) = E S"'(x)t1pl/2,
p
'\t(,\2(X» = E S7T(x)tl1T1/2,
",.
the last two summations being over partitions p = (at + 1,..., ap + 11 at,..., ap)
and 1T IE: (at - 1,..., ap - 11 at,..., ap).
11. Let Xt == ... =xN = t, xN+ t =xN+2 = ... == 0 in the formula of Example 4.
Then SA = (  )tlAI (3, Example 4) and hence, for each n  0,
'{"" (N) fi . f . ( )-N( 2)-N(N-t)/2
 ,\ =- coef IClent 0 tn In 1 - t 1 - t
IAI-n
= coefficient of tn in (1 - t) - N(N + 1)/2(1 + t) - N(N -1)/2.
Since this is true for all positive integers N, it is a polynomial identity, i.e.
E () == coefficient of tn in (1 - t)-X(x+t)/2(1 + t)-X(X-t)/2.
IAI-n

80
I SYMMETRIC FUNCTIONS
12. Let Xl = ... XN = tiN, XN+ 1 =XN+2 = ... = 0 in the identity of Example 4,
and let N -+ 00. Then from Example 11 we obtain
E h(A) -1 = coefficient of tn in exp(t + tt2)
IAI-n
where (4, Example 3) h( A) is the product of the hook lengths of A. From Example
2 it follows that the total number of standard tableaux of weight (In) is equal to n!
multiplied by the coefficient of tn in exp(t + tt2). This number is also the number
of permutations w e S" such that w2 = 1.
13. Let A be a partition. A plane partition of shape A is a mapping 11' from (the
diagram of) A to the positive integers such that 1T(Xl)  1T(X2) whenever X2 lies
below or to the right of Xl in A. The numbers 1T(X) are the parts of 11', and
11T1 = E 1T(X)
xe,\
is called the weight of 11'. Any plane partition 'TT determines a sequence A = A(O):)
A(1)::> ... of (linear) partitions such that 11'-1(i) = AO-1) - A(i) for each i  1.
If 11'(X1) > 1T(X2) whenever X2 lies directly below Xl (i.e. if the parts of 'TT
decrease strictly down each column) then 1T is said to be column-strict. Clearly 7T' is
column-strict if and only if each skew diagram 11'-1(i) = A(i-1) - A(i) is a horizontal
strip.
A plane partition 11' has a three-dimensional diagram, consisting of the points
(i,j,k) with integer coordinates such that (i,j)EA and 1 k 11'(i,j). Alterna-
tively, we may think of the diagram of 11' as a set of unit cubes, such that 7T'(x)
cubes are stacked vertically on each square x e A. As in the case of ordinary
(linear) partitions, we shall use the same symbol 11' to denote a plane partition and
its diagram.
If S is any set of plane partitions, the generating function of S is the polynomial
or formal power series
E ql1T1
'tT'eS
in which the coefficient of qn is the number of plane partitions of weight n which
belong to S.
(a) Consider column-strict plane partitions of shape A, with all parts  n. By (5.12)
the generating function for these is SA(qn,qn-J,...,q), which by 3, Example 1 is
1 - n+c(x)
qIAI+n(A) f1 q
h(x) ·
xe,\ 1 - q
(b) Let 1, m, n be three positive integers, and consider the set of plane partitions 11'
with all parts  n and shape A such that l( A)  1 and l( A')  m: that is, the set of
three-dimensional diagrams 1T which fit inside a box B with side-lengths /, m, n.
By adding 1 + 1 - i to each part in the i th row of 11', for 1  i  l, we convert 11'

5. SKEW SCHUR FUNCTIONS
81
into a column-strict plane partition of shape (m,..., m) = (ml) and largest part
 1 + n. From (a) above, the generating function for the plane partitions 7T' c B is
....:
therefore
(1)
n
xE (m')
1 - ql+" +C(X)
1 - qh(x)
In this form the result does not display the symmetry which it must have as a
function of 1, m, and n. It may be rewritten as follows: for each y = (i, j, k) E B,
define the height of y to be ht(y) = i + j + k - 2 (so that the point (1,1, 1) has
height 1). Then the generating function (1) may be written in the form
l_ql+ht(y)
E ql1rl_ n
'lTcR - yE R 1 - qhl(Y) ·
(2)
(c) We may now Jet any or all of 1, m, n become infinite. The most striking result is
obtained by letting all of I, m, n tend to 00: the box B is then replaced by the
positive octant, and for each n  1 the number of lattice points (i,j, k) with
i + j + k - 2 = nand i,j, k  1 is equal to the coefficient of tn-1 in (1 - t)-3,
hence to tn(n + 1). It follows that the generating function for all plane partitions
is
(3)
r1 ( 1 - qn: I ) n(n + 1)/2 _ DO
n-l 1 - q
n (1-4")-n.
n-l
(d) Likewise, the generating function for all plane partitions with largest part  m
is
(4)
00
n (1- q")-min(m,n).
n-l
14. From Example 13(a), by letting n  00, the generating function for all column-
strict plane partitions of shape A is
(1)
qIAI+n(A)HA(q) -1
where HA(q) = Ox e A(l - qh(x»).
Another way of obtaining this generating function is as follows. Let 'TT' be a
column-strict plane partition of shape A, and let S be the set of pairs ('TT'(i, j), j)
where (i, j) E A. The elements of S are all distinct, because 'TT' is column-strict. We
order S as follows: (r,j) precedes (r',j') if either r> r', or r = r' and j <j'. This is
a linear ordering of S. Define a standard tableau T( 'TT') of shape A as follows:
T(i,j) = k  ('TT'(i,j),j) is the kth element of S in the linear ordering defined
above. For example, if 'TT' is
33211
22
1

82
I SYMMETRIC FUNCTIONS
then S is the ordered set
(3,1),(3,2),(2,I)i(2,2),(2,3),(1,1),(1,4),(1,5),
and T( 'IT) is the standard tableau
12578
34
6
Conversely, let T be a standard tableau of shape A, and let 'IT be a column-strict
plane partition such that T( 'IT) :II: T. Let I AI ::II n, and for 1  k  n let ak be the part
of 'IT in the square occupied by k in T. Then at  ...  an  1 and ak > ak+ 1
whenever k e R(T), where R(T) is the set of integers k e [1, n - 1] such that k + 1
lies in a lower row than k in the tableau T. Now let
(ak-ak+1
ble == ak - ak+ 1 - 1
a -1
n
if k  R(T) and k:l= n
if keR(T)
if k == n
so that bk  0 for k == 1,2,..., n. Then we have
n n
E ale = n + r(T) + E kbk
k-l k-]
where
r(T) == E {k: k + 1 lies in a lower row than k in T}
and therefore the generating function for the column-strict plane partitions 'IT such
that T('IT) = T is
qn +r(T)'Pn(q)-1
where as usual 'Pn(q) == (1 - q). . . (1 - qn).
Hence the generating function for column-strict plane partitions of shape A is
(2)
qn(  qr(T») / f/Jn<q)
summed over all standard tableaux T of shape A.
From (1) and (2) it follows that
(3)
E q,(T) =qn(A)lpn(q)/HA(q).
T
15. Let S be any set of positive integers. From (5.12) and Example 4 it follows that
the generating function for column-strict plane partitions all of whose parts belong
to S is
n (l_qi)-l n (l_qi+i)-l.
ieS ;,jeS
;<j

s. SKEW SCHUR FUNCTIONS
83
(a) Take S to consist of all the positive integers. Then the generating function for
all column-strict plane partitions, of arbitrary shape, is
(1)
00
n (1 - qn) - [(II + 1)/2].
,,-I
(b) Take S to consist of all the odd positive integers. We obtain the generating
function
00
(2)
n (1 - q 2n - 1 ) -t (1 _ q 2,. ) - [,. /2] .
,,-I
Now the column-strict plane partitions with all parts odd are in one-to-one
correspondence with the symmetrical plane partitions 1f (i.e. such that 1f(i, j) =
",(j, i». For the diagram of a symmetrical plane partition may be thought of as a
sequence of diagrams of symmetrical (linear) partitions 17'(1) ::> 1f(2)::> ..., piled one
on top of the other; each 17"(i) is of the form (at,. . ., ap.l at,..., ap) in Frobenius
notation, and hence determines a linear, partition u(') == (2at + 1,..., 2ap + 1)
with odd parts, all distinct; and the u (,) can be taken as the columns of a
column-strict plane partition with odd parts. It follows that (2) is the generating
function for the set of all symmetrical plane partitions.
16. Let <I>(x],...,X,.) = nj(l-x;)-tn;<j(l-xixj)-1 as in Example 4. By setting
t == 0 in the identity of Chapter III, A5, Example 5 we obtain
(1) E ums.\(Xt,...,X,.) = E <I>(xi1,...,x:")/(l-unxp-e;)/2)
m,A &
where the sum on the left is over all partitions A = (At,..., A,.) of length  n, and
integers m > At; and the sum on the right is over all B = (s],..., B,.) with each
Bi = f: 1.
We shall rewrite (1) in the notation of root-systems. Let Vt,..., v,. be the
standard basis of R". Then the set of vectors
R = ( :l:vj(1  i < n), :t: vi:t: vj(l  i <j  n)}
is a root-system of type B", for which
R+= (v;(1  i  n), vi:t: vj(l  i <j  n)}
is a system of positive roots, so that
p == !«2n - l)V1 + (2n - 3)V2 + ... + V,.)
is half the sum of the positive roots. The subset Ro of R defined by
Ro = {v; - vj: i :# j}
is a subsystem of R of type An -1' and Rt == R + n Ro is a system of positive roots
for Ro. The Weyl group Wo of Ro is the symmetric group S,., acting by permuta-
tions of Vt,..., v,., and the Weyl group W of R is the semidirect product of Wo

84
I SYMMETRIC FUNCTIONS
with the group (of order 211) of transformations we: Vi ..-+ SiVi(l  i  n), where as
before S == (Sl'. . ., s,,) and each Si is :i: 1. In this notation,
<I>(e-V.,...,e-VII) = n (l-e-O)/ n (l-e-a)
aeRt aeR+
= ( E s(w)ewp)/( E s(w)ewp).
weWo weW
by virtue of Weyl's identity (Example 9). It follows that the right-hand side of (1)
(with Xi replaced bye-VI) may be written as a sum over W, and by equating the
coefficients of um on either side of (1) we arrive at the identity
(2)
E s.\(e-V1,...,e-UII) ==e-m81(m(J+ p)/l( p)
A
where (} = t(v1 + ... +v,,), and for any vector V
l(v) == E s(w )eWU,
wEW
and the sum on the left is over all partitions A such that leA)  nand leA')  m (i.e.
such that A c (mn».
(If preferred, the right-hand side of (2) can be written as a quotient of determi-
nants:
(2')
E s,\(Xl'''., x,,) = Dm/DO
A
where Dm==det(xi+2"-i_xJ-1)1<itj<,,, and the summation is as before over
partitions A c (mn ).)
This identity (2) is a polynomial identity in n independent variables e-u;. We
may therefore specialize it to obtain identities in one variable q, by replacing each
e-v1 by qli, where the /; are arbitrary integers. This means that each exponential
e-v is replaced by q<v,j), where f= E liVi and (v,/) is the standard scalar product
on Rn. In this way we obtain
(3)
E s(w )q-(m8+p,wJ)
 s (qll qlll) - qm(9,J)
"':.\ ,..., - Es(w)q-(Ptw!) ,
the sum on the left being over all partitions A c (mn).
17. In formula (3) of Example 16 let us take f = 2 p, the sum of the positive roots
of R, so that /; = 2n - 2i + 1. On the right-hand side, the alternating sum
E s(w)q-(m8+p,2wp)
is by Weyl's identity (Example 9) equal to the product
n (q-(m9+p,a) _q(m8+p,a»
aER+

5. SKEW SCHUR FUNCTIONS
85
and therefore the right-hand side of (3) is equal to
q(2m9+2pta} - 1
n+ q(2p,a) -1 ·
aER
In this product the positive roots Vi - vj(i <j) make no contribution, because they
are orthogonal to 8 = t E Vi. Hence we obtain the identity
(4)
n qm+2i-l_1
r s\(q2n-l,q2n-3,...,q) = n
/..J 1\ q2i-l - 1
Ac(m") i-I
n
1 <i<j<n
q2(m +i +j-l) - 1
q2(i+j-l) - 1 .
The left-hand side of (4) is the generating function for column-strict plane
partitions with odd parts  2n - 1, and with at most m columns and at most n
roWS; or equivalently (Example 15) it is the generating function for symmetrical
plane partitions 11' whose diagrams are contained in the box B = B(n,ntm) =
{(i,j, k): 1  i, j  n, 1  k  m}.
The right-hand side of (4) can be rewritten in a form analogous to that of
Example 13, formula (2), as follows. Let G2 be the group of two elements
consisting of the identity and the mapping (i, j, k) .-+ (j, i, k), so that the box B is
stable under G2. For each orbit T/ of G2 in B let IT/I (= 1 or 2) be the number of
elements of T/, and let
ht( T/) = E ht(y)
yE 11
where ht(i,j, k) = i + j + k - 2 as in Example 13. Then the generating function for
symmetrical plane partitions 'TT c B is
(5)
n
."eB/G2
1 - qht(1J)+ 11J1
1 - q ht('1) .
18. Let G'j be the group of three elements generate'd by (i, j, k) ...... (j, k, i) and let
Cn be the cube {(i,j, k): 1  i,j, k  n}. The formula (5) of Example 17 suggests the
following conjecture: the generating function for cyclically symmetric plane parti-
tions 'TT (i.e. those whose diagrams are stable under G3) contained in the cube Cn
should be
(6)
n
7Je clI/G)
1 - qhl('1)+I'11
1 - q ht(''1) ·
This conjecture has since been proved by Mills, Robbins, and Rumsey [MIl].
Next, let G6 be the group of all permutations of (i, j, k), and call a plane
partition completely symmetric if its diagram is stable under G6. The obvious
analogue of (5) and (6) for G6 is trivially false, because the rational function
1 - qht('1)+I'11
1 - qht(l1)
n
7JE CII/G6

86
I SYMMETRIC FUNCTIONS
is not a polynomial if n  3. However, it seems likelyt that the number of
completely symmetric plane partitions with all parts  n is correctly given by
setting q == 1 in this expression, Le. it is equal to
n
.,.,e C,,/G6
ht( 11) + 1111
ht( 11)
19. With the notation of Example 16, the set of vectors
RI == { f: 2v; (1  i  n), :t vi:t Vj (1  i <j  n)}
is a root system of type CII, for which
Rt.. {2v; (1  i  n), vi:i: vj (1  i <j  n)}
is a system of positive roots, so that
PI =nv1 + (n -1)v2+ ... +v,.
is half the sum of the positive roots. The Weyl group is the same group W as in
Example 16.
We shall take f PI in formula (3) of Example 16, so that e-vJ is replaced by
q"-i+l. As in Example 17, by virtue of Weyl's identity we have
E s(w)q-(m8+p,wP1) == n (q-(m8+p,a/2) - q(m8+p,a/2})
aeRt
and therefore the right-hand side of (3) is equal to
q<m8+p,a} -1
JJt q(p,a)-l ·
Again the roots Vi - vj (i <j) make no contribution to this product, and hence we
obtain
(7)
E SA(Q",...,q)== n
AC:(m") l<;<j<n
qm+i+j-l - 1
qi+j-l - 1 ·
The left-hand side of (7) is the generating function for column-strict plane
partitions with largest part  n and at most m columns, and the right-hand side
can be written in terms of the height function introduced in Example 13, namely as
(8)
1 - qhl(Y)+ 1
Xl 1 - qht(y)
where D is the prism {(i, j, k): 1  i <:j  n, 1  k  m}.
t This conjecture has recently been proved by J. Stembridge [S30].

5. SKEW SCHUR FUNCTIONS
87
20. (a) Let A = (aij) and B = (bij) be n X n matrices such that det A = 1. Let Cij
be the determinant of the matrix obtained from A by replacing its ith column by
the jth column of B, so that
n
Cij = E aikbkj,
k-l
where aik is the cofactor of aki in A, which since det A = 1 is equal to the (i, k)
element of A -1. It follows that C ij is the (i, j) element of A -1 B, and hence that
det(c;.) == det B.
No let r < n and suppose that for each j > r the jth column of A is equal to
the jth column of B. Then Cij = 8;j whenever j > r, and hence we have
(1)
det(cij)l <.i,j<.' = det B.
(b) Let ,\ = (aI' · · · , a, 1131, . . . , 13,) be a partition of length  n, in Frobenius
notation, and let
A (i ,j) = ( a 1 , .. . , a;, .. . , a, 1 /31'...' Pj, . . . , 13, )
for i, j = 1,2,. . . , r, where the circumflexes indicate that the symbols they cover are
to be omitted. Then the skew diagram [a; Il3j] = A - A(i,j) (which of course
depends on A as well as ai' I3j) is a border strip, and more precisely is that part of
the border of A consisting of the squares (h, k) such that h  i and k  j. With this
notation established, we have
(2)
SA = det(s[a 1 Q ])1 .. .
I} <'Id<"
(Let fk = k - Ak (1  k  n). The sequence  = (fk) is obtained from the sequence
( - aI'...' - a" 1,2,..., n) by deletion of /3, + 1,..., 131 + 1. Hence the correspond-
ing sequence for the partition A(;,j) is obtained from f by deleting - ai and
inserting /3j + 1. It follows that, up to sign, s[ ai' J3j] is equal to the determinant C;j
of the n X n matrix obtained from A = (h_f/+f) by replacing its ith column by the
(j + 1)th column of. the matrix B = (hA/-ij) = (h_ f;+j). Moreover, the sign
involved is ( -l)J3I-i+J, so that we have
(3)
det(s[all /3}]) = ( -1)1131 det(c;j).
On the other hand, by (1) above, det(cij) is equal to the determinant of the matrix
obtained from B by rearranging its columns in the sequence (/31 + 1,..., /3, + 1,
1,2, . . . ,n), and therefore
det(c;j) = ( _1)lfJl det B = ( -1)lfJls,\.
The formula (2) now follows from (3) and (4).)
(4)
21. Let 8, cp be two skew diagrams. Let a be the rightmost square in the top row
of 8, and let b be the leftmost square in the bottom row of cpo Let cpv (resp. cph)

88
I SYMMETRIC FUNCTIONS
denote the diagram obtained from cp by a shift sending b to the square immedi..
ately above a (resp. immediately to the right of a) and let (1 * cp (resp. () * 'P)
. U h
denote the skew diagram (1 U cpv (resp. () U cph).
(a) Show that
(1)
Se sq> = s(J . q> + s(J * q> .
U h
(From (5.12) it follows that
SeSq> = E xTxu
T,U
summed over all pairs of tableaux T, U of respective shapes (1, cpo Split up the set
of these pairs (T, U) into two subsets according as T(a)  U(b) or T(a) > U(b),
where T(a) is the symbol occupying the square a in T, and likewise for U(b ).)
(b) In view of (5.7), the repeated use of (1) enables us to express any skew Schur
function as a sum of skew Schur functions corresponding to connected diagrams. In
particular, we have
(2)
hi = E S(J
9
summed over the 2n-1 border strips (or ribbons) of length n. Taking the coeffi-
cient of Xl ... xn in both sides of (2) (or, equivalently, the scalar product with h)
we see that this decomposition describes the partition of the symmetric group Sn
into the subsets of permutations having a given set of descents.
(c) Let A = (a I f3) = (al'. .. , a, 1131, . . . , 13,) be a partition in Frobenius notation,
and let S(o 113)' S[a 1 f3] denote the r X r matrices
S(O 113) = (s(a;l f3j»'
( i+j )
S[all3]= (-1) s[a;l/3j]
in the notation introduced in Example 20. Show that
(3)
S(O 113) = Ha S[o 1 fJ]EI3
where Ha = (ho/- OJ)' Ef3 = (efJj- 13). (Use (1) above.)
By taking determinants in (3) and using 3, Example 9, we obtain another proof
of the formula (2) of Example 20.
22. (a) Let A = (at,..., a, I Ph...' 13,), J.-L = (1'1'...' 1's /61'...' 6s) be two partitions
in Frobenius notation. Define matrices
S(o 113) = (s(aj Il3j)l;,j"",
Ha 'Y = (ha.- 1'>1' 1. ,
I 'J';;'''', .;;}.;; s
Ea e = (en - & )1' 1. .
IJ, IJj j <,"s, <J'"

5. SKEW SCHUR FUNCTIONS
89
Then we have
(1)
s (S(QIP)
S Alp. = ( - 1) det E
p,e
HQIy)
o ·
When J.L = 0 (so that s = 0) this formula reduces to that of 3, Example 9.
(Choose n  max(l(A), l( J.L)) and let 1=( - at, . . ., - ar' 1,2, . . . ,n), J =
( _ ;'1'...' - 'Ys' 1,2,..., n). Then the sequence (i - Ai)l.; i.; n is obtained from the
equence I by deleting the terms f3i + 1 (1  i  r), and likewise the sequence
(j _ p.)t ." j "n is obtain.ed fro J by del eting e terms &J + 1 (1 :E; j :E; s). H ece
the matrIX (hAj- Jl- -;+j) IS obtained from the matrIx M = (hj-)(i,j)E Ix] by deletIng
the roWS with indices r + f3i + 1 (1  i :E;;; r) and the columns with indices s + 6j + 1
(1 j  s). Hence if U is the matrix of n + r rows and r columns which has 1's in
the positions (r + f3i + 1, i) (1  i  r) and zeros elsewhere, and if V is the matrix of
s roWS and n + s columns that has 1's in the positions (j, s + 6j + 1) (1 j  s) and
zeros elsewhere, we shall have
k
SAlp. == det(h"j_ p.j-i+ j) = (-1) det B
where B is the matrix (of N = n + r + S rows and columns) ( .l; ), and
k = ,2 + rs + s 2 + I f31 + 161.
Now let A be the N X N matrix whose first r + s rows are those of the unit
matrix IN' and whose last n rows coincide with the last n rows of the matrix B.
From the definition of M it follows that det A = 1. We now apply the method of
Example 20(a) (but with rows replacing columns): if C ij denotes the determinant of
the matrix obtained by replacing the jth row of A by the ith row of B, for
1  i,j  r + s, then det B = det(cij). By calculating the cij and paying careful
attention to the signs involved, we obtain the desired formula (1).)
(b) Let Her = (hQj_ Qj)' E{3 = (ef3j- p) as in Example 21(c). From (1) it follows that
_ _ s (H;lS(QI{3)Ei1 H;lHQ,y)
(2) S Alp. - ( 1) det _ 1 ·
Ep, eEf3 0
From Example 21(c) we have
H-1S E-l ( 1)i+i )
Q (er I P) {3 = - s[Q;I Pj] l';i,j<r.
Consider the matrix H;lHQ,y. Its (i,j) element, say Cij' is equal to the determi-
nant of the matrix obtained from HQ by replacing its ith column by the jth column
of Ha, y. There are three possibilities:
(0 if 'Yj = ak for some k, then cij = 8ik;
(ij) if Yj is not equal to any ak, and ai < Yj' then cij = 0;
(iii) if Yj is not equal to any ak' and aj  Yj' then cij is (up to sign) the Schur
function corresponding to the border strip consisting of those squares (a, b) in
the border of A such that Yj < b - a  ai.

90
I SYMMETRIC FUNCTIONS
Similar considerations apply to the matrix E/3, & E/3-1. Hence (2) leads to an
expression for the skew Schur function S A/p, as a determinant of Schur functions of
border strips. We shall leave the precise formulation as an exercise to the reader.
23. As in 3, Example 21, let SA(X/Y) denote the Schur functions associated with
the rational function
m n
Ex/y(t) == E e,(x/y)t' == n (1 + Xii) n (1 + yjt) -1.
r>O ;-1 j-I
If ClJy denotes the involution CIJ acting on symmetric functions of the y variables,
then SA(X/Y) is obtained from (JJySA(X, y) by replacing each Yj by -Yj. Hence by
(5.9) and (5.6) we have
(1)
SA(X/Y) = E( -l)IA-p,lsp,(x)s,\I/P,'(Y)
p.
and
(2)
SA(Y Ix) == ( -l)IAlsA,(x/y).
(a) Let fm,n denote the set of lattice points (i,j) E Z2 such that i,j  1 and either
i  m or j  n. By (5.8) and (1) above, SA (X /Y) =F 0 if and only if there is a partition
/-L c A such that I( J.L) < m and Ai - /-Li  n for all i  1, and this condition is
equivalent to A c r m, n.
(b) A bitableau T of type (m, n) and shape A - /-L (where A::> /-L) is a sequence of
partitions
(3)
J.L = A (0) c A (1) c ... c A (m + n) = A
such that the skew diagram 8(0 =- A(i) - A(i-l) is a horizontal strip for 1  i  m
and a vertical strip for m + 1  i  m + n. Graphica]]y, T may be described by
filling each square of lJ(i) with the symbol i, for 1  i  m, and each square of
o(m +j) with the symbol j', for 1 j  n. Thus the symbols follow the order
1 < 2 < ... < m < l' < 2' < ... < n', and the conditions on Tare
(0 in each row (resp. column) of T the symbols increase in the weak sense from
left to right (resp. from top to bottom);
(ij) there is at most one marked symbol j' in each row, and at most one unmarked
symbol i in each column.
With each such bitableau T we associate a monomial (x/y)T obtained by
replacing each symbol i (resp. j') by Xi (resp. -Yj) and then forming the product
of all these x's and -y's. It follows then from (2) above and (5.12) that
(4)
l'
SA/P,(X/Y) == E (x/y)
T
summed over all bitableaux T of type (m, n) and shape A - /-L.
(c) In (b) above the symbols followed the order 1 < 2 < ... < m < l' < 2' < ... < n',
Show that (4) will remain true if this order is replaced by any other total ordering

5. SKEW SCHUR FUNCTIONS
91
cof.the set {I, 2, .. . , , I' ,2' , . . . , n'}. (In A-ring terms, this is simply th observation
tha,t the summands In Xl + ... +xm - Yl - ... -Yn may be permuted In any way.)
24." (a) Let p be an integer  2 and let 'Pp: A -+ A be the ring homomorphism
defined as follows: 'Pp(hn) == hn/ p if P divides n, and 'Pp(hn) = 0 otherwise. (The
effect of 'Pp is to replace each variable Xi by its set of pth roots.)
-If ,\ is a partition of length  n, let A*  (A(r»o < r < p-l denote its p-quotient,
as in iI, Example 8. If J.L C A let
p-l
SA*/p,. = Il SA(r)/p.(r).
r-O
ShoW that 'Pp(SA/p,) = :tSA*/p,. if A, /L have the same p-core, and that 'PP(SA/P,) = 0
_ otherwise. (Even when A and /L have the same p-core, we do not necessarily have
p.(r) C A(r), so that SA*/p,. may be zero also in this case.)
(Let = A + 8, 11 = /L + 8, where 8:= (n - 1, n - 2,...,1,0), so that SA/p, =
det(hti- 111) by (5.4). For each , = 0,1,..., P - 1 let A, (resp. Br) be the set of
indices i between 1 and n such that i == ,(mod p) (resp. l1i == ,(mod p ». Then
cp (SA/P,) is equal to :t det M, where M is the matrix which is the diagonal sum of
die (not necessarily square) matrices M, = (h(fl- "'1)/ p), (i,j) EAr X Br. It follows
that 'Pp(SA/P,) = 0 unless IArl:= IBrl for each " i.e. unless A and /L have the same
p-core. If this condition is satisfied, then 'PP(SA/P,) = :t n det Mr:= :tSA./p._.)
(b) Let w = e2Tri/ p. Deduce from (a) that sA/p.(I, w,..., wp-1) is zero unless
0) A and /L have the same p-core;
(ij) A(r) J /L(r) for, = 0,1, ... ,p - 1;
(iii) each A(r) - /L(r) is a horizontal strip.
Conditions (0 and (ij) together mean that A -  is a union of border strips of
length p. We shall write A =p J.L to mean that conditions (i)-Gij) are satisfied.
If A = p /L then sA/p.(l, w,. . ., wp-1) = :t 1. The sign, up( A//L) say, may be
determined as follows. Let '; E [0, P - 1] be the remainder when ;:= Ai + n - i is
divided by p, and rearrange the sequence  = (1"." n) so that i precedes j if
and -only if either '; < 'i' or '; ='j and i > j' Let &p(A) be the sign of the
corresponding permutation, and define Bp( /-L) 1ikewise. Then O'p(A//L) =
Bp()..)Bp( /L).
25. We shall identify the elements of the ring A  A with the functions of two
sets of variables (x) and (y), symmetric in each set separately: thus f 0 g corre-
sponds to f(x)g(y). Clearly, for any f E A the function [(x, y) lies in A 0 A, and
we define the diagonal map (or comultiplication) 11: A -+ A  A by
(f)(x,y) =f(x,y).
Also the counit s: A -+ Z is the linear mapping which vanishes on An for each
n  1 and is such that B(l) = 1 (so that s(f) is the constant term of f).
It is easy to verify that this comultiplication and counit endows A with the
structure of a cocommutative Hopf algebra over Z (for the definition of such an
object see, for example, [Z2]; the most important axiom is that 11 is a ring
homomorphism).

92
I SYMMETRIC FUNCTIONS
(a) The definitions at once imply that
hn = E hj, fl} hq,
p+q-n
en = E ep fl} eq ,
p+q=n
 Pn = Pn fl} 1 + 1 fl} Pn
(n  1).
This last relation signifies that the Pn are primitive elements of A.
(b) From (5.9) we have
SA = E SA/p.  sp.
p.
for any partition A. Hence, with the notation of Example 3,
I = E sp..l. 1  sp.
p.
for any 1 E A, and more generally
/= E u:/ u
p.
whenever (uJA)' (uJL) are dual bases of A.
(c) The ring A fl} A carries a scalar product, defined by
(11 fl}gl,/2 fl}g2) = (/1,/2)(g1,g2)
for f17/2' g1' g2 E A. With respect to this scalar product, the diagonal map d: A-+
A 0 A is the adjoint of the multiplication map A fl} A  A (i.e. I fl) g  Ig). In other
words,
(/, g fl) h) = (I, gh)
for all I, g, h E A. (By linearity it is enough to check that (ds,\, sp.  SJI) = (SA' SJ.LSJI)'
which follows from (b) above together with (5.1) and (4.8).) Also the counit
6: A  Z is the adjoint of the natural embedding e: Z  A with respect to the
scalar product <m, n) = mn on Z.
These properties signify that the Hopf algebra A is self-dual.
(d) It follows easily from (c) that if I E A and I = E ai  bi then
f.l.(gh) = E a/(g )bi.l.(h)
i
for all g, h EA. In particular,
s,\.l.(gh) = E c;JlsJ.(g )sJI.l.(h).
p. II
(e) From (c) it also follows that an element p E An (n  1) is primitive (i.e.
 P = P  1 + 1 fl) p) if and only if P is orthogonal to all products Ig, where I, g E A
are homogeneous of positive degree. In particular, (p, h,\) = 0 for all partitions A

5. SKEW SCHUR FUNCTIONS
93
of n except A = (n). Since (h») and (mA) are dual bases of A, it follows that p is a
scalar multiple of Pn. Hence the Pn (n  1) are a Z-basis of the space of primitive
elements of A.
26. The identities (4.3) and (4.3') can be generalized as follows: for any two
partitions '\, J.L we have
(1) E Sp/A (x)sp/p.(Y) = f1 (1 - XiYj) -1 E SP./T(X)SA/T(Y);
p i,j T
(2)
E Sp/A'(X)Sp'/p.(Y) = n (1 + x;Yj) E SP.'/T(X)SA/T'(Y).
p i,j".
(By applying the involution (J) to the symmetric functions of the x variables we see
that (2) follows from (1), so it is enough to prove (1). Let P(x,Y) = ni,j(l-x;Yj)-l
and let (x), (y), (z), and (u) be four sets of independent variables. We shall
decompose the product
P =P(x, y)P(x, u)P(z, y)P(z, u)
in two different ways.
Firstly, from (4.3) and (5.9) we have
(a) P = E sp(x, z)sp(y, u) = E Sp/A(X)Sp/P.(Y)SA(Z)SIA-(u).
p p,A,
On the other hand, using (4.3) and (5.9) we see that
(b)
P =P(x,y) E su(x)su(u)sp(y)sp(z)S".(Z)ST(U)
U, JI,1"
=P(X,y) E c:,su(x)c;'Sp(Y)SA(Z)SP.(u)
u,JI,T,A,p.
= P(x, y) E SJL/T(X)SA/T(Y )SA (z,)slA-(u).
'T,A,p.
Now compare the coefficients of SA(Z)SIA-(u) in (a) and (b).)
In view of Example 25 we may say that (1) is obtained from (4.3) by applying the
diagonal map .
27. (a) By applying the same arguments as in Example 26 to the identities of
Examples 4 and 5 we obtain
(3)
E Sp/A = f1 (1-x;)-1 f1 (l-X;Xj)-l E SA/T'
P i i <j 1"
(4)
ESp/A = f1 (1 - X;Xj) -1 E SA/T'
P even i <.j T even
(5)
E Sp/A = f1 (1 - x;xj) -} E SA/T.
p' even i <j If' even

94
I SYMMETRIC FUNCTIONS
(b) Likewise, from the identities of Example 9 we obtain
(6) E S",/A = n (1.+ xixj) E SA'/p ,
w i<j P
(7) ESp/A" n (1 + XiXj) E SA'/," ,
P i<j '"
(8) E aUsU/A(x) = n (I-x) n (I-xixj) E ausA,/u( -x).
u i<j i <j u
Here, as in Example 9, 'IT runs through partitions of the form (ai - 1,..., ap -11
at,. . ., ap) in Frobenius notation; p through partitions of the form (a],..., a I
al - 1,..., ap - 1); u through self-conjugate partitions (al".., ap I ah..., a:);
and finally au = ( _I)(lul+p)/2.
28. Show that
 II n( .-In( . )-1)
(a) i..J q PSp/A(X)Sp/A(Y)= . (I-q') . 1-q'XjYk ,
p,A ,;>1 j,k
(b) E qlplsp'/A(X)Sp/A'(Y) = D (1- qi) -1 n (1 + qiXjYk)),
p,A £;>1 J,k
( 1 + 2i-lX. )
E ql pi Sp/A (x )Sp'/A (Y) = n (1 + q2i-l) n q 2i JYk .
p,A i1 j,k I-q XjYk
(Let F(x, y) denote the left-hand side of (a). Then it follows from equation (1) in
Example 26 that
(c)
F (x, Y ) = n (1 - qx j Y k) -1 . F (qx , Y)
j,k
and hence that
F(x,y)= n n(1-qiXjYk)-1.F(O,y).
;;> 1 j,k
But
F(O, Y) = E qIPISp/A(O)Sp/A(Y) = E qlpi
p,A p
= n (1 - qi) -1
i;> 1
and (a) is proved.
The identity (b) follows from (a) by applying the involution w to the symmetric
functions of the x's.
Finally, let G(x, y) denote the left-hand side of (c). Again from Example 26 we
obtain
G( x , y) = n (1 + qx j Y k) . wyG (qx , Y)
j,k

5. SKEW SCHUR FUNCTIONS
95
where wy is the involution w acting on symmetric functions in the Y variables, and
hence
_ 1 +qXjYk 2
G(x,y) - TI 2 .G(q x,y).
j,k 1-q XjYk
The proof may now be completed as in (a) above.)
29. With the notation of Example 3, let
EJ.(t) = E enJ. tn,
nO
H.1.(t)= E h;tn.
nO
Both E.l(t) and Ht) map A into A[t], and HJ.(t)w = wEJ.(t).
(a) Show that Et) and HJ.(t) are ring homomorphisms. (Use Example 25(a), (d).)
(b) Since h; (hn) = hn-m (with the usual convention that h, = 0 if r < 0) it follows
that
(1)
H.J.(t)(H(u» = E hn_mtmun
m,n;>O
= (1 - tu) -} H(u)
and hence, using (a) above, that
HJ.(t)(H(u)!) = (1 - tu) -1 H(u)Hl.(t)f
for all f E A, so that
(2)
Hl.(t) 0 H(u) = (1 - tu) -} H(u) 0 Hl.(t)
where H(u) is regarded as a multiplication operator.
Show likewise that
(3)
(4)
(5)
HJ.(t) 0 E(u) = (1 + tu)E(u) 0 HJ.(t),
EJ.(t) 0 E(u) = (1 - tu) -1 E(u) 0 EJ.(t),
EJ.(t) 0 H(u) = (1 + tu)H(u) 0 EJ.(t).
(c) For all f E A we have
HJ.( t) f( Xl' X 2' . . .) = f( t , Xl' X 2' · · · ).
(By (a) above, it is enough to prove this when f = hn, in which case it follows from
(1).)

96
I SYMMETR.IC FUNCTIONS
(d) Now define
B(t) = E Bntn = H(t) 0 £ol.( -t-1),
neZ
Bol.(t) = E Bnol.tn = E( -t-1)0 Hol.(t),
neZ
so that
Bn = E ( -l)rhn+r 0 e;- ,
r;> 0
B nol. = E ( - 1) r e r 0 h; + r .
r;> 0
Each Bn is a linear endomorphism of A, and Bn.l. is the adjoint of Bn with respect
to the usual scalar product on A. Also Bnol. = ( -l)nwBn w for all n E Z.
Since
H(t) = exp( E  pntn),
nl n
( (_l)n-l )
E(t) = exp E Pntn ,
n;>l n
we have
(6)
B(t)=exp( E '::Pntn)oexp( E -lpnl.rn).
n;>l n nl n
More generally, if t1,..., tn are independent variables, let
B (t 1 , .. . , t n) = H (t 1 ) . . . H (t n ) £.l.( - t 11 ) .. . Eol.( - t;- 1 ) .
Deduce from (5) that
B(tt)...B(tn) = fl(l-t;-ltj).B(tt,...,tn)
i<j
and hence that
(7)
B ( t 1 ) . . . B (t n ) 1 = fl (1 - t i- 1 t j ) H ( t 1 ) . . . H ( t n ) .
i<j
Let A be a partition of length  n. By equating the coefficients of t A on either
side of (7), show that
BA ... B). (1) = fl (1 - Rij)h).
1 " ..
l<}
and hence by (3.4") that
SA = B). ... B). (1).
1 "

5. SKEW SCHUR FUNCTIONS
97
(e) Let q;(tu) = En E Z (nun. Show that
H(t )E( - u -1) cp(tu) = cp(tu)
(8)
and deduce from (2), (4), and (8) that
B(t)BJ.(u) + tuBJ.(u)B(t) = cp(tu),
or equivalently
B,Bs.l. + B/-l Br-l = Srs
for all r, s E Z.
30. With the notation of 3, Example 21 define, for any two partitions A and 
- - d t( p..-j+ l(h »)
s AI,." - e cp '} Aj - IA-j - i + j 1 <, i I j < m
where m  max(l(A), I( J.L». (Thus sAID = SA as defined loco cit.). Show that sAIJL = 0
unless A:J J.L.
(a) Assume then that A:J J.L, and let 9 = A - J.L. The function s(} = sAIJL depends not
only on the skew shape 9, but also on its location in the lattice plane. For each
(p, q) E Z 2 let "PI q denote the translation (i, j) .--+ (i + p, j + q). Show that
i"'o,tUJ) = cps(J,
· -1-
STl,O(tn = cp S(}
and hence that
. - q-p.
ST (8) - cp So .
p,q
In particular, $8 is invariant under diagonal translation (p = q).
A
(b) Let () be the result of rotating () through 1800 about a point on the main
diagonal. Show that s8 = 8S(}, where 8 is the involution defined by cpshr.--+ cpl-r-sh,.
(c) Let 9' = A' - J.L' be the reflection of () in the main diagonal. Show that
s(}' = lAJ$(J, where w is the involution (3, Example 21(c» defined by w( cpshr) = cp -ser
for all '., s. Equivalently,
d t ( - 1£" + j - 1 ( ) )
s AI 1£ = e cp J e A'j - 111- i + j ·
(The proof is the same as that of (2.9), using the relation E = H-1 of 3, Example
21(a).)
(d) Extend the results of Examples 20, 21, and 22 to the present situation.
(Example 20 goes throJlgh unchanged; so does Example 21(a), provided that the
contents (1) of the squares a, b are related by c(b) = c(a) + 1. In Example 21(c),
the matrices H a and EI3 should now read
H = ( cpaj+lh )
a aj- aj ,
E (-13--1 )
13 = cp I e13j-13j.

98
I SYMMETRIC FUNCTIONS
In Example 22(a), the matrices Ha,,,, and Ep, & should now read
H - ( "'J + 1 h .)
a,..,- cP cxl-"'j'
E - ( - el-l ) )
fJ,e- cP eI3J-et.
31. As usual let H(t) = En  0 hntn, and assume that the h" are real numbers (with
ho = 1). The power series H(t) is called a P-series if SA = det(hA;_i+j)  0 for all
partitions A. In particular, the coefficients h" are  o.
(a) If H(t) is a P-series, so also is H( - t) -1.
(b) The product of P-series is a P-series. (Use (5.9) and the fact that the
coefficients C;II are  0.)
(c) If H(t) is a P-series and hn = 0 for some n, then H(t) is a polynomial of degree
< n. (For hn+l <; h1h1l since s(n,l)  0, and hence hn+l == 0.)
(d) Every P-series has a positive radius of convergence. (We may assume that H(t)
is not a polynomial, hence hn > 0 for all n  1, by (c) above. Since S(1I2)  0 we have
hn+l/hn<:hn/hn-l' and hence the sequence of positive real numbers hn+l/hll
converges.) .
(e) Let En  1 xn be a convergent series of positive real numbers. Then H(t)::
O(1-xnt)-1 is a P-series (by (5.12».
(f) If a > 0, eat is a P-series. (For SA = aIA1/h(A) > 0, by 3, Example 5.)
From (a), (b), (e), and (0 it fol1ows that any H(t) of the form
co 1+xnt
H(t) = eat Il ,
n - 1 1 - Ynt
where a > 0 and Ex n' Ey n are convergent series of positive terms, is a P-series.
Conversely, every P-series is of this form (but this is harder to prove: see [T3]).
Notes and references
Example 2. The fact that the number of standard tableaux of shape A is
equal to n!/h(A) is due to Frame, Robinson, and Thrall [F8]. For a purely
combinatorial proof, see [F9], and for a probabilistic proof see [G15] or
[ S2].
Example 3. The operators enJ., h; were introduced by Hammond [H4],
and the S A1. by Foulkes [F5], in both cases defined as differential operators.
See also Foulkes [F7].
Example 4. This identity is usually ascribed to Littlewood [L9], p. 238;
however, in an equivalent form it was stated by Schur in 1918 (Ges.
Abhandlungen, vol. 3, p. 456). Bender and Knuth found an elegant combi-
natorial proof (reproduced in [S23], p. 177), using the properties of Knuth's
correspondence.
Examples 5 and 9. These identities are all due to Littlewood [L9]. The
observation that the identities of Example 9 follow naturally from Weyl's
identity for the classical root systems is, I believe, new.

6. TRANSITION MATRICES
99
Examples 13, 14, and 15. Plane partitions were first investigated by
MacMahon [M9], and the generating functions (1), (3), and (4) of Example
13 are due originally to him, but proved differently. The application of
Schur functions to these problems is due to Stanley [S23], who gives more
details and references.
Examples 16 and 17. MacMahon [M9] conjectured the generating func-
tion (Example 17, (4)) for symmetrical plane partitions, but was unable to
prove it. It remained a conjecture until proved by Andrews [A4].
Example 18. For an account of recent developments in the study of
generating functions of plane partitions satisfying prescribed symmetry
conditions, see Stembridge [S31] and the references given there.
Example 19. The generating function (7) was established by Gordon (see
Stanley [S23], p. 265) who did not publish his proof. It too was proved by
Andrews [A4]. .
Examples 20, 21, and 22. These results are due to A. Lascoux and P.
Pragacz [L2], [L3]. Examples 21(a) and (b) were contributed by A. Zelevin-
sky to the Russian translation of the first edition of this book.
Examples 25, 26, and 27. These examples were also contributed by A.
Zelevinsky. The Hopf algebra structure on A (Example 25) is discussed in
[G6], and it is shown in [Z2] that the whole theory of symmetric functions
can be systematically derived from this structure. The results of Examples
26 and 27 appear to have been discovered independently by various people
(Lascoux, Towber, Stanley, Zelevinsky).
Example 29. The operators Bn were introduced by J. N. Bernstein (see
[Z2], p. 69).
6. Transition matrices
In this section we shall be dealing with matrices whose rows and columns
are indexed by the partitions of a positive integer n. We shall regard the
partitions of n as arranged in reverse lexicographical order (1), so that (n)
comes first and (In) comes last. It follows from (1.10) that A precedes IL if
A) JL (but not conversely). A matrix (MAJ..) indexed by the partitions of n
will be said to be strictly upper triangular if MAJ.t = 0 unless A  IL, and
strictly upper unitriangular if also MAlt = 1 for all A. Likewise we define
strictly lower triangular and strictly lower unitriangular.
Let Un (resp. U) denote the set of strictly upper (resp. lower) uni-
triangular matrices with integer entries, indexed by the partitions of n.
(6.1) u", U are groups (with respect to matrix multiplication).
Proof. Suppose M, N E Un. Then (MN)ltJ.t = Lv MAVJl. is zero unless there
exists a partition v such that A  v  J.L, i.e. unless A  J.L. For the same
reason, (MN)A).. = M)..)..N)..).. = 1. Hence MN E Un'

100
I SYMMETRIC FUNCTIONS
Now let ME Un' The set of equations
(1) E M).p,xp, = YA
J.I.
is equivalent to
(2)
E (M-1 ) Ap, Y p, = x A ·
For a fixed '\, the equations (1) for Yv where v  '\, involve only the xIJ. for
J..t  J), hence for J.L  A. Hence the same is true of the equations (2), and
therefore (M-1 )Ap, = 0 unless J.L  A. It follows that M-1 E Un. I
Let J denote the transposition matrix:
{I if A' = J..t,
lAp, = 0 otherwise.
(6.2) M is strictly upper triangular (resp. unitriangular) if and only if JM] is
strictly lower triangular (resp. unitriangular).
Proof. If N = JMJ, we have NAp, = MA,p," By (1.11), A'  J..t' if and only if
J.L  A, whence the result. I
If (uA), (v).) are any two Q-bases of An, each indexed by the partitions of
n, we denote by M(u, v) the matrix (MAp,) of coefficients in the equations
uA = E MAp,vp,;
J.I.
M(u, v) is called the transition matrix from the basis (u).) to the basis (VA).
It is a non-singular matrix of rational numbers.
(6.3) Let (u).), (VA)' (w,\) be Q-bases of An. Then
(1) M(u,v)M(v,w)=M(u,w),
(2) M(v,u)=M(u,V)-l.
Let (u), (v) be the bases dual to (u,\), (v,\) respectively (with respect to the
scalar product of 4). Then
(3) M(u',v') =M(v,u)' =M(u,v)*
(where M' denotes the transpose and M* the transposed inverse of a
matrix M).
(4) M(wu,wv)=M(u,v)
where w: A --+ A is the involution defined in 2.
All of these assertions are obvious.
Consider now the five Z-bases of An defined in 2 and 3: (eA), (f,\),

6. TRANSITION MATRICES
101
(h ), (mA), (SA). We shall show that all the transition matrices relating pairs
o/these bases can be expressed in terms of the matrix
K=M(s,m)
and the transposition matrix J.
Since (mA) and (hA) are dual bases, and the basis (SA) is self-dual (4.8),
we have
M(s, h) = K*
by (6.3X3). If we now apply the involution wand obselVe that
M(_ws,s) =J,
by virtue of (3.8), we have
M(s, e) = M( ws, h) = M( ws, s)M(s, h) = JK*
by (6.3X1) and (4). Finally, by (6.3X3) again,
M(s,!) =M(s,e)* = (JK*)* =JK.
We can now use (6.3X1) and (2) to complete the following table of
transition matrices, in which the entry in row u and column v is M(u, 1):
Table 1
e h m f s
e 1 K'JK* K'JK K'K K'J
h K' JK* 1 K'K K'JK K'
m K-1JK* K-1K* 1 K-1JK K-1
f K-1K* K-1JK* K- 1 JK 1 K-1J
s JK* K* K JK 1
Some of the transition matrices in Table 1 have combinatorial interpre-
tations. From (5.13) it follows that
(6.4) KAp. is the number of tableaux of shape A and weight IL. I
The numbers KA,.,. are sometimes called Kostka numbers. By (6.4) they
are non-negative. Moreover,
(6.5) The matrix (KA,.,.) is strictly upper unitriangular.
Proof. If T is a tableau of shape A and weight IL, then for each r  1 there
are altogether ILl + . .. + IL, symbols  r in T, which must all be located in
the top r rows of T (because of the condition of strict increase down the
columns of a tableau). Hence ILl + . .. + IL,  Al + . .. + A, for all r  1, i.e.
#J.  A. Hence KA,.,. = 0 unless A  J..L, and for the same reason KAA = 1. I

102
I SYMMETRIC FUNCTIONS
From Table 1 and (6.5) we can read off:
(6.6) (i) M(s, h) and M(h, s) are strictly lower unitriangular.
(ii) M(s, m) and M(m, s) are strictly upper unitriangular.
(iii) M(e, m) = M(h, f) and is symmetric.
(iv) M(e, f) = M(h, m) and is symmetric.
(v) M(e, h) = M(h, e) = M(m, f)' = M(f, m)'.
(vi) M(h, s) = M(s, m)'.
(vii) M(e, s) = M(s, f)'.
In fact (Example 5 below) M(e, h) is strictly upper triangular (and
therefore by (6.6Xv) M(m, f) is strictly lower triangular).
(6.7) (i) M(e, m)AIL = Ev KVAKv'J.L is the number of matrices of zeros and ones
with row sums Ai and column sums J..Lj'
(ii) M(h, m)AJ.L = Ev KVAKvJ.L is the number of matrices of non-negative inte.
gers with row sums Ai and column sums J..Lj'
Proof. (i) Consider the coefficient of a monomial x IL (where J..L is a
partition of n) in eA = eAt eA2 . .. . Each monomial in eA, is of the form
OJ xj'J, where each aij is 0 or 1, and Ej aij = Ai; hence we must have
n XiJ = n xfLJ
..} .}
" J }
so that Ei aij = J..Lj' Hence the matrix (aij) has row sums Ai and column
sums J..L '.
For (ii) the proof is similar: the only difference is that eA is replaced
by hA, and consequently the exponents aij can now be arbitrary integers
 O. I
Remark. From (6.4) and (6.7Xi) it follows that the number of (0, I)-matrices
with row sums Ai and column sums J..Lj is equal to the number of pairs of
tableaux of conjugate shapes and weights A, J..L. In fact one can set up an
explicit one-one correspondence between these two sets of objects (Knuth's
dual correspondence [K12], [S23]).
Likewise, there is an explicit one-one correspondence, also due to
Knuth, between the set of matrices of non-negative integers with row sums
Ai and column sums J..Lj' and pairs of tableaux of the same shape and
weigh ts A, J..L.
We shall now consider transition matrices involving the Q-basis (p»).
For this purpose we introduce the following notation: if A is a partition of
length r, and f is any mapping of the inteIVal [1, r ] of Z into the set N + of
positive integers, let f(A) denote the vector whose ith component is
f(A);= E Aj'
I(j) - i

6. TRANSITION MATRICES
103
for each i  1.
Let L denote the transition matrix M(p, m):
(6.8) PA = E LAp,mp,'
p.
With the notation introduced above we have.
(6.9) LAP, is equal to the number off such that f(A) = IL.
Proof. On multiplying out PA = PA PA ... , we shall obtain a sum of mono-
A.\ . 12. [ ()] +
mials xf(l)x/b.)... , where f is any mapping 1, I A  N . Hence
PA = E Xf(A)
f
from which the result follows, since LAP, is equal to the coefficient of x IL in
PA' I
If IL = f(,\) as above, the parts JLi .of IL are partia sums of the Aj;
equivalently, ,\ is of the form U i> I A(J), where each ,\(1) is a partition of
#-tit We say that A is a refinement of IL, and write A R IL to express this
relationship between A and JL. Clearly, R is a partial order on the set 9"n
of partitions of n, for each integer n  O.
(6.10)
A R J.L  A  J.L.
Proof. Let Ik = f([I, k]) for 1  k  l(A). Since lLi = 1:f(j)-i Aj' it follows
that
'\1 + ... + Ak  E J.Li  ILl + ... + J.Lk ,
ielk
the last inequality because Ik has at most k elements. I
Remark. The converse of (6.10) is false, since for example two distinct
partitions A and I-L of n such that l( A) = l( J.L) are always incomparable for
the relation R' but may well be comparable for .
From (6.9) and (6.10) it follows that LAP, = 0 unless A R IL, hence
unless A  IL. The matrix L is therefore strictly lower triangular.
The transition matrices M(p,e), M(p,f), and M(p,h) may all be
expressed in terms of L:
(6.11) (i) M(p, e) = szL*,
(ii) M(p, f) = sL,
(iii) M(p,h)=zL*,

104
I SYMMETRIC FUNCTIONS
where L* is the transposed inverse of L, and 8 (resp. z) denotes the
diagonal matrix (8A) (resp. (z A)).
Proof. Since the bases dual to (hA) and (PA) are respectively (m») and
(zA" 1 PA)' it follows from (6.3) that
M(p,h) =M(z-lp,m)* = (z-lL)* =zL*.
Hence
M(p,e) =M(wp, we) =M(ep,h) = 8zL*,
and
M(p,f) =M(wp, wf) =M(8p,m) = 8L.
Finally, we have
(6.12)
M(p,s) =M(p,m)M(s,m)-l =LK-1.
We shall see in the next section that M(p, s), restricted to partitions of n,
is the character table of the symmetric group Sn.
Finally, the relations between the six bases e, f, h, m, s, and pare
summarized in the graph below, in which the symbol attached to a directed
edge uv is the transition matrix M(u, v). (In the cases where M(u, v) =
M(v, u), the edge uv carries no arrow.) For the sake of clarity, the
diagonals of the hexagon have been omitted.
s
e
h
K'K
"
K'K
f
m
p

6. TRANSITION MATRICES
105
Examples
1. M(h, m».JJ. is equal to the number of double cosets S).wSI-'- in Sn, where
SA ::;: S).1 X S).2 X ... , SI-'- = SI-'-l X SIJ-2 X... .
2. For certain choices of A or J.L there are closed formulas for the Kostka number
K A 1£ ·
(a) For any partition A of n, K).,(l") is by (6.4) the number of standard tableaux of
shape A, so that by 5, Example 2 we have K>.,(l") = n!/h(A), where h(A) is the
product of the hook-lengths of A.
(b) Let A = (a, 1 b) be a hook. Then for any partition J.L of n = a + b we have
KAJ> = (l( t-t - 1 ). For the tableaux of shape A and weight t-t are determined by
the entries in the first column, which must all be distinct, and the top entry (i.e. the
number in the corner square of A) must be 1.
(c) Suppose that A and J.L are partitions of n such that each part of J.L is  A2. In
any tableau T of shape A and weight J.L, the l's must occupy the first J.Ll squares in
the top row, and hence T is determined by its entries in the remaining rows, which
constitute a tableau of shape v = (AZ' A3,...) and arbitrary weight. Hence it follows
from (5.12) and 3, Example 4 that KAI'- = ( l( t-t! ,- 1 ).
3. The (infinite) matrix K is the diagonal sum of matrices Kn (n  0), where
Kn = (K).I-'-)>', I-'- E !J1',,' We have Ko = Kl = (1), and for n = 2,3,...,6 the matrices Kn
are shown on p. 111.
The matrices Kn and their inverses have the following stability property, when
their rows and columns are ordered in reverse lexicographical order, as in the
examples on p. 111. Let m = Er  n /2 p(r), where p(r) is the number of partitions
of r; then the principal m X m submatrix in the top left-hand corner of Kn (resp.
K; 1) is equal to the corresponding principal subm atrix of K n' (resp. K;,l), for all
n' > n.
This is a consequence of the following fact: if J.Ll' A2, then K).p, == K).+(r), ,..,+(r)
for an r  1. (Each tableau of shape A + (r) = (Al + r, AZ' A3,...) and weight
J.L + (r) = ( J.L 1 + r, ILz, IL3' . . .) can be obtained from one of shape A and weight IL
by moving the top row r squares to the right and inserting r l's in the squares
vacated.)
4. Let Ki; 1) denote the (A, J.L) entry of the inverse Kostka matrix K-1. Thus
Kl; I) is the coefficient of h). in sIJ- = det(hl-'-i-i+j)' and is also the coefficient of 9,..,
in m). expressed as a sum of Schur functions. We have Kl; 1) = 0 unless A  J.L, by
(6.1) and (6.5), and Kt 1) = 1.
(a) Let r, n be positive integers. Then
" K(-I) = (_l)b+r-l( b )
i...J ).1-'- r - 1
I( ).) - r
I).I-n
if J.t is of the form (a,lb), where a + b = n; and is zero othelWise. (This follows
from 4, Example 10.)

106
I SYMMETR-IC FUNCTIONS
(b) Suppose J.L == (a, 1m) is a hook. Then
(-1) _ _ l(A)+l(J.L) (l(A) -I)! ,
K"". - ( 1) TI () Au'
mi A !
i 1
(By expanding the determinant det(h}£;_;+j) along the top row, we obtain
slJ.=haem -ha+1em-l +... +(-1)mha+m
from which it follows that slJ. is the coefficient of tn (where n = a + m == I J.LI) in
(1)
(_I)m E( -t) E h,tr == (_I)mH(t)-l E h,tr.
r'> a
r'>a
Since
H(t)-l = (1 + E hrtr)-)
r'> 1
= E (_l)n( E hrtrf
n,>O r,>1
it follows that (1) is equal to
() l(v)l
(_l)m E hrt' E ( _1)/(1') , hvtlvl,
r'>a II TImi(v).
; 1
the right-hand sum being over all partitions v. Now K1; 1) is equal to the
coefficient of h A in this expression.)
(c) Suppose J.L is a partition of the form (at,..., a" 1m), where r;>.; 1, ai -.ai+ 1 
r - 2 for 1  i  r - 1, and a,. '> r. Then for any partition A of weight m +
a] + ... + ar we have
(-1) _ _ I(A)+/(j.L) (l( A) - r)! , _ .
K"u - (1) TI () det(A..-i+J" l + 1).. .
r- mi A ! I .......,).....r
; '> 1
(By expanding the determinant of the matrix (hJ.L;-i+j) down the first column, we
shall obtain
r
 ( );-1
SIJ. == i..J -1 hai-i+ 1 SJ.L(I) ,
;-1

6. TRANSITION MATRICES
107
where p,(i) = (al + 1,..., ai-I + 1, ai+ I' ..., ar, 1m), From this it follows that
K(-I)- (-1);-IK(-1) .
A,.., -  A - (al-; + 1), 1£(')
where the sum is over integers i == 1,2,..., r such that ai - i + 1 is a part of A, and
A - (OJ - i + 1) denotes the partition obtained by deleting this part. The formula
JI1ay noW be proved by induction on r; the starting point r == 1 of the induction is
the formula of (b) above.)
(d) Let JL be a partition of length 1. By expanding the determinant det(h,ul-i+j)
down the last column, we shall obtain
I
" ( )/-ih
s,.., == i...J -1 I£I+I-isv(i)
i-I
(i) - ( 1 1) Th . · (i) 11
where v - 1-'1'.'.' J-Li-l' J.Li+ 1 - ,..., J.LI - . ese partitIons v are a con-
tained in J-L, and the skew diagram I-' - v(i) is a border strip (or ribbon) of length
p-; + 1- i and height 1 - i which starts at the square (r, 1), i.e. intersects the first
column. Hence if we define a special border strip to be one that intersects the first
column, we have
(2)
K1; 1) == E ( _1)ht(S)
S
summed over all sequences S == (J-L(O), J.L(1),..., J-L(r» of partitions such that r == leA),
0= p.(O) C J-L(1) C ... C J-L(r) = J-L, with each 8j = J.L(i) - J-L<i - 1) a special border strip,
and such that the lengths of the 8; are the parts of A in some order; and finally
ht(S) = Ei-l ht(8;),
(e) The combinatorial formula (2) above may be used to derive closed formulas for
K1- 1) in certain cases, of which the following is a sample:
0) if I-' = (a, 1m) is a hook (as in (b) above), 81 mut be a hook of length Ai  a,
and 82,.." 8, are vertical strips. Hence the formula (2) leads to the result of
(b) above.
(ii) K&) ::; (-I)b if J-L == (a, 1 b), and is zero otherwise.
(Hi) If A == (a, Ib) with a > 1, then Ki; 1) is equal to (-I)a+l(b + 1) if J.L == (la+b»);
to (_1)a-a if J-L = (a,213, 1')') with a + f3  a; and is zero othelWise.
(iv) If A is of the form (rS), then K1; 1) = 0 or :t 1, because there is at most one
choice for S.
5. A domino is a connected horizontal strip, i.e. a set of consecutive squares in the
same row. If A and J-L are partitions of n, a domino tabloid of shape ,\ and type J.L
is a filling of the diagram of A with non-overlapping dominos of lengths J-Ll, J.L2'.':'
dominos of .the same length being regarded as indistinguishable. Let d>..pt denote
the number of domino tabloids of shape A and type J-L. Then we have
M(e,h).1£ =M(h,e)A,.., == Bl£d).1£
where as usual Gpo == (_1)1,..,1-1(1£).

108
I SYMMETRIG FUNCTIONS
(Since
E(t) =H(-t)l = (1- E (_1)i-lhiti)-t
i 1
- E ( E (_1)i-l hiti)'
, 0 i 1
it follows that, for each integer n  0,
ell = E 6o:ha
a
summed over all finite sequences a = (aI' . . . , aT) of positive integers such that
E a; == n, where Ba == (_l)E(aj-l). Hence for any partition A we have
e). = E Bh
/3
summed over all sequences {3 = ( f31,. . ., f3s) of positive integers that can be
partitioned into consecutive blocks with sums A1' A2, . .. . Such sequences are in
one-one correspondence with domino tabloids of shape A.)
It follows that M(e, h)AJ£ == 0 unless J.1. R A, and that M(e, h)AA = BA. Thus the
matrix M(e, h)B is a strictly upper unitriangular matrix of non-negative integers.
6. Since by (6.7Xv) the transition matrix M(f, m) is the transpose of M(e, h), it
follows from Example 5 that the' forgotten' symmetric functions fA are given by
fA = 6A L. dJ£A mIL
J.L
so that BAtA is the generating function for domino tabloids of type A. In particular:
(a) 1(1") == E mJ£ = hn, since for each partition J.1. of n there is just one domino
tabloid of shape  and type (In).
(b) If A == (,1't-,) where ,  2 we have
fA = (_l)r-l E cp.m,.,.
I-L
where cp. = p!, + p!,+ 1 + ... . For a domino tabloid of shape J.L and type (rin -') is
determined by the position of the single domino of length r, which can lie in any
row of IL of length J.1.;  r, and has J.1.i - , + 1 possible posi tions in that row. Hence
c,.,. == EJ£I>'( i -, + 1) = E;>, Ji;.
7. A domino tableau of shape  and type A is a numbering of the squares of the
diagram of JL with positive integers, increasing along each row, and such that for
each i  1 the squares numbered i form a domino (Example 5) of length Ai- In this

6. TRANSITION MATRICES
109
t rl11inology, (6.9) may be restated as follows: the number of domino tableaux of
ape p. and type .\ is equal to L >'1'" (For each such domino tableau determines a
apping I: [1, l( .\)] -+ [1, l( IL)] such that 1(.\) = IL by the rule that I(i) = j if the
quares numbered i lie in the jth row of J.L; and conversely each such mapping
etermines a domino tableau of shape J.L and type A, in which the Ai squares
numbered i lie in row !(i).)
8. The weight of a domino tabloid (Example 5) is defined to be the product of the
lengths of the leftmost dominos in the successive rows. Let wAJ.t denote the sum of
the weights of all domino tabloids of shape A and type J.L. Then we have
(1)
M(p, h)AJ.t = BASJ.tW)..J.t.
(Since
P(t) =H'(t)H(t)-l
= .E (_1)i( E h,t'f( E rh,t,-l)
IO r;.1 r;.1
it follows that, for each integer n ;> 1,
=  ( _ 1) l( a) - 1 h
Pn  a) a
a
summed over all finite sequences a = (at, .. . ,ar) of positive integers such that
E a; = n, where l( a) is the length r of the sequence. Hence for any partition A we
have
-  (_I)I(!3)-I()..) h
PA -  up fJ
13
summed over all sequences 13 = ( 1317 . . . , 13s) of positive integers that can be
partitioned into consecutive blocks (131'...' 13i1)' (Pi1 + 1"." 13i2)".' with sums
At' '\2'." ; where ul3 is the product 131 13i1 + I 13;2+ I ... of the first terms in each
block. As in Example 5, such sequences are in one-one correspondence with
domino tabloids of shape A, and u13 is the weight of the tabloid.)
From (1) and (6.11) it follows that M(p,e).J.t = sJ.tw)..J.t' M(f,p»)..J.t = B).Z;lw).J.t and
M(m,p»)..J.t = S)..BJ.tZ;lwJ.t)... Hence
SA!).. = E z; lW)..J.tpJ.t
IL
is a polynomial in the power-sums with positive rational coefficients. Also sL -lsZ
is a matrix of non-negative integers, with (A, J.L) entry wJ.t)...
9. From (6.11) we have M(h,m)=L'z-lL and M(m,!)=L-1sL. Comparison
with Table 1 shows that the matrices K and L are related by
K'K=L'z-IL,
K-1JK=L -isL.

110
I SYMMETRIC FUNCTIONS
Hence the matrix X = LK-1 = M(p, s) satisfies
XX' =z
,
x-1sx=].
10. For each partition J.L let
Up. = [] m;( J.L)!
i;;. 1
and let U denote the diagonal matrix (u,u). Thenlthe matrix LU-1 is a strictly lower
unitriangular matrix of non-negative integers.
(Let A, J.L be partitions of n and let EAp. denote the set of mappings f: [1, I(A)] -.
[1, l( J.L)] such that f(A) = J.L, so that by (6.9) we have LAp. = IE,\p.I. Also let S:;
nsm.{p.) be the subgroup of St(p.) that fIXes J.L. If fEE,\p. and WESJL it is clear
that wf E EAp.' so that S p. acts faithfully on E,\p.. Hence L,\p. = IE,\p.1 is divisible by
IS P.I = up.. Finally if A = J.L we have Ep.p. = S p., whence Lp.p. = up..)
The matrix Lu-1 is the transition matrix M(p, in) between the power-sum
products and the 'augmented' monomial symmetric functions mJL = up.mp.. We have
m- =  X ""1 X "'"
P. '1 ... 'I
where 1= l( J.L) and the sum is over all sequences (i1,..., it) of pairwise distinct
positive integers. Since Lu -1 is unitriangular, (mp.) is a Z-basis of the subring
Z[ PI' P 2' . . .] 0 fA.
11. Let A = (At,..., An) be a partition of n. Show that
m,\ = E sa
a
summed over all derangements a of A. (Multiply mA(X1,...,xn)=Exa by
as(x1"..' xn).) Hence Ki; 1) = r - s, where r (resp. s) is the number of derange-
ments a of A such that sa = sp. (resp. - Sp.).
Notes and references
The relations between the various transition matrices contained in Table 1
and (6.6) were known to Kostka [K16], who also computed the matrices Kn
and K;1 up to n = 11 [KI7]. See also Foulkes [F7]. The formulas in
Example 4(a), (b), and (c) are taken from [KI7], and those in (d) and (e)
from [EI]. The results in Examples 5-8 are due to Egecioglu and Remmel
[E2]. See also [D6].

6. TRANSITION MATRICES
111
-
2 12
-
2 1 1
-
12 1
-
3 21 13
3 1 1 1
21 1 2
13 1
I""'"
5 41 32 312 221 213 15
-
5 1 1 1 1 1 1 1
41 1 1 2 2 3 4
32 1 1 2 3 5
312 1 1 3 6
221 1 2 5
213 1 4
15 1
4 31 22 212 14
4 1 1 1 1 1
31 1 1 2 3
22 1 1 2
212 1 3
14 1
6 51 42 412 32 321 313 23 2212 214 16
6 1 1 1 1 1 1 1 1 1 1 1
51 1 1 2 1 2 3 .2 3 4 5
42 1 1 1 2 3 3 4 6 9
412 1 0 1 3 1 3 6 10
32 1 1 1 1 2 3 5
321 1 2 2 4 8 16
313 1 0 1 4 10
23 1 1 2 5
2212 1 3 9
214 1 5
16 1
.._,"::'0./;:.
J.;' \\\

112
I SYMMETRIC FUNCTIONS
7. The characters of the symmetric groups
In this section we shall take for granted the elementary facts about
representations and characters of finite groups.
If G is a finite group and f, g are functions on G with values in a
commutative Q-algebra, the scalar product of f and g is defined by
1  -1
(f, g)G = TGi '-- f(x)g(x ).
xEG
If H is a subgroup of G and f is a character of H, the induced character
of G will be denoted by ind(/). If g is a character of G, its restriction to
H will be denoted by res(g).
Each permutation W E Sn factorizes uniquely as a product of disjoint
cycles. If the orders of these cycles are PI' P2" · ., where PI  P2  ...,
then p(w) = (PI' P2'..') is a partition of n called the cycle-type of w. It
determines w up to conjugacy in Sn, and the conjugacy classes of Sn are
indexed in this way by the partitions of n.
We define a mapping I/J: Sn  An as follows:
t/1 (w) = pp(w).
If m, n are positive integers, we may embed Sm X Sn in Sm +n by making
Sm and Sn act on complementary subsets of {1, 2,. .., m + n}. Of course
there are many different ways of doing this, but the resulting subgroups of
Sm+n are all conjugate. Hence if v E Sm and wE Sn' V X W E Sm+n is
well-defined up to conjugacy in Sm+n' with cycle-type p(v X w) = p(v) u
p(w), so that
(7.1)
I/J(v X w) = l/J(v)t/1(w).
Let Rn denote the Z-module generated by the irreducible characters of
Sn, and let
R = €a Rn,
nO
with the understanding that So = {1}, so that RO = Z. The Z-module R has
a ring structure, defined as follows. Let f E Rm, g E Rn, and embed
Sm X Sn in Sm+n. Then fXg is a character of Sm X Sn, and we define
I. g = indmX'Js (Ix g),
m n
which is a character of Sm+n' i.e. an element of Rm+n. Thus we have
defined a bilinear multiplication Rm X Rn  Rm+n, and it is not difficult to
verify that with this multiplication R is a commutative, associative, graded
ring with identity element.

7. THE CHARACTERS OF THE SYMMETRIC GROUPS 113
Moreover, R carries a scalar product: if f, g E R, say f = E fn' g = E g n
with In' gn ERn, we define
(f,g) = E (fn,gn)Sn.
nO
Next we define a Z-linear mapping
ch: R  Ac = A  C
as follows: if fERn, then
1
ch(f)=(f,lf;)sn=, E f(w)lf;(w)
n. weS,.
(since If; ( w ) = .p (w -1 ». If fp is the value of f at elements of cycle type p,
we have
(7.2)
ch(f) = E z;lfppp.
Ipl-n
ch(f) is called the characteristic of f, and ch is the characteristic map. From
(7.2) and (4.7) it follows that, for f and g in Rn,
(ch(f),ch(g» = E z;lfpgp= (f,g)sn
Ipl-n
and hence that ch is an isometry.
The basic fact is now
(7.3) The characteristic map is an isometric isomorphism of R onto A.
Proof Let us first verify that ch is a ring homomorphism. If f E Rm and
gERn, we have
ch(f.g) = (indsm+n (fXg), If;)s
SmXSn m+n
= (f X g, res::, n(.p) )sm XSn
by Frobenius reciprocity,
= (f, If; )sm( g, t/J )Sn = ch(f) · ch(g)
by (7.1).
Next, let 17n be the identity character of Sn. Then
ch(1Jn)= E z;lPp=hn
I pl=:n

114
I SYMMETRIC FUNCTIONS
by (7.2) and (2.14'). If now A = (At, A2,...) is any partition of n, let 11
denote '71'\1. '71A2. ... · Then '71,\ is a character of S n' namely the characte
induced by the identity character of S,\ = S Al X S A2 X..., and we have
ch('71,\) = hA.
Now define, for each partition A of n,
(7.4)
X,\ = det( 1]'\1- i+j)1.. i,j.. n ERn,
i.e. X,\ is a (possibly virtual) character of Sn' and by (3.4) we have
(7.5)
ch ( X ,\) = SA ·
Since ch is an isometry, it follows from (4.8) that ( X'\, X p.) = 8,\p. for any
two partitions A, /L, and hence in particular that the X A are, up to sign,
irreducible characters of S n. Since the number of conjugacy classes in S 11 is
equal to the number of partitions of n, these characters exhaust all the
irreducible characters of S n; hence the X,\ for I AI = n form a basis of RII,
and hence ch is an isomorphism of Rn onto An for each n, hence of R
onto A. I
(7.6) (i) The i"educible characters of Sn are xA(IAI = n) defined by (7.4).
(ii) The degree of X,\ is K,\,(t"), the number of standard tableaux of shape A.
Proof. From the proof of (7.3), we have only to show that X A and not - X 
is an irreducible character; for this purpose it will suffice to show that
X'\(l) > O. Now we have from (7.5) and (7.2)
s,\ = ch( XA) = E Z;1XpApp
P
where X: is the value of X,\ at elements of cycle-type p. Hence
(7.7) Xp'\ = (S,\, pp)
by (4.7), and in particular
X '\(1) = X(") = (S,\, p)
so that
h7 = p = E XA(1)s,\
IAI-n
and therefore X A(1) = M(h, s)(t"),'\ = K,\,(ln) from Table 1.
(7.8) The transition matrix M(p, s) is the character table of Sn' i.e.
Pp = E Xp'\SA.
A
Hence Xp'\ is equal to the coefficient ofxA+8 in aspp.

7. THE CHARACTERS OF THE SYMMETRIC GROUPS 115
This is a restatement of (7.7).
Remark. From Table 1 in 6 we have
hA = SA + E KP.ASJ.t'
p,>A
eA, = SA + E Kp.' A'Sp..
p,<A
Hence
71A=XA+ E Kp.AXIL,
p,>A
BA'=XA+ E Kp.'A'XIL.
p,<>'
These relations give the decomposition of the induced modules HA =
ind"(l) and EA, = ind,<B) respectively. It follows that HA and EA, have a
unique common irreducible component, namely the irreducible Sn-module
with character XA. This obselVation leads to a simple construction of the
irreducible Sn-modules: see Example 15 below.
Let '\, J.L, ." be partitions of n, and let
1
i'p.Av == (x A, XV)Sn = -, E X.\(w) XP.(W)XV(W)
n. wES
,.
which is symmetrical in A, JL, )). Then we have, for two sets of variables
x = (x l' X 2' · . .) and y = (y], Y 2 , .. . )
(7.9)
SA(XY) = E i'p.:sp.(x)sv(y),
p,. v
where (xy) means the set of variables XiYj. (Compare (5.9).)
Proof. For all partitions p we have pp(XY) = pp(x) pp(Y) and hence from
(7.8)
E x AS.\(XY) = E XVSp.(x)sv(y)
>. p" v
so that SA(XY) is the coefficient of X A in the right-hand side.
Let f, g E An, say f = ch(u), g = ch(v) where u, v are class-functions on
Sn. The internal product of f and g is defined to be
f * g = ch(uv)

116
I SYMMETRIC .FUNCTIONS
where uv is the function w t-+ u(w)v(w) on Sn. With respect to this
product, An becomes a commtative and associative ring, with identity
element hn.
It is convenient to extend this product by linearity to the whole of A
A )
and indeed to its completion A (2); if
1= E I(n),
n;>O
g = E g(n)
n;;:...O
with I(n), g(n) E An, we define
I*g= E I(n)* g(n).
nO
A A
For this product, A is a commutative ring with identity element 1 = E h",
and A is a subring (but does not contain this identity element).
If A, JL, )) are partitions of n, we have
(7.10)
SA * sp. = E YASJI
V
so that by (7.9) and the symmetry of the coefficients I'A
(7.11) SA(.1}') = E sP.(X)(SA * Sp.)(y).
JL
Also we have
(7.12) PA * Pp. = 8AjLZAPA
so that the elements Z-;lPA E AnQ are paiIwise orthogonal idempotents, and
their sum over all partitions A of n is by (2.14) the identity element hn of
An.
Finally, for all I, g E A we have
(7.13)
(I, g) = (/* g)(l)
where (I * g )(1) means 1 * g evaluated at (Xl' X2,...) = (1,0,0,...). (By
linearity it is enough to verify (7.13) when 1 = PA and g = Pp.' and in that
case it follows from (7.12) and (4.7), since PA(l) = 1 for all partitions A.)
Examples
1. X(n) = '1n is the trivial character of S n' and X(ln) = Bn = B is the sign character.
(Compare (7.10) with (2.14').)
2. For any partition A of n, XA' = Bn XA. For
XpA' = (SA"Pp) = (s,\, Bppp) = BpX/
since w(SA') = s,\ and w(pp) = Bppp. Hence the involution w on A corresponds to
multiplication by Bn in Rn. Equivalently, ell * f = w(f) for all f E An.

7. THE CHARACTERS OF THE SYMMETRIC GROUPS 117
3 Corresponding to each skew diagram A - IL of weight n, there is a character
X.A/P- of Sn defined by ch( XA//J-) = sA//J-. If I ILl = m we have from (5.1) and (7.3)
( X A//J- , X v ) S II = ( X 1\ X /J- . X v ) S m + II
= (resSm XSIIXA X/J- X X V)
Sm+II' SmxSII
by Frobenius reciprocity, and therefore the restriction of X A to Sm X Sn is
E X p, X X A//J-.
I I" m d f A//J-. I ( n ) K . h b f
The egree 0 X IS equa to sA//J-' el = A//J-, (1")' I.e. to t e num er 0
standard tableaux of shape A - J.L.
4. Let G be a subgroup of S n' and Jet c( G) be the cycle indicator of G (2,
Example 9). Then c(G) = ch( XG), where XG is the character of Sn induced by the
trivial character 1G of G. For ch( XG) = ( Xc' t/J )s" = (Ie, t/J I G)e (by Probenius
reciprocity) = c(G).
If G, H are subgroups of Sn' (c(G), c(H) is the number of (G, H) double
cosets in S n .
5. From 3, Example 11 and (7.8) we obtain the fol1owing combinatorial rule for
. A.
computing Xp.
XpA = L ( _l)ht(S)
S
summed over all sequences of partitions S = (A(O), A(1), . . . , A(m» such that m = l( p),
0= }.,(O) C A(l) C ... c A(m) = A, and such that each A(i) - A(i-l) is a border strip (3,
Example 11) of length Pi' and ht(S) = E i ht(A(i) - A(i-l».
6. The degree fA = X A(l) of X A may also be computed as follows. By (7.8), it is the
coefficient of XA+6 in (Ex)nEwe s e(w)xw6. If we put IL = A + 8 (so that IL; =
"
,\. + n - i, 1  i  n), this coefficient is
I
L e(w )n!/n (ILj - n + w(i))!
weS 1= 1
n
which is the determinant n! det(l/( ILi - n + j)!), hence equal to
n!
- det( ILi( ILi - 1)... ( ILi - n + j + 1))
IL!
n! . n!
= -, det( IL?-J) = -, L\( ILl'...' ILn)
IL . IL .
where JL! = Oi JLi! and L\( J.Lh...' ILn) = 0; < j( ILi - ILj).
7. Let P = (r, In-r), so that XpA is the value of the character X A of Sn at an r-cycle
(1  r  n). By (7.8), XpA is the coefficien t of x /J- = X A + 8 in
(E xiXE xi)n-rEw e S e(w)xw8. From the result of Example 6, this coefficient is
"
L (n - r) ! ( ILl' · .. , ILj - r, · .. , ILn)
i IL 1 ! . · . ( ILi - r )! . . . J.Ln !

118
I SYMMETRIC FUNCTIONS
and therefore
A A (n - r)! 
Xp /f = , £.,
n.
/-Lj! TI /-Lj - /-Lj - r .
( J.Li - r )! j  1 J.Li - J.Lj
i-I
If we put lp(X) = n (x - J.Li) and hp = n!/zp = n!/(n - r)!T, this formula becomes
n
-T2hpXpA//A= E J.Li(J.Li-1)...(J.Li-r+ l)lp(J.Li-r)/cp'(J.L;)
i-I
which is equal to the coefficient of x -1 in the expansion of
x(x -1)... (x - , + l)lp(x - T)/lp(X)
in descending powers of x.
In particular, when r == 2 we obtain
hp XpA /fA = n( A') - n( A).
8. If A is a partition of n, let
fA(q) = qn(A)lpn(q)/ TI (1- qh(x».
xE ,\
Let W E Sn be an n-cycle and let, be a primitive complex nth root of unity. Then
for all T  0 we have
(1)
X A ( W r) = f). ( ( r) .
(Since fA (q) = lpn{q )SA (I, q, q2, . . . ) (3, Example 2) it folJows that
E f..\(q)sA(x)
/,\I-n
is equal to the coefficient of tn in lpn{q)O;,j(1-x;qj-1t)-1, and hence by the
q-binomial theorem (2, Example 4) is equal to
E ga(q)xa
/al-n
where ga(q)=lpn(q)/ni>llpa,(q), and the sum is over all sequences a=
(at, a2,...) of non-negative integers such that lal = E a; = n. Now,r is a primi-
tive sth root of unity, where n/s is the highest common factor of T and n. Show
that g a( , r) = 0 unless each a; is divisible by s, and that if a = s f3 then ga (,') is
equal to the multinomial coefficient (n/s)!/O f3;! Deduce that
(2)
E fA( ")SA(X) = (Ext)n/s == p:/s.
IAI-n

7. THE CHARACTERS OF THE SYMMETRIC GROUPS 119
on the other hand we have by (7.8)
(3) E X,\( wr)s,\ (x) == P:/ s,
IAI-n
and (1) now follows from comparison of (2) and (3).)
9. (a) Let A"+ c An denote the semigroup consisting of the non-negative integral
linear combinations of the Schur functions SM I AI = n, and let A+::: EBn;. 0 An+. If
IE A, then fE A+ if and only if (/, g)  0 for all g E A+.
By (7.5) and (7.6), A"+ consists of the characteristics of ordinary (not virtual)
characters of Sn. From (7.3) it follows that An:. . A"+ c A+n for all m, n  0, so
that A+ is closed under multiplication as well as addition. (Equivalently, the
coefficients c., =:!J < S,\, s,."s.,) are non-negative.) In particular, h A and e A lie in A +
for all partitions A. From Examples 2 and 3 it follows that A + is stable under the
involution Cd, and that s,\/,." E A+ for all partitions A, J.L.
(b) Define a partial order on A by letting /  g if and only if / - g E A +. Show that
the following conditions on partitions A, J.L of n are equivalent:
(a) h). < h,.,,; (a') eA < e,.,,; (b) s,\  h,.,,; (b') SA' < ep'; (c) M(e, m)A'p. > 0; (d)A  J.L.
(Since A+ is stable under ro, we see that (a)  (a') and (b)  (b'). Next, the
relation h,\ = E K,."ASp. (6.7) shows that s,\ < hA and hence that (a)  (b). Since
eA' SA we have M(e, m),\,,.,, == (e,\" h,.,,>  (SA' hp.)' whence (b)::::> (c). The next im-
plication (c) => (d) follows from (6.6) (n. Finally, to show that (d) => (a) we may by
(1.16) assume that A = Rij J.L, and then we have
h,." - h A = h., (h"",hp.) - hp.,+ 1 h,.,,)-l) == h.,s( ""it P.)  0,
where JI is the partition obtained from J.L by deleting J.Li and J.Lj.)
The equivalence (c)  (d) is known as the Gale-Ryser theorem: there exists a
matrix of zeros and ones with row sums Ai and column sums P-j if and only if
A' > P- (use (6.6».
Another combinatorial corollary is the following: if A  P- then KSA  K(Jp. for
any skew diagram 8. (Take the scalar product of both sides of (a) with ss, and use
(5.13).) In particulr, we have K"p. > 0 whenever A  p- (because KAA == 1).
10. (a) We have
ES,\*SA= n(l-Pk)-l
A k>l
where the sum on the left is over all partitions A. For
2
S" * SA == E Z;l( XpA) Pp
P
and
2
E(xpA) =zp
,\

120
I SYMMETRIC FUNCTIONS
by orthogonality of characters, so that
E SA * SA = E Pp
IAI-n Ipl-n
which is equivalent to the result stated.
(b) We have
rI (1 -XiYjZk)-l = E SA(X)SI£(Y)(SA * SI£)(Z),
i,j,k A,p..
rI (1 +XiYjZk) = E SA(X)SI£,(Y)(SA * SI£)(Z).
i,j,k A,p..
11. Let cP = EIAI-n X'\ If w E SrI has cycle-type p, then
cp(w) = E XpA = E (SA'Pp) = (S,pp)
A A
where (5, Example 4)
S = rI (1 -Xi) -1 rI (1 -XiXj)-1.
i i<j
By calculating log s, show that
( 2 ) 2
Pn Pn Pn
S= rI exp -+- rI exp-
n odd n 2n n even 2n
and hence that cp(w) = ni 1 aml(p», where am) is the coefficient of tm in
exp(t + tit2) or exp(tit2) according as i is odd or even. In particular, cp(w) = 0 if w
contains an odd number of 2r...cycles, for any r  1.
12. Let Cn be the cyclic subgroup of Sn generated by an n...cycle, and let (J be a
faithful character of C n. Show that the induced character CPn = ind( 8) is indepen-
n
dent of the choice of (J, and that
1
ch( CPn) = - E J.L(d) Pd I d
n din
where J.L is the Mobius function.
(Let V be a finite...dimensional vector space over a field of characteristic 0, and
let L(V) = EBn  0 Ln(V) be the free Lie algebra generated by V. Then for each
n  0, Ln is a homogeneous polynomial functor of degree n, and a(Ln) = 'Pn in
the notation of Appendix A, (5.4).)
13. For each permutation wand each integer r  1, let a,( w) denote the number
of cycles of length r in the cycle decomposition of w. The a, are functions on the
disjoint union of the symmetric groups Sn. As such they are algebraically indepen-
dent over Q, for if f is a polynomial in r variables such that f(a](w),..., a,(w)) = 0

7. THE CHARACTERS OF THE SYMMETRIC GROUPS 121
 r all permutations w, then t(ml"'.' m,) = 0 for all choices of non-negative
itegers mi. and hence f is the zero polynomial. Hence
A = Q[al,a2,...]
's a polynomial ring. (The multiplication in A is pointwise multiplication of
unctiOns: (tg Xw) = t(w )g(w), not the induction product defined in the text.)
(a) For each partition p = (1 ml 2 m2 . . .) let
(a) (a ) a(m,)
=[1 , =[1
p rl m, rl mr! '
where am,) is the 'falling factorial'
am,) = ar(a, - 1)... (ar - m, + 1).
Since the aT are algebraically independent, the monomials a;nl ai2 . .. form a
Q-basis of A, and hence the polynoials (:) form another Q-basis.
Define a linear mapping (J: A -+ A Q by
(J(f) = E ch(f I Sn)
n>D
for lEA. If f= (:) then
cMfl Sn) =  E (a )(W)r/J(W).
n! wE S P
n
If w has cycle-type T = (ln, 2n2...) we have a,(w) = n, and hence (: )(W) =
n,;. 1 ( ::, ), which is equal to ZT/ zp z" if T = P U a for some partition a (i.e. if p
is a subsequence of T) and is zero othelWise. It follows that
8(:) = Z;lz;lppp"
-1 H
= Zp Pp ,
where
H= Ez;lpu= E hn.
0' n>O
(b) Define a linear mapping cp: AQ -+ A by
cp(pp) =Zp(:) = [1 ,m'am,)
r> 1

122
I SYMMETR.IC FUNCTIONS
for each partition p =- (1m! 2m2...). From (a) above it follows that (hI': AQ -. A .
multiplication by H. The mapping tp is bijective and hence defines a new protfu:
f* g on A by the rule
f* g == (()«({)-l(f)({)-l(g».
We have
(a) (a) ZPUq( a )
p * u == ZpZq P U u
or equivalently
a(m) * a(n) == a(m +n)
for any two finite vectors m == (mh m2,...), n == (nt, n2,..') of non-negative inte..
gers, where a(m) == a\m1) am2) ... , and likewise for a(n).
(c) Let
p== n (1 +Pr)a,.
r> 1
Then cp(f) == (P,f) for all fE A.
(Introducing variables x == (Xl' X2'...)' we have
(1)
P(X) == n (1 + Pr(x»Q,
r> 1
( ar ) m
== n E m Pr(X) r
r> 1 m,> 0 r
= E(:)p/x).
p
Let C(X, y) == ni.j(l - XiYj)-) == Lp Z; Ipp(X)pp(Y), as in 4, Example 9. Then it
follows from (1) and the definition of (() that P(x) == ({)yC(x, y), where 'Py acts on
symmetric functions in the y's. Hence (loc. cit.)
(P,f) = ({)y(C(x,y),f(x» == ({)yf(y) == ({)(f).)
14. Let A == (AI' A2, . . .) be a partition of n and let (N, A) == (N) U ,\ =
(N, At, A2, . . .) where N is any integer  Ato From (7.8) it follows that, for each
WESN+n, X(N.A)(W) is equal to the coefficient of x&,+nxtJ+n-1...x:" in the
product
(1)
Q6(Xo, Xl"'" Xn) n Pr(XO' Xl'.'" xn)Q,(w)
r> 1
where, as in Example 13, ar(w) is the number of r-cycles in the cycle decomposi-
tion of w. Since the polynomial (1) is homogeneous in xo, x l' .. . , X n' nothing is lost

7. THE CHARACTERS OF THE SYMMETRIC GROUPS 123
b setting Xo == 1, which shows that X(N, A)(W) is equal to the coefficient of
Y +n - 1 X All in
Ai · · · n
Xt
n
a8(xl'''.' xn) n (1 - Xi) n (1 + p,(x1,..., xn))Qr(W)
i- 1 r 1
and therefore is equal to the coefficient of SA in
n (1 - Xi) n (1 + p,)Qr(w).
i> 1 r 1
It follows that (as functions on S N + n) we have
(2)
x(N, A) = (EP, SA)
where
E:=: n (1 - Xi) = E (-1)' e,
i 1 r 0
and
(3)
P = n (1 + p,)Qr = E (a )pp
r 1 p P
as in Example 13.
The scalar product (EP,SA) on the right-hand side of (2) is a polynomial
XA eA == Q[al, a2' .0..] called the character polynomial corresponding to the parti-
tion A. It has the property that XA I SN+n = x(N, A) for all N  At.
There are various explicit formulas for the polynomials X A:
(a) Since by (2.14)
E = E (_1)/(0') z; lpo
q
it follows from (3) that
(4)
xII. - E ( _1)1("') z; IXpAu.,. (:)
p,CT
summed over partitions P, 0' such that I pi + 10' I = I AI.
(b) Alternatively, we have
xA= E (-I)'(e,P,sA)
rO
== E (-1)' (P, SA/(tr»
rO
= E ( _1)IA- #£I( P, sJL)
JL

124
I SYMMETRJC FUNCTIONS
summed over partitions IL c A such that A - IL is a vertical strip. Hence
(5)
xl. = L (_1)11.-1'1 E x:(:)
p.. p
summed over IL as above, and p such that I pi = I ILl.
(c) Let uA = E.l.s,\ == L, __ o( -1)"s,\/(1'). From (5.4) it follows that
1 1 1
h Al - 1 h'\l h,\} +n-1
(6) uA = h '\2 - 2 h'\2 -1 h'\2+n-2 .
hA -n h,\ -n+l hA
n rr n
From Example 13(c) we have
X A = < EP , SA> = < P, u,\ > = cp (u,\)
where cp: AQ  A is the mapping defined in Example 13(b). Hence if we define
1T,=cp(h,)= E (:)EA
Ipl-r
for all r (so that "', = 0 if r < 0), it follows from (6) that
*
1 1 1
"''\1 - 1 "',\ 1 7T,\ 1 + n - 1
(7) XA= "''\2 - 2 "''\2 - 1 "''\2 +n - 2
7T,\ - n "',\ - n + 1 7T,\
n rr n
where the asterisk indicates that the multiplication in A is that defined in Example
13(b).
(d) By subtracting column from column in the determinant (7) we obtain
(8)
X'\ = det*( 7T;.-i+j' )11" 'I'
I ...I,j...n
where ",: = "',. - 7T,.-1' and again the asterisk indicates that the determinant is to be
expanded using the * -product.
15. In general, if A is a ring and x, yEA, then Axy is a submodule of Ay and is
the image of .Ax under the homomorphism a ..... ay (a E A), hence is isomorphic to
a quotient of Ax.
(a) Let A be a partition of n and let T be any numbering, of the diagram of A with
the numbers 1,2,..., n. Let R (resp. C) denote the subgroup of Sn that stabilizes

7. THE CHARACTERS OF THE SYMMETRIC GROUPS 125
each roW (resp. column) of T, so that R =:. SA and C =:. SA'. Let A = Q[Sn] be the
grOUP algebra of S n and let
a = E s(u)u,
ueC
s = E v.
veR
Then As is the induced module ind(l), isomorphic to HA, and likewise Aa :::: E).,.
Let e = as EA. Since R n C = {I}, the products uv (u E C, V E R) are all
distinct, and hence
(1)
e= 1+... +0.
From the obselVation above, MA = Ae is a submodule of As and is isomorphic to a
quotient of Aa. From the remark following (7.8) it follows that M). is the
irreducible A-module (or Sn-module) with character XA.
(b) Let cp: A  A be right multiplication bye. Then ep(MA) = MAe =Ae2 cAe = MA.
Since MA is irreducible, it follows from Schur's lemma that epl MA is a scalar c E Q,
and therefore e2 = epee) = ceo Hence ep2 = cep, so the only eigenvalues of ep are 0
and c, and the eigenvalue c has multiplicity equal to the dimension of M).. Hence
(2)
trace lp = c dim M). = cnI/h(A)
by (7.6) and 5, Example 2. On the other hand, it follows from (1) above that for
each w E Sn the coefficient of w in ep(w) = we is equal to 1; hence relative to the
basis Sn of A the matrix of ep has all its diagonal elements equal to 1, and
therefore
(3)
trace ep = n!
From (2) and (3) it follows that c = h(A) and hence that e = h(A)-las is a primitive
idempotent of A affording the character X A.
(c) With the notation of (a) above, let my E Q[x),..., xn] denote the monomial
xt(1) 0.. x:<n), where d(i) = r - 1 if i lies in the rth rpwof T, and let IT denote the
product n(xi - Xj) taken over all pairs (i, j) such that j lies due north of i in T.
Thus IT is the product of the Vandermonde determinants corresponding to the
columns of T, and mT is its leading term, so that IT = amTo
Let 8: A -+ Q[Xl'...' xn] be the mapping u  um1'. Since d(i) = d(j) if and only
if i and j lie in the same row of T, it follows that the subgroup of Sn that fixes mT
is the row-stabilizer R, and hence that 8(A) =:.As. Consequently 8 I MA is an
isomorphism, and we may therefore identify MA with its image O(M).) = Aasmr =
AamT = At T. In this incarnation M). is the Specht module corresponding to the
partition A: it is the Q-vector space spanned by all n! polynomials IT' for all
numberings T of the diagram of A, and the symmetric group acts by permuting the
x's.
(d) The dimension of M). is equal to the number of standard tableaux of shape A,
by (7.6). In fact the polynomials IT' where T is a standard tableau, are linearly
independent over any field, and hence form a basis of MA.
(Order the monomials x«, a E Nn as follows: xQ <x{3 if and only if a precedes
13 in the lexicographical order on Nn.

126
I SYMMETRIC FUNCTIONS
(0 Suppose that i <j and d(i) < d(j) in T, and let w be the transposition (ij).
Then mT < mwT.
(ii) Deduce from (0 that if T is standard then IT = mT + later monomials.
(iii) Let Tt,..., T, be the standard tableaux of shape A. The monomials mTt'.." m
are all distinct, and we may assume that mT1 < ... < mT," Use (ij) to show that th
IT; are linearly independent: if E C il T{ == 0, the coefficient of mTt in the left-hand
side is equal to Cl' hence Cl == 0; repeating the argument gives C2 == 0, and so on.)
(e) From (d) it follows that each IT is a linear combination of the fT, say
IT =: Ec;IT{ with coefficients ci e Q. Show that each ci is an integer. (Let m be the
common denominator of the rational numbers Ci' and let Ci == mi/m where the m.
are integers. Then we have r
(4)
mlT == E miIT{.
If m > 1, let p be a prime dividing m, and reduce (4) mod.p. Since not all the m.
r
are divisible by p, we conclude that the IT. are linearly dependent over the field of
.
p elements, contrary to (d) above.)
Hence for each permutation w e S n the entries in the matrix representing w,
relative to the basis (IT) of MA' are all integers.
16. For each partition A of n, let RA be an irreducible matrix representation of SII
with character X A, such that RA(w) is a matrix of integers faT each w e Sn.
(Example 15(c) provides an example.) For each partition p of n, let cp denote the
sum (in the group ring Z[Sn]) of a11 elements of cycle-type p in Sn. Then c
commutes with each we Sn, and hence by Schur's lemma RA,(cp) is a scala;
multiple of the unit matrix, say
(1)
RA(cP) == wpA1d'
where wpA is an integer and d == n!/h(A) is the degree of XA. By taking traces in (1)
we obtain
n' n'
· · A
;- XpA == h(A) Wp
p
and therefore
(2)
h(A)
wA==_XA
p Z P
p
is an integer for all A, p.
Let Cn denote the centre of the group ring Z[Sn]: it is a commutative ring with
Z-basis (cp)lpl-n. For each partition A of n, the linear mapping wA: Cn -+ Z defined
by wA(cp) == wpA is a ring homomorphism, since RA(cpcq) == RA(cp)RA(cq). Moreover
the w\ IAI == n, are a Z-basis of Hom(Cn,Z).
17. (a) For each partition A, the 'augmented Schur function' SA is defined by
..,  h(A) A  A
SA ==h(A)SA == '-' - XpPp == '-' wpPp
p zp p

7. THE CHARACTERS OF THE SYMMETRIC GROUPS 127
. the notation of Example 16. Thus SA is a polynomial in the power sums Pr with
nteger coefficients, i.e. SA E V, where V = Z[Pl' pz,.. .] is the subring of A
In nerated by the power sums. Moreover, SA is the smallest integer multiple of SA
ge · "T,.. th ffi . f n. ..... 1
that lies In y, since e roe IClent 0 PI In SA IS ·
(b) Let I be a prime number (the letter P being preempted) and let 'I!; denote the
subring of 'I' generated by the Pr with r prime to I. Show that SA E 'I!; if and only
if ,\ is an I-core (fi1, Example 8).
(If ,\ is an i-core, all the hook-lengths in A are prime to I, and hence all the
bOrder strips of A have lengths prime to 1. From this observation and Example 5 it
followS that XpA = 0 if p has any earts divisible by I, and hence that SA E 'I!;.
Conversely, if A is not an I-core, let A and  * denote the I-core and I-quotient of A
(1, Example 8). Let I A *1_ b  1 and let I AI = a. Then for the partition p = (lb1a)
we have XpA = :i:a!b!lh(A)h(A*) =1= 0, and hence SA  '1';.)
(c) Let X be an indeterminate and BX: 'I! -+ Z[X] the homomorphism defined by
&x(Pr) == X for all r  1. Then
&X{SA) = c,\{X)
where cA{X) is the content polynomial of A (iI, Example 11).
18. Let A be a partition of n, let I be a prime number, and let I< be the I-core of
A. Then
(1)
SA == SK(P - PI)r (mod. I)
in'l'=Z[PI,P2'''.], where r=(n -11(1)1/ and SA is as defined in Example 17.
The proof of (1) uses some concepts from modular representation theory, for
which we refer to [P4]: namely the notion of the defect group of a block (or
equivalently of a 'central character) and the Brauer homomorphism.
Let F denote the field of I elements. For any finite group G, let C( G) denote
the centre of the group algebra F[ G]. For each partition A of n we have a
character 1I7'A: C(Sn)..... F of the F-algebra C{Sn), obtained from wA (Example 16)
by reduction modulo I. Each defect group DA of wA is a Sylow I-subgroup of the
centralizer of an element of cycle-type (1m) in Sim C Sn' where 0  m  nil ([19],
6.2.39). It follows that if DA =1= {1}, then DA contains a subgroup Q of order I,
generated by an I-cycle. On the other hand, if D A = {I}, there is a partition p of n
for which both zp and wp'\ are prime to 1, and hence (Example 16) h{A.)X/ = zp wpA
is prime to I. Consequently h( A) is prime to I and therefore (1, Example 10) A is
an I-core.
Assume now that A is not an I-core, so that DA contains Q as above. Let
H:: Q x Sn-l be the normalizer of Q in Sn. Then the mapping cp: C{Sn) -+ C(H)
= C(Q) F C(Sn-l) defined by restriction to H is an F-algebra homomorphism
(the Brauer homomorphism), and fJ7 A factors through cp: say w'\ = (B  W 11-) 0 cp,
where J.L is some partition of n -I, and B is the unique (trivial) character of C(Q).
If now p is a partition of n, the conjugacy class cp in Sn meets H only if p is of
the form (1/) U 0' or (I) U 0' for some partition 0' of n -I. If p == (1/) U 0' we have
tuA(cp) == wll-(cu)' and if; p = (I) u 0' we have fJ7A(Cp) = (I - l)wll-(cu) =
- 'GT"''(cu). Since w A(Cp) is the reduction modulo I of wpA (Example 16) it fol1ows

128
I SYMMETRIC FUNCTIONS
that modulo i we have wpA == wf if p = (11) U u, wpA == - wf if p = (i) U a) and
wpA == 0 for all other p. Hence
SA = E wpApp = ( E w':Pu }(p{ - PI)
p (7
so that
SA = S IL (p { - PI) (mod. 1).
If J.L is not an i-core we may repeat the argument. In this way we shall obtain
(2)
,., ,., (I )q
S,\ == Sv Pt - PI
for some integer q  1 and some I-core II. Now apply the specialization 8X: Pr  X
(r  1) (Example 17(c)): we obtain
c,\(X) == cv(x)(X' - X)q (mod. i)
== c1T(X)
where 'IT = II + (lq) has i-core II. From  1, Example II(c) we now conclude that the
partitions A and 'IT have the same I-core, so that II = K and q = r, completing the
proof.
19. Let A, J.L be partitions of n, and let i be a prime number. Then with the
notation of Example 17, S). == SIL (mod. I) if and only if A, J.L have the same I-core
('Nakayama's conjecture'). (If S,\ == SIL' then c).(X) = clL(X) by Example 17(c), and
hence A, J.L have the same I-core by 1, Example 11(c). Conversely, if A, I-L have the
same i-core, then it follows from Example 18 that S). == SIL.)
20. As in 5, Example 25 we shall identify each 10 g E A 0 A with f(x)g(y),
where (x) and (y) are two sets of independent variables. Define a comultiplication
A*: A -+ A 0 A and a counit 8*: A -+ Z by
A*f= f(xy)
where exy) is the set of all products XiYj' and
8*1=/(1,0,0,...)
for all IE A.
With respect to A* and 8*, A is a cocommutative Hopf algebra over Z; both !:l*
and 8* are ring homomorphisms, and (1  8*) 0 A* is the identity mapping.
(a) Show that, for all n  1,
(1)
A*hn = E s,\ 0 S,\,
IAI-n
8*h = 1.
n ,
(2)
*en = E s).  SA"
IA/=n
8*en = ln;
(3)
A*Pn = Pn 0 Pn'
8*Pn = 1.

7. THE CHARACTERS OF THE SYMMETRIC GROUPS 129
Also we have (7.9)
(4)
fl*s" = E 'Y,."s,.,,  sv.
p." J.I
(b) As in 5, Example 25 define a scalar product on A  A by
(II gl' 12 g2) = (/1,/2)(gt,g2)
for 11,/2' gl' g2 E A. With respect to this scalar product, fl* is the adjoint of the
internal product: in other words, we have
(5) (fl*l, g f8 h) = <f, g * h)
for all f, g, h E A. (By linearity we may take 1= s'" g = s,.", h = sv' and then both
sides of (5) are equal to 'Yp.,Av by (4) above.)
21. For any commutative ring A, let G(A) denote the set of all unital ring
homomorphisms a: A -+A. Each such homomorphism is determined by the formal
power series
aH(t) = E a(hi)ti EA[[t]]
i 0
with constant term a(ho) = 1, and we may therefore identify G(A) with the set of
formal power series in A[[t]] with constant term equal to 1.
(a) The comultiplication fl: A -+ A  A defined in 5, Example 25 induces an
abelian group structure on G(A) as follows. If a, 13 E G(A), we define
a+l3=mAo(a{3)ofl
where mA: A A -+A is the multiplication in A. We have then (IDe. cit.)
(a+{3)hk= E a(hi){3(hj)
i+j-k
so that
(a + (3 )H(t) = (aH(t))( (3H(t)),
the product of the power series aH(t) and (3H(t) in A[[t]].
Next let w: A -+ A be the involution defined by w(h) = ( -l)ie; (i  1), so that
on An, 'Uf is ( - l)nw. Then define
-a=aow;
we have (-a)H(t) = a(E( -t)) = (aH(t))-l, so that (-a)(H(t)) is the inverse in
A[[t]] of the power series aH(t).
FinaHy, the zero element 0 of G(A) is induced by the counit 8: namely
O=eAoe, where eA:Z-+A is the unique homomorphism of Z into A. Since
e(h) = 0 for each i  1, it follows that OH(t) = 1.
(b) The comultiplication fl* of Example 20 induces a multiplication in G(A) by the
rule
a{3=mA o(a (3)ofl*.

130
I SYMMETRIC -FUNCTIONS
This product may be described as follows: if we formally factorize the power series
aH(t) and /3H(t), say
aH(t) = f1 (1 + it),
/3H(t) = f1 (1 + 17it),
then
(a/3)H(t) = f1 (1 + i17jt),
i ,j
The element 1 e G(A) defined by
1 = eA 0 8*
with 8* the eounit defined in Example 20, is the identity element for this
multiplication, and I(H(t»:=: (1- t)-l.
(c) With addition and multiplication as defined in (a) and (b), G(A) is a commuta-
tive ring, with zero element 0 and identity element 1. If (,0: A -+ B is a homomor-
phism of A into a commutative ring B, then G«(,O): G(A) -+ G(B) defined by
G( (,0 ) a = (,0 0 a is a ring homomorphism, Thus G is a covariant functor on the
category of commutative rings.
22. Define an internal product on A 0 A by
(fl 0 f2) * (gl 0 g2) = (fl * gl)  (f2 * g2)
for fl,f2,gt,g2 eA. Show that l1(f* g) = (&f)*(g) for all f,g E A, but that in
general *(f * g) + (d*f) * (d*g).
23. (a) Let f, g, h e A. Then the scalar product (f * g, h) is symmetrical in J, g;'
and h.
(b) Let (u.\),(vA) be dual bases of A, and let fe A. Then
*f= E (u.\ * f) 0 VA'
A
h.\ * slA- = E f1 SIA-(i)/IA-(/-l)
i 1
(For (*f, g 0 u.\) = (f, g * UA) = (UA * f, g) by (a) above and Example 20(b).)
(c) Let f,g,h e A and let h = Eai  bi. Then
(fg) * h = E (f * ai)(g * bi)'
In particular,
(fg) * SA = E (f * SA/IA-)(g * slA-)'
I'"
(d) Let A, J.L be partitions. Then
summed over all sequences (J.L(O), J.L(1),...) of partitions such that 0 = J.L(O) c
. .. C J.L and I J.L(i) - J.L(i-l)1 = Ai for each i  1. (Use (c).)

7. THE CHARACTERS OF THE SYMMETRIC GROUPS 131
(e) If M == (mij) is a matrix of non-negative integers, let hM = ni,j hm;( If A and J.L
are partitions, show that
h).*hp,= EhM
summed over all matrices M of non-negative integers with l( A) rows and l( J.L)
columns and row sums Ai' column sums J.Lj.
24. The symmetric group S" embeds naturally in S,,+ 1 as the subgroup that fixes
n + 1. The union
SCQ = U S"
nO
is the group of permutations of the set of positive integers that fix all but a finite
subset. If w E Sn has cycle-type P, where P = (Pt,..., p,) is a partition of length r,
then when regarded as an element of S"+k the permutation w has cycle type
(PI"..' p" 1,...,1) = p U (lk). We are therefore led to define the modified cycle-type
of w to be the partition (PI - 1,. . . , P, - 1). This modified cycle-type is stable
.<, under the embedding of SI1 in Sn+k.
For each partition A, let C). denote the set of all W E SCQ whose modified
...cycle-type is A. As A runs through all partitions, the C). are the conjugacy classes of
tJte group SCQ. For example, C(l) is the class of transpositions, and Co consists of the
. . identity permutation.
.'; For each n  0, let Zn denote the centre of the group ring Z[S n]' and for each
partition A let c).(n) E Z" denote the sum of an w E S" whose modified cycle-type
is A, i.e. the sum of all W E Sn () CA. We have c,\(n) =1= 0 if and only if IAI + l(A)  n.
." Now let A, J.L be partitions. The product c;..(n)clL(n) in Z" will be a linear
..".":combination of the c.,(n), say
:.r._":" _"<:.
:"'.','(1)
c,\(n)clL(n) = E aflL(n)c,,(n)
v
:,,'::,\J{ith coefficients aflL(n) EN, and zero unless I vi  IAI + I ILl. For example, when
'F,":"!A= IL == (1), c(l)(n)2 is the sum of all products (ijXkl) of two transpositions in S",
!}:, .:-:a.rid a simple calculation gives
iCht7) c(I)(n)2 = 3c(2)(n) + 2c(II)(n) + tn(n - l)co(n).
t...;..-....- '".-.
.':.,.:L...':'
r:,i.:..:Igeneral (see [F2]) the coefficients aflL(n) are polynomial functions of n, and are
:'_:.i,i.ndependent of n if and only if I vi = I AI + I ILl. We may therefore, following [F2],
K':'<.'f.9DStruct a ring as follows. Let R be the subring of the polynomial ring Q[ t]
';;,:!1sisting of polynomials that take integer values at all integers (the binomial
"jlr,;fficients (), n  0, form a Z-basis of R), and let F be the commutative
:%f,':,"'algebra with R-basis (c.\) indexed by partitions A and multiplication defined by
I!!) CACIL = E aflLcv'
h;L-\;#'".':' V
tf:,,,.e aflL E R takes the value aflL(n) at the integer n, and the sum is over
Ii-.j!p.titions v such that I vi  I AI + I ILl. If we assign each c,\ the degree I AI, F is not a

132
I SYMMETRIC.FUNCTIONS
graded ring, because the right-hand side of (3) is not homogeneous; but it is a
filtered ring: if Fr is the subspace of F spanned by the c,\ such that I AI  r, then
pr. ps cpr+s. We may therefore form the associated graded ring GR = Gr(F): G
is the direct sum ED,. > 0 GR, where GR = pr / pr-l, and the multiplication in GR i
induced from that in F. The effect of passing from F to GR is simply to throw out
the terms of lower degree in (3): in G R the multiplication is defined by
C,\Cp. = E afp.cv
Ivl-I'\I+II
and (as remarked above) the structure constants afp. such that I vi = I AI + I p,1 are
non-negative integers. (For example, in GR we have ca) = 3C(2) + 2C(11)' from (2)
above.) It follows that GR = R 0z G, where G is the free Z-module with basis (c.\)
and multiplicative structure given by (4).
Let us write cr in place of c(r)' r  1 (co = 1 is the identity element of P and G).
Show that
(4)
(a) if IAI +, = m, then
(5)
( (m + 1),!/[I miCA)!
a(m) - . 0
'\(r) - ,;>
o
if l( A)  r + 1,
otherwise,
where mo(A) =, + 1 -leA);
(b) if IAI +, = Ivl then
(6)
v _ '" (V;)
a '\(r) - i..J ap.(r)
summed over pairs (i, J.L) such that p, U v = A U (vi). Deduce that ar(r) = 0 unless
v  A u (r), and that a:(}r) > o.
From (b) it follows that, for each partition A = (At, A2,...), CA1CA2... is of the
form
C'\lC'\2...= E d,\p.cp.
p.;>'\
with d,\,\ > o. Hence c], C2'''' are algebraically independent elements of G, and
generate Gover Q, i.e. G  Q = Q[c1, C2'''']. Moreover, the multiplicative struc-
ture of G is uniquely determined by (a) and (b).
25. Let «/1 be the involution on A defined in 2, Example 24. With the notation of
that Example, the h = t/J(h,\) form a Z-basis of A. Let (g,\) be the dual basis, so
that (g,\, h) = 8,\p.. Equivalently, g,\ = t/f-(m», where t/f- is the adjoint of «/J
relative to the scalar product. We shall show that, in the notation of Example 24
above, the linear mapping q;: A  G defined by q;(g,\) = c,\ for all partitions A is a
ring isomorphism.
(a) From 2, Example 24 we have
h = -hn + E u(n)p.hp.
«n)

7. THE CHARACTERS OF THE SYMMETRIC GROUPS 133
for suitable integers u(n)p,' and hence h! is of the form
h = ( -1)/('\)h.\ + E uAp,hp,.
JL<A
This shows that the transition matrix M(h*,h) (=M(h,h*» is strictly upper
triangular, with diagonal elements (-1)/(.\). By (6.3X3) we have M(g, m) =
M(h, h*)', hence
( )/(.\) "
g.\ = -1 m.\ + '-' up,.\mp,.
JL>A
In particular, g(n) = -m(n) = -Pn.
(b) Let b{p, = (gAgp" h) be the coefficient of gv in gAgp,. Since the g(n) = -Pn
generate AQ' in order to prove that lp defined above is a ring isomorphism it will
be enough to show that b{p, = afp, whenever JL is a one-part partition (r), and for
this purpose it will suffice to show that the b's satisfy the counterparts of the two
relations (5) and (6) of Example 24.
Consider first
(1)
b1<:1 = - (gAP" h) = - (gA' p/" h:')
in the notation of fi5, Example 3. From 2, Example 24, h is the coefficient of tm
in
1 m 1 1 ( " p,t')
H(t)- - = exp -(m + 1) '-' - .
m + 1 m + 1 , 1 r
Hence -p/-h:" = -rah*",jap, is equal to the coefficient of tm in t'H(t)-m-l,
that is to say it is the residue of the differential
1
t'dtj(tH(t»m+l = - d(tr+1)/um+1
r+1
(2)
where U = tH(t) as in 2, Example 24. Now
(r+1 = ur+1( E h:U")'+l
(r + I)!
= E h*ul.\I+,+l
n m i ( A) ! .\ ,
i__ 0
summed over partitions A of length l(A)  r + 1, where mo(A) = r + 1 -l(A).
Hence from (2) we obtain
r!(m + 1)
-p/-h:' = E h*
fl m;(A)! .\
iO
summed over partitions A such that l( A)  r + 1 and I AI = m - r. From (1) it now
follows by comparison with relation (5) of Example 24 that b1(:. = a(:. whenever
IAI+r=m.

134
I SYMMETRIC FUNCTIONS
(c) Finally consider
bM.r) = -(gPr,h:) = - (g>.,p/-h:).
Since p,.l. = r a/a p, is a derivation (5, Example 3) we have
.1. h* - "h* .1. h*
p, v - '-' v(l)p, v.,
,
where v(;) = (v1'. . . , v;, . . .), and the sum is over i  1 such that V;  r. Hence
(3)
b,) = - E (g >.' h:(l) p,.l. h:i),
in which the scalar product on the right-hand side is the coefficient of h! in
h:(;>pr.l. h:i, hence is zero unless A = v(i) U J.L for some partition J.L, i.e. J.L U v = A U
(v;); and then is equal to the coefficient of h in p,.l. h:;, which is (gJL' p,.l. h:) ==
- b by (1). Hence (3) takes the form
b,) = E b(
summed over pairs (i, J.L) such that J.L U v = A U (Vi). We have thus established the
counterpart of relation (6) of Example 24, and the proof is complete.
26. Let 8: R -+ R  R be the comultiplication on R that corresponds to il: A 
A  A (5, Example 25) under the characteristic map (so that (ch  ch) 0 8 = A 0 ch).
If [e Rn, show that
8[= E9 flSpXSq.
p+q-n
Notes and references
The representation theory of finite groups was founded by Frobenius in a
series of papers published in the last years of the nineteenth century, and
reproduced in Vol. 3 of his collected works; in particular, he obtained the
irreducible characters of the symmetric groups in 1900 [FI0], and our
exposition does not differ substantially from his.
The internal product f * g occurs first (as far as I am aware) in the 1927
paper of Redfield [Rl], and later in [Lll]. (Littlewood calls it the inner
product: we have avoided this terminology, because inner product is
sometimes taken as synonymous with scalar product.)
Example 5 is due to Littlewood and Richardson [L13], but is commonly
known as the Murnaghan-Nakayama rule ([MI8], [Nl]). Examples 6 and 7
are due to Frobenius (loc. cit.). Example 9 was contributed by A. Zelevin-
sky.
Examples 13 and 14. Character polynomials occur already in Frobenius'
1904 paper [Fll]. The formulas (7) and (8) are due to Specht [819].
Examples 24 and 25. For proofs of Example 24(a), (b) see [F2]. A better
proof of the result of Example 25 will be found in [G9].

8. PLETHYSM
135
8. Plethysm
In this section we shall study briefly another sort of multiplication in A,
called plethysm or composition, and defined as follows. Let I, g E A, and
write g as a sum of monomials:
g=EuaxQ.
a
Now introduce the set of fictitious variables y; defined by
(8.1)
n (1 + y;t) = n (1 +xat)ua
a
and define
(8.2)
fog = f(Yl' Y2' · · · ).
If f E Am and g E A", then clearly fog E Amn. Also e1 acts as a two sided
identity: f 0 e1 = e1 0 f= f for all fE A.
From the definition (8.2) it is clear that
(8.3) For each g E A, the mapping f  fog is an endomorphism of the ring
A. I
By taking logarithms of both sides of (8.1) we obtain
Pn(Y) = E ua(xa)n
(n  1)
a
so that
(8.4)
Pn 0 g = g 0 Pn = g(xf , x,. . . )
for ,all g EA. In particular,
(8.5)
Pn 0 Pm = Pm 0 Pn = Pmn.
-.' From (8.4) it follows that
(8i6) For each n ;;a= 1, the mapping g  Pn 0 g is an endomorphism of the
ring A. I
. Plethysm is associative: for all I, g, h E A we have
(8.7)
(log) 0 h = 10 (g 0 h).
lioof. Since the Pn generate AQ (2.12), by virtue of (8.3) and (8.6) it is
enough to verify associativity when f= Pm and g = Pn' in which case it is
obvious from (8.4) and (8.5). I

136
I SYMMETRIC FUNCTIONS
For plethysm involving Schur functions, there are the following formu-
las: from (5.9) it follows that
SA 0 (g + h) = E C;II(SP. 0 g )(sv 0 h)
(8.8)
JL, v
= E (SA/p. 0 g )(Sp. 0 h)
JL
and from (7.9) that
(8.9)
SA 0 (gh) = E Yp.(sp. 0 g )(sv 0 h).
JL, v
The sum in (8.8) is over pairs of partitions J.L, v c'\, and in (8.9) over
pairs of partitions J.L, v such that I J.LI = I v I = 1'\1.
Finally, let '\, J.L be partitions. Then SA 0 sp. is an integral linear combina-
tion of Schur functions, say
(8.10)
SA 0 sJL = E afp.s'7T
'IT
summed over partitions 71' such that t7T'1 = 1'\1.1 MI. We shall prove in
Appendix A that the coefficients afp. are all  o.
Remarks. 1. We have obseIVed in (3.10) that to each f E A there corre-
sponds a natural operation F on the category of A-rings. In this correspon-
dence, plethysm corresponds to composition of operations: if f, g E A
correspond to the natural operations F, G, then to g corresponds to
FoG.
2. Plethysm is defined in the ring R of 7 via the characteristic map: for
u, v E R, U 0 v is defined to be ch-1(ch u och v). If u (resp. v) is an
irreducible character of Sm (resp. Sn)' then u 0 v is a character of Smn
which may be described as follows: if U (resp. V) is an Sm-module with
character u (resp. an Sn-module with character v), the wreath product
Sn ,..., Sm (which is the normalizer of S::' = Sn X ... X Sn in Smn) acts on U
and on the mth tensor power Tm(v), hence also on U  Tm(v); and u 0 v
is the character of the Smn-module induced by U  Tm(v). See Appendix
A to this Chapter.
Examples
1. (a) Let IE Am, g E An. Show that
{/O (wg)
w(/o g) = (wf) 0 (wg)
if n is even,
if n is odd,

8. PLETHYSM
137
and that
foe -g) = (-l)m(wl)og.
(b) If '\, JL are partitions, let ,\ 0 JL( = JL 0 ,\) denote the partition whose parts are
Ai P-j" Then we have
PA 0 Pp, = Pp, 0 PA == PA 0 P. .
(c) Show that w(h, 0 Ps) == ( -1),(s-l)e, 0 Ps.
2. Since SA/J.L == Sp,J... (SA) in the notation of 5, Example 3, it follows from (8.8) that
I 0 (g + h) = E ( ( S J. I) 0 g ) (s p. 0 h)
JL
for all I, g, h E A. Also
I 0 (gh) = E « s Jl. * I) 0 g ) (s p, 0 h).
JL
3. We have
hn o(fg) = E (SA 0 I)(SA 0 g),
1i\I-n
en 0 (fg) = E (SA 0 1)(s).1 0 g).
1i\I-n
These formulas are particular cases of (8.9) (and are consequences of (4.3) and
(4.3'».
4. Let A be a partition of length  n, and consider (SA 0 s(n -l»)(Xt, X2)' By
definition this is equal to s).(xf -1, X1- 2 X2'" ., X -1), i.e. to
xn-l)I).ISA(qn- t, qn-2,..., 1),
where q = x 1 X 2" 1. On the other hand, by the positivity of the coefficients in (8.10),
(sA 0 sn -lXXl' X2) is a linear combination of the S1f(x1, X2) with non-negative
integer coefficients, where 7T = (7T), 7T2) and 7Tl + 7T2 = (n - l)IAI = d say. Now
S1T(X1' X2) == xi1xi2 + xi1-1x2'2+1 + ... +xi2xi1
=X(q1Tl + q1T1-1 + ... +q1T2).
Hence sA(qn -1 , qn - 2,. . . , 1) is a non-negative linear combination of the poly-
nomials q'ITl +q1T1-1 + ... +q1T2, where 7T)  7T2 and 7T) + 7T2 =d. It follows that
s,\(qn-l, qn- 2,...,1) is a unimodal symmetrical polynomial in q, i.e. that if ai is the
coefficient of qi in this polynomial, for 0  i  d, then a; + ad-; (symmetry) and
ao  at  ...  a[d/2]
(unimodality ).

138
I SYMMETRIC FUNCTIONS
From A3, Example 1, it follows that the generalized Gaussian polynomial
[n ], = 1 - qn-c(x)
A xQ l_qh(X)
is symmetrical and unimodal for all nand A.
5. Let G be a subgroup of Sm and H a subgroup of Sn, so that G,..", H is a
subgroup of the wreath product Sm""" Sn C Smn. Then the cycle-indicator (2
Example 9) of G ,..", H is '
c(G,..", H) = c(H) 0 c(G).
6. Closed formulas for the plethysms h, 0 h2, h, 0 e2, e, 0 h2, e, 0 e2 may be derived
from the series expansions of 5, Examples 5 and 9:
(a) h, 0 h2 = E s, summed over an even partitions J.L of 2r (i.e. partitions with all
parts even).
(b) h, 0 e2 = Eit sit" summed over even partitions p., as in (a).
(c) e, 0 e2 == E1T s", summed over partitions 'TT of the form (at - 1,. . ., cxp -11
at,..., ap), where al > ... > ap > 0 and al + ... + ap = ,.
(d) e, 0 h2 ;:: L1T sf(" summed over partitions 'TT' as in (c).
7 Let he,) ==p 0 h == h 0 P so that
· n , II n "
h)(Xl' X2'. ..) = hn(xt, X2'. · · )
which is the coefficient of tn' in
,
n(l-xit,)-t= n n(l-x;wjt)-\
i;.1 ;l j-l
where CIJ = e2frij,. By (4.3) this product is equal to
E s(x)s(l, w,..., w,-l )tllLl.
IJ.
Now (A3, Example 17)
s(l, w,..., w'-l) = u,( J.L) = :f: 1
if l( J.L) , and J.L is an '-core, and sJ£(1, w,. . . , w'- 1) = 0 otheIWise. It follows that
p, 0 hn = E u,( J.L)SIL
IL
summed over ,-cores J.L such that l( J.L)  r and I J.LI = nr.
8. More generally, if p is any partition, let hp) = Pp 0 hn, so that
hp) = n (p 0 hn);:: n hPj)
jlJl Pj jlJl

8. PLETHYSM
139
is the coefficient of tnp == tiPltiP2 . .. in
(1)
Pj
n (l-xfitPj)-t == n n (l-Xiwt.)-I,
.. J '.k 1 JJ
Ipj I,) -
where Wj = exp(21Tij Pj). By (4.3) this product is equaJ to
(2) E s(x)sJ.£(tty(1),..., tmy(m»
"",
summed over partitions J.t of length  I pi, where y(j) denotes the sequence
({J)f>t<.k<'PJ' and m is the length of p.
By (5.11) we have
(3)
m
«1) (m» -  n (J1(J)- J1(j-l)1 ( (j»
S t1y ,..., tmy - '-' tJ SJI(J)/J1(J-l) Y
j-I
summed over all sequences (v(O), v(1), . . . , v(m» of partitions such that 0 = v(O) C
p(1) C ... C v(m) = J.t. Now from 5, ExampJe 24(b) we have
(4)
. { u. (v(j) j v(j - 1»
S"u)/,,<I-I)(y(J)) = OPI
if v(j) = v(j -1)
Pj ,
otherwise.
From (1)-(4) we can pick out the coefficient of each s in hp). The result may
be stated as follows: define a generalized tableau of type P, shape J.L, and weight
np:s (np1' nP2'...' np,,) to be a sequence T == (v(O),..., v(m» of partitions satisfy-
ing the following conditions:
(0 0 == v(O) C v(l) C ... C v(m) = J.L;
(ii) Iv<j) - vj 1)1_ n Pj for 1 j  m;
(Hi) v(j) = p. v(J-l) (1 j  m). (5, Example 24.)
(When P =J (1m) these are tableaux in the usual sense, of weight (nm).)
For such a tableau T let
m
u(T) = n up'<v(j)/v(j-l» = :1:1,
. 1 J
)-
and define
K(p) =  u(T)
, np '-'
T
summed over all generalized tableaux T of type P, shape J.L, and weight n p. The
integers KJp may be regarded as generalized Kostka numbers.
With these definitions, we have
Po h == h(p) =  K(p) S
p n n '-' J.£,np J.£'

summed over partitions J.L such that I J.LI == nl pi and I( J.L)  I pl.

140
I SYMMETRIC F.UNCTIONS
9. Since
_ '" -1 A
SA -. i.J Zp Xp Pp
p
it follows from Example 8 that
SA 0 hn = E Z;lX/pp 0 hn
p
=  ( z-lvAK( p) ) S
i.J i.J p /\.p ,n P Il
IL P
the outer sum being over partitions J.L such that I J.LI = nl AI and l( J.L)  I AI.
(a) When I AI = 2 we have (Example 7)
h) = E u2( J.L)sJL
IL
summed over I J.LI = 2n, l( J..L)  2, so that J.L = (2n - j, j); U2( J.L) = s(l, -1) ==
( - 1)1, and therefore
(1)
fa
h(2) -  ( - 1 )j
n - i.J s(2n - j,j) .
j-O
On the other hand,
(2)
fa
h(12) - h2 -  S
n - n - i.J (2 n - j ,j) .
j=O
From (1) and (2) it follows that
h2 0 hn = E S(2n-j,j)'
j even
e 2 0 h n = E S(2 n - j, j) ·
jodd
By duality (Example 1) we obtain
h2oen= E S(n+k,n-k)"
k even
e2oen= E S(n+k,n-k)'.
kodd
(b) When I AI = 3 we have
KiJ3) = number of tableaux of shape J.L and weight (n3)
= 1 + m( J.L)
where m( J.L) = min( ILl - J.L2' J.L2 - J.L3)' and l( J.L)  3.

8. PLETHYSM
141
Next, since h) == h (mod 3), it follows that
KiJ.n) == 1 + m{ p..) (mod 3)
and since KZ.n) = 0 or ::f: 1, it is determined by this congruence.
Finally,
n
h(21) - h(2)h -  { -1)j h
n - n n - i..J S (2 n - j I j) n'
j-a
from which we obtain
K(21) _ { 0
#J.(2 n In) - (_ 1) #J.2
if m{ p..) is odd,
if m{ p..) is even.
From these values we obtain
h3 0 hn = E ([!-m{ IL)] + 8{ p..»)S#J.'
JL
S21 0 hn = E {tm{ p..)}S#J. '
JL
e3 0 hn = E ([im{ p..)] + 8{ J.L) - ( -1)JL2)s#J.'
,."
summed in each case over partitions p.. such that I p..1 = 3n and l{ p..)  3, where
B( J.L) = 1 if m{ /-L) and p..2 are even or if m{ p..) == 3 or 5 (mod 6), and e{ p..) = 0
otherwise; and where {x} = - [ -x] is the least integer  x.
10. Foulkes conjectured that hm 0 hn  hn 0 hm whenever m  n, with respect to
the partial ordering on A defined in 7, Example 9(b). The results of Example 6
and Example 9(a) show that this is true for m = 2 and all n  2.
11. From 5, Example 4 it follows that
E SA = E h,{hs 0 e2) = E e,{hs 0 h2).
A r,s r,s
On passing to representations of S n' these formulas give
E XA = E 17, { 17s 0 82) = E 8,{ 17s 0172)
IAI-n
where r + 2s = n in the second and third sums. Now 17s 0 82 is the character of S2s
induced from the sign character of the wreath product S2  Ss (the hyperoctahe-
dral group of rank s), hence 11,{11s 0 82) is the character of Sn induced from the
character TI, X (lIs 0 82) of the group 5, X (S2  5s)' which is the centralizer in Sn of
an involution of cycle-type (2S1'), i.e. a product of S disjoint transpositions. It
follows that
(1)
E X A = E c
IAI-n c

142
I SYMMETRIC FUNCTIONS
where c runs over the conjugacy classes of elements w E S n such that w2 = 1, and
c is the induced representation from the centralizer of an element wEe just
described. The other sum in (1) may be interpreted analogously.
Notes and references
Plethysm was introduced by D. E. Littlewood [L9]. His notation for Our
SA 0 S,.,. is {J.L}  {A}. Many authors have computed (or have described
algorithms to compute) SA 0 s,.,. for particular choices of either A or J.L. For
their work we refer to the bibliographies in Littlewood [L9] and Robinson
[R7], and to [C2]. Examples 9(a) and 9(b) are due to R. M. Thrall [T5]. For
the next case (IAI = 4) see H. O. Foulkes [F6] and R. Howe [HI0].
9. The Littlewood-Richardson rule
If J.L and v are partitions, the product s,.,. Sv is an integral linear combina-
tion of Schur functions:
SS., = E C:"SA
A
or equivalently
(9.1)
SA/J.L = E c;., s., ·
II
The coefficients c;'v are non-negative integers, because by (7.3) and (7.5)
c:v = ( X'\, X"" . X., ) is the multiplicity of X A in the character X,.,.. X.,; also
we have c:v = 0 unless 1,\1 = I J.LI + I v I and J.L, v C A.
This section is devoted to the statement and proof of a combinatorial
rule for computing c:v, due to Littlewood and Richardson [LI3].
Let T be a tableau. From T we derive a word or sequence w(T) by
reading the symbols in T from right to left (as in Arabic) in successive
rows, starting with the top row. For example, if T is the tableau
1 1 2 3
2 3
1 4
w(T) is the word 32113241.
If a word w arises in this way from a tableau of shape A - J.L, we shall
say that w is compatible with A - J.L.

9. THE LITTLEWOOD-RICHARDSON RULE 143
A word w = a1a2 ... aN in the symbols 1,2,..., n is said to be a lattice
permutation if for 1  r  Nand 1  i  n - 1, the number of occurrences
of the symbol i in at a2 . . . a, is not less than the number of occurrences of
i + 1.
We can now state the Littlewood-Richardson rule:
(9.2) Let A, J.L, v be partitions. Then c;v is equal to the number of tableaux T
of shape A - J.L and weight JJ such that w(T) is a lattice permutation.
The proof we shall give of (9.2) depends on the following proposition.
For any partitions A, J.L, '1T' such that A C J.L, let Tab( A - J.L, 'IT) denote the
set of tableaux T of shape A - J.L and weight 'IT, and let TabO(A - J.L, 'IT)
denote the subset of those T such that w(T) is a lattice permutation. From
(5.14) we have
(9.3)
I Tab(A - J.L, 7T) I = KA- It.'" = (SA/It' hfr).
We shall prove that
(9.4) There exists a bijection
,.,.,
Tab( A - J.L, 'IT) --+ U (Tab ° (A - J.L, v ) X Tab( JJ, 'IT » .
v
Before proving (9.4), let us deduce (9.2) from it. From (9.4) and (9.3), we
have
(sA/It,h",) = E ITabO(A - J.L, v)l(sa"h",)
v
for all partitions 'IT, and therefore
SA/It = E I Tab ° (A - J.L, v ) I S II ·
v
Comparison of this identity with (9.1) shows that c;v::= ITabO(A - J.L, v) I.
To construct a bijection as required for (9.4), we shall follow the method
of Littlewood and Robinson [R5], which consists in starting with a tableau
T of shape A - J.L and successively modifying it until the word w(T)
becomes a lattice permutation, and simultaneously building up a tableau
M, which serves to record the sequence of moves made.
If w = a1a2... aN is any word in the symbols 1,2,..., let m,(w) denote
the number of occurrences of the symbol r in w. For 1 p  Nand r  2,
tbe difference m,(at... ap) - mr-t(at ... ap) is called the r-index of ap in
w. Observe that w is a lattice permutation if and only if all indices are
,Q .

144
I SYMMETRIC "FUNCTIONS
Let m be the maximum value of the ,-indices in w, and suppose that
m > O. Take the first element .of w at which this maximum is attained
(clearly this element will be an ,), and replace it by , - 1. Denote the
result of this operation by S,_ 1,,( w) (substitution for , - 1 for ,). Observe
that Sr_I,,(W) has maximum ,-index m -1 (unless m = 1, in which case it
can be - 1).
(9.5) The ope,ation Sr-l" is one-to-one.
Proof. Let w' = S r- 1,,( w). To reconstruct w from w', let m' be the
maximum ,-index in w'. If m'  0, take the last symbol in ' with r-index
m', and convert the next symbol (which must be an , - 1) into r. If m' < 0
,
the first symbol in w' must be an r - 1, and this is converted into r. In
either case the result is w, which is therefore uniquely determined by w'
and,. I
(9.6) Let w' = S,_I,r{W). Then w' is compatible with A - J.L if and only ifw is
t'ompatible with A - J.L.
,
Proof. Let w = weT), w' = weT'), where T and T' are arrays of shape
A - J.L. They differ in only one square, say x, which in T is occupied by r
and in T' by , - 1.
Suppos that T is a tableau. If T' is not a tableau there are two
possibilities: either (a) the square y immediately to the left of x in T is
occupied by r, or (b) the square immediately above x is occupied by r-1.
In case (a) the symbol , in square y would have a higher r-index in
weT) than the, in square x, which is impossible. In case (b) the square x
in T will be the left-hand end of a string of say s squares occupied by the
symbol r, and immediately above this string there will be a string of s
squares occupied by the symbol , - 1. It follows that weT) contains a
segment of the form
(, _ 1)s . .. ,s
where the unwritten symbols in between the two strings are all either > r
or < r - 1, and the last r is the one to be replaced by , - 1 to form w'.
But the ,-index of this , is equal to that of the element of w immediately
preceding the first of the string of (, - 1)'s, and this again is impossible.
Hence if T is a tableau, so also is T'.
The reverse implication is proved similarly, using the recipe of (9.5) for
passing back from w' to w. I
Suppose now that the word w has the lattice permutation property with
respect to (1,2,..." - 1) but not with respect to (, - 1, ,), or in other
words that all the's-indices are  0 for 2  s  , - 1 but not for s = r. This
is the only situation in which we shall use the operator S,-1". The effect of

9. THE LITTLEWOOD-RICHARDSON RULE 145
replacing r by r - 1 in W as required by S r- 1, r may destroy the lattice
permutation property with respect to (r - 2, r - 1), i.e. it may produce
some (r - I)-indices equal to + 1. In this case we operate with Sr- 2,r-l to
produce
Sr-2,r(W) = Sr-2,r-1Sr-l,r(W).
At this stage the (r - I)-indices will all be  0, but there may be some
(r - 2)-indices equal to + 1, and so on. Eventually this process will stop,
and we have then say
Sa,r(w) = Sa,a+ 1 ... Sr-l,r( w)
for some a such that 1  a  r - 1, and the word Sa ,( w) again has the
,
lattice property with respect to (1, 2, . . . , r - 1), and maximal r-index strictly
less than that of w.
At this point the following lemma is crucial:
(9.7) If w, Sa,,(w) = w' and Sb,r(W') = wIt all have the lattice property with
respect to (1,2,..., r - 1), then b  a.
Proof. Let w=X1x2X3.... We have to study in detail the process of
passing from w to w'. This starts by applying S r- 1, r' i.e. by replacing the
first symbol r in W with r-index m, where m is the maximum of the
r-indices, by r - 1. Suppose that this happens at xPo. Then for each s  1,
the (r - I)-index of Xs is unaltered if s < Po' and is increased by 1 if
s  Po. The element on which S r- 2, r- 1 operates is therefore in the PIth
placet where PI is the first integer  Po for which xPI has (r - I)-index in
w equal to O. Likewise the element on which Sr-3,r-2 operates is in the
P2 th place, where P2 is the first integer  P 1 for which x P2 has (r - 2)-index
zero, and so on.
In this way we obtain a sequence
Po Pl  ... Pr-a-l
with the property that, for each i  1, xPi is the first element not preceding
Xp for which the (r - i)-index is O. Observe that in w' the element in the
1- I
pith place still has (r - i)-index zero, for each i  1 (though it will no
longer be the first with this property).
Now consider the passage from w' = YIY2Y3". to w". In w' the maxi-
mum r-index is m - 1 (which by assumption is still positive) and occurs
first at say Y qo' where qo < Po. (This is because the ,-index can by its
definition only go up or down in single steps, and therefore the r-index
m - 1 occurs first in W at some element to the left of x ; and the
elements to the left of the Poth are the same in w' as in w,)°In w'the
(r - I)-index of Yp is zero, and is therefore + 1 in S r- 1 r( w'). Hence
1 ,
S'-l,r(W') admits the substitution Sr-2,r-l' which will operate on the

146
I SYMMETRIC FUNCTIONS
element in the qlth place, where ql is the first integer  qo for which the
(r - I)-index of Yq1 in w' is 0, o that qo  ql < Pl. Continuing in this Way
we get a sequence
qo  ql <: q2 <: ...  q,-a-I
with qi<Pi for all iO, and w' admits the operator Sa".
If Sa,,(w') = wIt, then b = a; if not, then Sa,,(w') admits further substi-
tutions Sa-I,a'...' until w" = Sb,,(W') is attained, so that b < a in this
case. In either case we have b  a, as required. I
We shall now describe the algorithm of Littlewood and Robinson which
constructs from a tableau T of shape A - J.L and weight 7T, where A, j.L, 7r
are partitions, a pair (L, M) where L E TabO(A - J.L, v) for some partition
v, and ME Tab(v, 7T).
If A is any array-not necessarily a tableau-and a, r are positive
integers such that a < r, we denote by Ra, reA) the result of raising the
right-hand element of the rth row of A up to the right-hand end of the
ath row.
The algorithm begins with the word w1 = weT) and the array M}
consisting of 7T1 1's in the first row, 7T2 2's in the second row, and so on
(i.e. M I is the unique tableau of shape 7T and weight 7T).
Operate on WI with SI2 until there are no positive 2-indices, and
simultaneously on M1 with R12 the same number of times: say
w2 = S(Wl)'
M2 = R'f'2(M).
Next operate on w2 with S23 or S13 as appropriate until there are no
positive 2- or 3-indices, and simultaneously operate on M2 with R23 or
R13: say
W3 = ... Sa2,3Sat,3(W2),
M3 = ... Ra 3Ra 3(M2)
2, t,
where each ai' a2 is 1 or 2.
Continue in this way until we reach (wI' MI), where I = I( 'IT). Clearly
from our construction WI is a lattice permutation. From (9.6) it follows that
wI is compatible with A - J.L, so that WI = w(L) where L E TabO(A - J.L, v)
for some partition v. Next, it is clear from the construction that at each
stage the length 1;(M,) of the ith row of the array M, is equal to the
multiplicity m;(wr) of the symbol i in the corresponding word w" so that
the final array M = M, has shape v and weight 'IT.
We have to show moreover that MI is a tableau. For this, we shall prove
by induction on r that the first r rows of M, form a tableau. This is clear
if r = 1, so assume that r > 1 and the result is true for r - 1.

9. THE LITTLEWOOD-RICHARDSON RULE 147
Consider the steps that lead from Mr-1 to Mr: we have, say,
Mr==Ra r...Ra r(Mr-l);
m' l'
let us put
Mr-l,; = Ra;,r... Ra1,r(Mr-1)
(1  i  m)
and likewise
W r - 1 ,. = Sa r... Sa r ( Wr - 1 ) ,
, I' l'
where each word wr- 1, i has the lattice property with respect to (1, 2, . . . ,
, - 1). Each array Mr-1,; is obtained from its predecessor Mr-l,;-l (or
M,- 1 if i = 1) by moving up a single symbol r from the r th row to the a i th
roW. By our construction the length lj(Mr-l,i) of the jth row of M,-I,i is
equal to the multiplicity mj(wr-l,i) of j in W,-I,i' for each j  1; and since
each word wr- 1, i has the lattice property with respect to (1,2,.. . , r - 1), it
follows that
ll(Mr-1,;)  ...  1,-I(M,-I,i).
Also, by (9.7), the integers a; satisfy a1  ...  am. It follows that no two
symbols r can appear in the same column at any stage, and consequently
the first r rows of Mr form a tableau.
The algorithm therefore provides a mapping
Tab(,\ - JL, 'IT)  U TabO(,\ - JL, v) X Tab( v, 'IT).
JJ
To complete the proof of (9.4) we have to show that this mapping is a
bijection. For this purpose it is enough to show that, for each r  1, we can
unambiguously trace our steps back from (Wr, Mr) to (W,-l' Mr-1). With
the notation used above, we have
Wr = Sa r'.. Sa, ( Wr - 1 ) ,
m' l'
and the sequence (al,..., am) can be read off from the array Mr, since the
Q; are the indices < r of the rows in which the symbols r are located in
M" arranged in descending order: a1  a2  ...  am (by virtue of (9.7)).
Since by (9.5) each Sa,r is reversible, it follows that (w,-1,Mr-1) is
uniquely determined by (w" Mr). Finally, by (9.6), if w, is compatible with
,\ - J.L, then so also is wr-l' and the proof is complete. Q.E.D.

148
I SYMMETRIC FUNCTIONS
Remark. A lattice permutation w = a1a2.'. aN of weight v may be de..
scribed by a standard tableau T( w) of shape v, in which the symbol r
occurs in the a,th row, for 1  .,  N (the fact that w is lattice ensures that
T( w) is a tableau). Hence the algorithm described above constructs from a
word w a pair of tableaux T(w/) and M1 of the same shape v, the first of
which is standard and the second of weight 7T. It may be verified that this
algorithm coincides with one described by Burge [B9] (see also Gansner
[G 1]).
Notes and references
The Littlewood-Richardson rule (9.2) was first stated, but not proved, in
[LI3] (p. 119). The proof subsequently published by Robinson [R5], and
reproduced in Littlewood's book ([L9], pp. 94-6) is incomplete, and it is
this proof that we have endeavoured to complete.
Complete proofs of the rule first appeared in the 1970s ([87], [T4]).t
Since then, many other formulations, proofs and generalizations have
appeared, some of which are covered by the following references: Berg-
eron and Garsia [B4]; James [J7]; James and Peel [JI0]; James and Kerber
[J9]; Kerov [K8]; Littelmann [L7], [L8]; White [W3]; and Zelevinsky [Z2],
[Z3 ].
t Gordon James [J8] reports that he was once told that 'the Littlewood-Richardson rule
helped to get men on the moon, but it was not proved until after they had got there. The first
part of this story might be an exaggeration.'

APPENDIX A: Polynomial functors
and polynomial representations
1. Introduction
Let k be a field of characteristic 0 and let  denote the category whose
objects are finite-dimensional k-vector spaces and whose morphisms are
k-linear maps. A (covariant) functor F:  -+  will be said to be a
polynomial functor if, for each pair of k-vector spaces X, Y, the mapping
F: Hom(X, Y) -+ Hom(FX,.FY) is a polynomial mapping. This condition
may be expressed as follows:
(1.1) Let fi: X -+ Y (1  i  r) be morphisms in , and let Al"'" A, E k.
Then F(Alfl + ... + A,f,) is a polynomial function of A],..., A" with coeffi-
cients in Hom(FX,.FY) (depending on f1"'" f,).
If F(Alfl + ... + A,t,) is homogeneous of degree n, for all choices of
11'." , f" then F is said to be homogeneous of degree n. For example, the
nth exterior power An and the nth symmetric power sn are homogeneous
polynomial functors of degree n.
Each polynomial functor F is a direct sum EBn ;. 0 Fn' where Fn is
homogeneous of degree n (2). We shall show that each Fn determines a
representation of the symmetric group Sn on a finite-dimensional k-vector
space En' such that
Fn(X) = (En rn)S';
functorially in X, and that Fn t---+ En defines an equivalence of the category
of homogeneous polynomial functors of degree n with the category of
finite-dimensional k[Sn]-modules. In particular, the irreducible polynomial
functors correspond to the irreducible representations of symmetric groups,
hence are indexed by partitions.
The connection with symmetric functions is the following. Let u: km -+
km be a semisimple endomorphism, with eigenvalues At,..., Am (in some
extension of k). Then trace F(u) is a symmetric polynomial function of
AI'.'" Am' say Xm(F)(At,..., Am), where Xm(F) E Am' As m -+ 00, the
Xm(F) determine an element X(F) E A. If F = Fp. is irreducible (where I-L
is a partition), it will appear that X(Fp.) is the Schur function sp.'
Notation. If X E 58 and A E k, we shall denote by Ax (or just A if the
context permits) multiplication by A in X.

150
APPENDIX A: POLYNOMIAL FUNCTORS
2. Homogeneity
Let F be a polynomial functor on , and let A E k. By (1.1), F(Ax) is a
polynomial function of A with coefficients in End(F(X)), say
(2.1) F(Ax) = E Un(X)An.
n>O
Since F«ApJx) = F(Ax J..Lx) = F(Ax)F( #-Lx), we have
E un(X)(AjL)n = ( E Un(X)An)( E un(X)jLn)
nO n>O n>O
for all A, #-L E k, and therefore (because k is an infinite field) Un(X)2 ==
un(X) for all n  0, and um(X)un(X) = 0 if m =F n. Also
E un(X) =F(lx) = IF(x)
n>O
by taking A = 1 in (2.1). It follows that the un(X) determine a direct sum
decomposition
(2.2)
F(X) = EB Fn(X)
n>O
where Fn(X) is the image of un(X): F(X)  F(X). Since F(X) is finite-
dimensional, all but a finite number of the summands Fn(X) in (2.2) will
be zero, for any given X.
Moreover, if f: X  Y is a k-linear map, we have fAx = Ay f for all
A E k, and hence F(f)F( A x) = F( Ay )F(f). From (2.1) it follows that
F(j)un(X) = un(Y)F(f) for all n  0, so that each un is an endomorphism
of the functor F. Hence F(f) defines by restriction k-linear maps
Fn(f): Fn(X)  Fn(Y), and therefore each Fn is a functor, which is clearly
polynomial. Consequently we have a direct decomposition
(2.3)
F = EB Fn
n>O
in which each Fn is a homogeneous polynomial functor of degree n.
Remarks. 1. The direct sum (2.3) may well have infinitely many non-zero
components, although for any given X E  we must have Fn(X) = 0 for
all sufficiently large n. An example is the exterior algebra functor A.
If Fn = 0 for all sufficiently large n, we shall say that F has bounded
degree.
2. Fo is homogeneous of degree 0, so that Fo('\) = 1 for all A E k, and in
particular Fo(O) = Fo(I). It follows that for all morphisms f: X  Y we
have Fo(t) = Fo(O), which is therefore independent of f and is an isomor..
phism of Fo(X) onto Fo(Y). Hence all the objects Fo(X) are canonically
isomorphic.

3. LINEARIZATION
151
More generally, let r be a positive integer and ' = 58 X . .. X  the
category in which the objects are sequences X = (XI'...' X,) of length r
of objects of , and Hom(X, Y) = n; Hom(X;, 1i). As before, a functor
F: '   will be said to be polynomial if F: Hom(X, Y) -+ Hom(FX, FY)
is a polynomial mapping for all X, Y E '. If F is polynomial and (A) =
(At,..., A,) Ek', then F«A)x)=F«A])x1,...,(A,)X) will be a polynomial
function of At,..., A, with coefficients in End(F(X), say
(2.4) F«A)x) = E um....m,(X1,...,X,)A'r.... >"';".
ml,...,m,
Exactly as before, we see that the um....m, are endomorph isms of F, and
that if we denote the image of um....m,(Xt,..., X,) by Fm....m,(Xt,..., X,),
then the Fml...m, are subfunctors of F which give rise to a direct decompo-
sition
(2.5)
F= EB F
ml...m,.
ml,...,m,
Each Fm....m, is homogeneous of multidegree (mt,..., m,), i.e. we have
Fm.... m,(At,. .., A,) = A;". .. . A'.
3. Linearization
Again let F:  -+,  be a polynomial functor. In view of the decomposition
(2.3), we shall assume from now on that F is homogeneous of degree
n> O. The considerations at the end of 2 apply to the functor F': 58n  
defined by F'(Xt,..., Xn) = F(X1 e ... ED Xn), and show that there exists a
direct sum decomposition, functorial in each variable,
F ( Xl ED · · · ED X n) =  F;". .. . m,. ( X t , · · · , Xn)
where the direct sum on the right is over all (m],..., mn) E Nn such that
m] + ... +mn = n.
Our main interest will be in the functor FL..t, the image of the
morphism ul.. .t. For brevity, we shall write L F and v in place of F...I and
ul...1' respectively. We call Lp the linearization of F: it is homogeneous of
degree 1 in each variable.
To recapitulate the definitions of LF and v, let Y=X] ED ... EBXn. Then
there are monomorphisms ia: Xa -+ Y and epimorphisms Pa: Y -+Xa
(1  a  n), satisfying
(3.1)
Paia=1X'
a
P a i {j = 0 if a =1= J3,
E iaPa = 1y.
a
For each >.. = (At,..., An) E kn, let (A)y or (>..) denote the morphism
L AaiaPa: Y -+ Y, so that (>..) acts as scalar multiplication by >"a on the

152
APPENDIX A: POLYNOMIAL FUNCTORS
component Xa. Then v(Xt,..., Xn) is by definition the coefficient of
AI... An in F«A)y), and Lp(XI,..., Xn) is the image of v(X1,..., Xn), and
is a direct summand of F(XI ED  .. ED Xn).
(3.2) Example. If F is the nth exterior power An, we have
F(XI e ... eXn) = EB Am1(X1) ED ... e Amn(Xn)
m}t...,m"
summed over all (m1,..., mn) E Nn such that m1 + ... +mn = n, and hence
Lp(XI,..., Xn) - Xl  ...  Xn.
4. The action of the symmetric group
Let F as before be homogeneous of degree n > 0, and let
L<p>(X) = Lp(X,..., X).
For each element s of the symmetric group Sn' let Sx or s denote the
morphism Eis(a>Pa:xnxn, where Xn=XED...EDX, so that Sx per-
mutes the summands of Xn. For any A = (A1,..., An) E kn we have from
(3.1)
sx(A) = L Aais(a>Pa = (SA). Sx
a
where sA = (As-l(1)'...' As-l(n», and hence
(4.1)
F(s)F«A» = F«sA». F(s).
By picking out the coefficient of A1... An on either side, we see that
(4.2)
F(s)v = vF(s)
from which it follows that F(s) defines by restriction an endomorphism
F(s) of Lc.;>. Explicitly, if
(4.3) j=jx:L<;>(X)-+F(Xn),
q = qx: F(Xn)  L<;>(X)
are the injection and projection associated with the direct summand
L<;>(X) of F(Xn), so that qj = 1 and jq = v, then
(4.4)
F(s) = qF(s)j.
From (4.2) and (4.4) it follows that F(st) = F(s)F(t) for s, t E Sn' so that
s H F(s) is a representation of Sn on the vector space L<;)(X), functorial
in X.

4. THE ACTION OF THE SYMMETRIC GROUP 153
We shall now show that this representation of Sn determines the functor
F up to isomorphism, and more precisely that there exists a functorial
isomorphism of F(X) onto the subspace of Sn-invariants of L<P)(X).
Example. In the example (3.2) we have L<P)(X) = Tn(x), the nth tensor
power of X over k, and the action of Sn on L<;)(X) in this case is given by
F(s)(x1 ... xn) = s(s)xs-J(t) .o. Xs-l(n)
where e(s) is the sign of s E Sn' Hence L<p){X)Sn is the space of skew-
symmetric tensors in Tn(x), which is isomorphic to An(x) since k has
characteristic O.
Let i = Eio:: X --+Xn, p = LPa: xn --+X. Then we have
(4.5)
vF{ip)v = E F(s)v.
SESn
Proof. Consider linear transformations f: Xn --+ xn of the form f =
EalJ3 gaf3iaPf3' with gaf3 E k; F(f) will be a homogeneous polynomial of
degree n in the n2 variables gaP' with coefficients in End{F(Xn» depend-
ing only on X (and F). For each S E Sn, let w., denote the coefficient of
gS(l)l ... gs(n)n in F(f).
We have F(s)v = vF(s)v by (4.2) (since v2 = v), hence F(s)v is the
coefficient of At... An J.Lt · · · ILn in
F«A»F(s )F« /J-» = F«A)S( /J-» = F( E As(a) /J-ais(a)Pa )
a
and therefore F(s)v = ws'
On the other hand, vF(ip)v is the coefficient of At... An ILt . . . ILn in
F«A»F(ip)F«/J-» =F«A)ip(/J-» =F( E A,./J-iaP)
a,13
and this coefficient is clearly
E Ws = E F(s)v.
sE Sn S
We now define two morphisms of functors:
g = qF{i): F --+ L<;),
TI = F(p)j: L<;)  F.
(4.6) We have 'TJg = n! (i.e. scalar multiplication by n!) and 'TJ = Ls e S P(s).
n

154
APPENDIX A: POLYNOMIAL FUNCTORS
Proof. TI'= F(p)jqF(i) = F(p)vF(i) is the coefficient of At...'\ in
F(p)F«A))FCi) = F(p(A)i). Now p(A)i: X -+ Y is scalar multiplicatio by
At + ... + An' so that F(p(A)i) is scalar multiplication by (At + ... + A )"
and the coefficient of At... An is therefore n!. II ,
Next, we have TI = qF(i)F(p)j, so that by (4.3) and (4.5)
jTlq = vF(ip)v = E F(s)v
s
and hence TI = Ls qF(s)j = Ls pes) by (4.4). I
Let L<p>(X)S" denote the subspace of Sn-invariants in L<p>(X). From
(4.6) it follows that u = (n!)-tTI is idempotent, with image L<p>(X)S". Let
B: L<;>(X)s" -+ L<p>(X), 'IT: L<p>(X) -+ L<p)(x)Sn
be the associated injection and projection, so that 'lTB = 1 and B7r = 0'.
From (4.6) we have immediately
(4.7) The morphisms
' = 'IT: F(X) -+ L<p>(X)s",
TI' = TIE: L<p>(x)Sn -+ F(X)
are functorial isomorphisms such that 'TI' = n! and TI'' = n!.
5. Classification of polynomial functors
It follows from (4.7) that every homogeneous polynomial functor of degree
n is of the form X  L(X,..., X)S", where L: n -+  is homogeneous of
degree 1 in each variable. The next step, therefore, is to find all such
functors.
We shall begin with the case n = 1, so that L:  -+  is homogeneous
of degree 1. From (1.1), L is clearly additive: L(/t + f2) = L(f1) + L(!2)'
(5.1) There exists a functorial isomorphism
L(X) =L(k) x.
Proof. For each x E X let e(x): k -+ X denote the mapping A  AX. Let Y
be any k-vector space, and define
t/ix: Hom(L(X), Y) -+ Hom(X,Hom(L(k), Y»)
by t/ix(fXx) = f 0 L(e(x». Clearly t/lx is functorial in X, and to prove (5.1)
it is enough to show that t/ix is an isomorphism.

5. CLASSIFICATION OF POLYNOMIAL FUNCTORS 155
Since L is additive we have L(Xt EB X2) = L(Xt) EB L(X2). It follows
that if r/1X1 and .px2 are isomorphisms, .px1 fJX2 is also an isomorphism.
f{ence it is enough to verify that .pk is an isomorphism, and this is
obvious. I
Now let L: 58" -+ 58 be homogeneous and linear in each variable.
(5.2) There exists a functorial isomorphism
L(Xt,...,Xn) =L(n)(k) Xt  ... Xn
where L(n)(k) = L(k,. .., k).
Proof. By repeated applications of (5.1),
L(Xt,...,Xn) =L(Xt,...,Xn_l,k) Xn
=L(XI,...,Xn_2,k,k) Xn-l Xn
=L(k,...,k) Xt X2  ... Xn.
From (5.2) and (4.7) we have immediately
(5.3) Let F be a homogeneous polynomial functor of degree n. Then there
exists an isomorphism of functors
F(X) = (L<P)(k)  Tn(x»s",
where Tn(x) = X  .. .  X is the nth tensor power 'of x.
Let " denote the category of homogeneous polynomial functors of
degree n, and 58 s the category of finite-dimensional k[Sn]-modules.
"
(5.4) The functors a:  n -+ 58 S , fJ: 58 S -+  n defined by
" "
a(F) = L<P)(k),
fJ(M)(X) = (M  Tn(x»S,.
constitute an equivalence of categories.
Proof. We have {3a == It'} by (5.3), and we have to verify that afJ = I . . If
M E  sand f3(M) == F," then SIt
,.
F(XI ED ... EDXn) = (M  Tn(XI ED ... EDXn»s"

156
APPENDIX A: POLYNOMIAL FUNCTORS
and therefore
LF(X1,...,Xn) = (M@ ( ED XS(l) @ ... @XS(n»))Sn.
seS,.
Hence
af3(M) =L<p)(k) = (Mk[Sn])s" =M. I
It follows that a and f3 establish a one-one correspondence between
the isomorphism classes of homogeneous polynomial functors of degree n
and the isomorphism classes of finite-dimensional k[Sn]-modules. '
In particular, the irreducible polynomial functors (which by (2.3) are
necessarily homogeneous) correspond to the irreducible representations of
the symmetric groups Sn, and are therefore naturally indexed by partitions.
For each partition A of n, let MA be an irreducible k[Sn]-module with
character X A. From (5.4) the irreducible polynomial functor FA indexed by
A is given by
(5.5)
FA(X) = (M)..  Tn(x))s".
Now if G is any finite group and U, V are finite-dimensional k[G].
modules, there is a canonical isomorphism (U.  V)G = Homk[G](U, V),
functorial in both U and V, where U* = Homk(U, k) is the contragredient
of U. In the present context we have M)..* = M>.., and therefore
(5.5')
F>..(X) = Homk[s"](M>.., Tn(x)).
Consider Tn(X) as a k[Sn]-module. Its decomposition into isotypic
components is
Tn(x) = e MA  Homk(s"](M>.., Tn(x))
,\
so that
(5.6)
Tn(x) = E9 M).. FA(X)
A
functorially in X.
6. Polynomial functors and k[Sn]-modules
We shall need the following facts. Let G, H be finite groups, M a
finite-dimensional k[G]-module, and N a finite-dimensional k[H]-
module. Then M  N is a k[ G X H ]-module, and we have
(A)
MG NH = (MN)GXH

6. POLYNOMIAL FUNCTORS AND k[Sn]-MODULES 157
as subspaces of M  N.
Suppose now that H is a subgroup of G, and let indN = k[G] [H) N
be the k[G]-module induced by N, and res:fM the module M regarded as
a k[H]-module by restriction of scalars. Then
(B)
NH = (indN)G.
This is a particular case of Frobenius reciprocity:
Homk[H](resM, N) == Homk[G](M,indN)
in which M is taken to be the trivial one-dimensional k[G]-module.
Suppose again that H is a subgroup of G. Then
(C)
indG (N  resH M) =::: (indG N) M
H G - H ·
For both sides are canonically isomorphic to k[ G] k[H] N k M.
Finally, suppose that H is a normal subgroup of G, that M is a
finite-dimensional k[ G]-module and L a finite-dimensional k[ G / H]-
module, hence a k[ G]-module on which H acts trivially. Then
(D)
(L MH)G/H = (L M)G
as subspaces of L  M.
For L  MH = (L  M)H since H acts trivially on L, hence
(L MH)G/H = «L M)H)G/H = (L M)G.
Now let E, F be homogeneous polynomial functors, of degrees m and n
respectively. Then
E F: XE(X) F(X)
is a homogeneous polynomial functor of degree m + n, hence corresponds
as in 5 to a representation of Sm+n.
Suppose that E = /3(M), F = /3(N) in the notation of (5.4). Then
(E  F)(X) = (M  Tm(x))sm  (N  Tn(x))sn
:::(MNTm+n(x))smxSn by (A)
::: (indsm+n (M  N)  Tm +n(x))Sm+n by (B) and (C)
SmXSn
so that E  F corresponds to the k[Sm+n]-module
(6.1)
M. N = ind:)tsn(M  N)
which we call the induction product of M and N.

158
APPENDIX A: POLYNOMIAL FUNCTORS
Since tensor products are commutative and associative up to isomor_
phism, so is the induction produc.t.
Consider next the composition of polynomial functors. With E, F as
before we have
(E 0 F)(X) = E«N  Tn(x»sn)
= (M  rm«N  rn(X))Sn)) Sm
::: (M  (Tm(N)  Tm,,(X»s:,) Sm by (A)
where S:' = Sn X ... X Sn as a subgroup of Smn. Now the normalizer of S:'
in Smn is the semidirect product S::' X Sm, in which Sm acts by permuting
the factors of S;': this is the wreath product of Sn with Sm, denoted by
Sn  Sm. Using (D) it follows that
(E 0 F)(X) ::: (M  rm(N)  Tmn(x»)S,,-Sm
::: (ind}.-:. Sm (M  Tm(N»  Tm,,(x) )Sm.
by (B) and (C). Hence Eo F corresponds to the k[Smn]-module
(6.2)
M 0 N = indm s (M  Tm(N»),
n m
which we call the composition product or plethysm (Chapter I, fi8) of M
with N. Plethysm is linear in M:
(6.3)
(M1 EDM2) 0 N:: (M1 0 N) ED (M2 0 N)
and distributive over the induction product:
(6.4)
(Mt 0 N) . (M2 0 N) ::: (MI. M2) 0 N.
For the corresponding relation for the functors is
(E1 0 F)  (E2 0 F) = (E1 E2)o F
which is obvious.
7. The characteristic map
If  is any abelian category, let K() denote the Grothendieck group of
&.

7. THE CHARACTERISTIC MAP
159
Let lS denote the category of polynomial functors of bounded degree
(2) on . From (2.3) and (5.4) it follows that t} is abelian and semisimple,
and that
K(t}):: e K(t}n):: e K(s ).
II
n;>O n;>O
Moreover, K( SIt) = Rn in the notation of Chapter I, 7.
The tensor product (6) defines a structure of a commutative, associa-
tive, graded ring with identity element on K(t}). In view of (6.1), this ring
structure agrees with that defined on R = EaRn in Chapter I, 7, so that
we may identify K(tf) with R.
K(lS) also carries a scalar product. If E, F are polynomial functors, then
Hom(E, F) is a finite-dimensional k-vector space, and we define
(E, F) = dimk Hom(E, F).
Again by (5.4), it is clear that this scalar product is the same as that
defined on R in Chapter I, 7.
We shall now give an intrinsic description of the characteristic map
ch: R --+ A, defined in Chapter I, 7. Let F be a polynomial functor on .
For each A = (AI"'" A,) E k', let (A) as before denote the diagonal
endomorphism of k' with eigenvalues ;.1\1"'" Ar' Then trace F«A» is a
polynomial function of At,..., Ar, which is symmetric because by (4.1)
trace F«s A» = trace F(s( A)S-1) = trace F(s )F«A»F(s)-1
= trace F«A»
for all s E Sr' Since the trace is additive, it determines a mapping
Xr: K(tJ) --+ Ar'
namely X,(FX AI' . . . , Ar) = trace F« A»; and since the trace is multiplica-
tive with respect to tensor products, X, is a homomorphism of graded
rings. Moreover, it is clear from the definitions that X, = Pq" 0 Xq for
q  r, in the notation of Chapter I, 2, and hence the X, determine a
homomorphism of graded rings
(7.1)
X: K() --+ A.
To see that this homomorphism coincides with the characteristic map
ch: R  A defined in Chapter I, 7, we need only obselVe that X(An) = en
and that An corresponds to the sign representation Bn of Sn in the,
correspondence (5.4).

160
APPENDIX A: POLYNOMIAL FUNCTORS
Hence from Chapter I, (7.5) and (7.6) it follows that
(7.2) If FA: 58 --+ 58 is the i"educible polynomial functor co"esponding to the
partition A, then X(FA) is the Schur function SA. I
If F is any polynomial functor, the decomposition (2.5) applied to the
functor F': (Xh..., X,)  F(Xt ED ... ED Xr) shows that the eigenvalues of
F« A» are monomials Ail... A';', with corresponding eigenspaces
F, I .. . m,( k, .. · , k), and therefore
Xr(F) = E dim Fl...mr(k,..., k)xrl... x;"'.
mt....,m,
From this and the definition of plethysm in Chapter I, AS, it follows that
(7.3)
X(E 0 F) = X(E) 0 X(F)
for any two polynomial functors E, F.
In particular, if A and J.L are partitions, the functor  0 FJI. is a direct
sum of irreducible functors F'tt' so that in K() we have
FA 0 FJI. = E a'fJl.F",
'1r
with non-negative integral coefficients a'fJl.. By (7.2) and (7.3) it follows that
(7.4)
SA 0 sJI. == E a'fJl.s",
'1r
with coefficients a'fJl.  o.
Example
It follows from 7 that a polynomial functor F is determined up to isomorphism by
its trace X(F). Hence identities in the ring A may be interpreted as statements
about polynomial functors. Consider for example the identities of Chapter I, i5,
Examples 5, 7, and 9. From Example 5(a) we obtain
(1)
Sk(S2V):: E9 FJI.(V)
p.
summed over even partitions JL of 2k, and from Example 5(b)
(2)
Sk( A2V):: EB F.,(V)
II
summed over partitions II of 2k such that II' is even (i.e. all columns of II have
even length). Likewise Example 7 gives
(3)
Sk(VEB A2V) = EB FA(V)
A

8. THE POLYNOMIAL REPRESENTATIONS OF GLm(k) 161
summed over partitions A such that IAI + c(A) = 2k, where c(A) (loc. cit.) is the
number of columns of odd length in A, or equivalently the number of odd parts of
A'. Finally from Example 9(a) and 9(b) we obtain
(4)
tf( f?V) == EB F1T(V)
'If'
summed over partitions 1T = (a] - 1,..., ap - 11 a},..., ap) in Probenius notation,
and
(5)
N'(S2V) == $ Fp(V)
p
summed over partitions p = (at + 1,..., ap + 11 al'...' ap).
8. The polynomial representations of GLm(k)
Let G be any group (finite or not) and let R be a matrix representation of
G of degree d over an algebraically closed field k of characteristic 0, the
representing matrices being R(g) = (Rij(g )), g E G. Thus R determines
d2 functions Rij: G -+ k, called the matrix coefficients of R. If we replace R
by an equivalent representation g...... AR(g)A -] , where A is a fixed matrix,
the space of functions on G spanned by the matrix coefficients is unal-
tered. It follows that if R is reducible, the Rij are linearly dependent over
k, because in an equivalent representation some of them will be zero.
Again, if Rand S are equivalent irreducible representations of G, the
matrix coefficients of S are linearly dependent on those of R.
(8.1) Let R(l), R(2), . .. be a sequence of matrix representations over k of a
group G. Then the following are equivalent:
(i) All the matrix coefficients Rg), RJ)'... are linearly independent;
(ii) The representations R(1), R(2),... are i"educible and pai1Wise inequivalent.
We have just seen that (i) implies (ii). The reverse implication is a
theorem of Probenius and Schur (see [CS], p. 183).
Now let V = km and let G = GL(V) = GLm(k). Let xij (1  i, j  m) be
the coordinate functions on G, so that xij(g) is the (i, j) element of the
matrix g E G. Let
p= EB pn=k[xij:li,jm]
n;>O
be the algebra of polynomial functions of G, where pn consists of the
polynomials in the Xij that are homogeneous of degree n. A matrix
representation R of G is said to be polynomial if its matrix coefficients
are polynomials in the xij. Clearly, each polynomial functor FA such that

162
APPENDIX A: POLYN.OMIAL FUNCTORS
leA)  m (SO that FA(V) =1= 0) gives rise to an equivalence class of polyno..
mial representations RA of G, .in which g e G acts as FJ,.(g) on FA(V).
We shall show that
(8.2) The representations R A such that I( A)  m are inequivalent i"educible
polynomial representations of G, and conversely every i"educible polynomial
representation of G is equivalent to some RA.
Proof. By (7.2), the dimension of FA (V) is equal to SA (x t, . · . , X m) evaluated
at Xt = ... = xm = 1, and hence by Chapter I, (4.3)
(1) dn = E (dim (V»2
I,\I-n
is equal to the coefficient of tn in (1 - t)-m2, so that
(2)
dn = dimk pn.
On the other hand, the decomposition (5.6)
Tn(v)  e MA (V)
I,\I-n
shows that the representation of G on Tn(v) is equivalent to the diagonal
sum of dim MA copies of R A, for each partition A of n of length  m. Now
the matrix coefficients of Tn are the monomials of degree n in the Xij'
hence span Pn. Consequently the Rt also span pn; but from (1) and (2)
above it follows that the total number of these matrix coefficients is
dn = dimk pn. Hence the Rt such that IAI = nand L(A)  m form a k-basis
of pn, and it follows from (8.1) that the RA are irreducible and pailWise
inequivalent.
Finally, if R is any polynomial representation of G, the argument of 2
shows that R is a direct sum of homogeneous polynomial representations.
So if R is irreducible, its matrix coefficients R;j are homogeneous of
degree say n, i.e. Rij epn. Hence by (8.1) R is equivalent to some RA. I
In the course of the above proof we have shown that
(8.3) The matrix coefficients RO, where A ranges over all partitions of length
 m, form a k..basis of P. I
If f E A is any symmetric function, we may regard f as a function on G
by the rule
I(x) = f(E],..., m)
where 1'...' gm are the eigenvalues of x E G. For example, e,(x) =
trace( f{ x) is the sum of the principal r X r minors of the matrix x, and

8. THE POLYNOMIAL REPRESENTATIONS OF GLm(k) 163
hence is a polynomial function of x. Since each f E A is a polynomial in
the e" it follows that f is a polynomial function on G.
(8.4Xi) The character of the representation R'\ of G is the Schur function S,\.
(in A polynomial representation of G is determined up to equivalence by its
/
character.
Proof. (i) From (7.2), s,\(x) is the trace of F,\(x) if x EGis a diagonal
matrix, and hence also if x is diagonalizable. Since the diagonalizable
matrices are Zariski-dense in G (because x is diagonalizable whenever
Vex) =1= 0, where D.(x) is the discriminant of the characteristic polynomial
of x) and both s,\(x) and trace F,\(x) are polynomial functions of x, it
follows that s,\(x) = trace F,\(x) for all x e G.
(ii) This follows from (i), since the Schur functions s,\ such that leA)  m
are linearly independent. I
The group G acts on P (and on each pn) by the rule
(gp)(x) = p(xg)
for peP and g, x E G. Hence
gRt(x) = Rt(xg) = E Rt(x)R(g),
r
so that
gRt = E R(g)Rt.
T
This equation shows that for each i = 1,2,..., d,\, where d,\ = dim F,\(V) is
the degree of R'\, the subspace of P spanned by the R (1 <j < d,\) is an
irreducible G-module affording the representation R'\, and hence is iso-
morphic to F,\(V). Consequently, if P,\ is the subspace of P spanned by all
the Rt, we have
(8.5)
P= ED P,\
,\
where A ranges over all partitions of length  m, and P,\ = F,\(V)dA.
Examples
1. (a) Let A be a partition of n and let M). (A7, Example 15) denote the Specht
module corresponding to A. It has a basis (It) indexed by the standard tableaux t
of shape A, where (Ioe. cit.) It is a product of Vandermonde determinants on the
columns of t, so that
(1)
wIt == e(w)lt
if W E S n lies in the column stabilizer C, of t.

164
APPENDIX A: POL YNO-MIAL FUNCTORS
If v E Sn' then vlt is a Z-linear combination of the Is, say
(2)
vii  E Ps(v)ls
s
summed over standard tableaux s of shape A, and the mapping v .-+ ( Ps(v» is an
irreducible representation of 5 n by integer matrices.
(b) Let X be a k-vector space with basis Xl'.'" Xq. Then Tn(x) has a basis
consisting of the tensors
X'T = X'T(I)  . o.  x'T(n)
where T runs through all mappings [1,n] --. [l,q]. If wE 5n we have w(X'T) =XTW-l.
Hence M).  Tn(x) has as basis the products!, 0xT, where t is a standard
tableau of shape A, and T: [1, n]  [1, q]. Now consider F).(X) = (M). 0 Tn(X))SII:
clearly it will be spanned by the elements
(3)
XI,T= E W(f,XT)= E wft0xTW-lo
weSn weSn
Given If as above, we may choose v E S n so that U = TV - I is an increasing
mapping of [1, n] into [1, q]. We then have
Xt,T = E wit 0Xuvw-l
weSn
= E wvft 0Xuw-l
weSn
and hence by (2) and (3) we have
(4)
Xt,T = E Ps(v )xs, u
.
and therefore F).(X) is spanned by the elements Xt T such that t is a standard
tableau of shape A and T: [1, n] --. [1, q] is an increasig mapping.
(c) We regard a standard tableau t of shape A as a bijective mapping of (the shape
of) A onto [1, n], so that t(i, j) is the integer occupying the square (i, j) E A. Then
T = Tot is a mapping A  [1, q], i.e. it is a filling of the squares of A with the
numbers 1,2,..., q. If If is increasing, then T is increasing (in the weak sense)
along rows and down columns, and we have XI, T = 0 unless T is a (column-strict)
tableau. For if there are two squares a, b in the same column of ,\ such that
T(a) = T(b), i.e. T(t(a» = T(t(b », let w E S n be the transposition that inter-
changes t(a) and t(b). Since w E Ct it follows from (1) above that wI, = -It, and
since also wx'T =xT we have w(f, 0 XT) = -It 0xT, and consequently xt,T = O.
(d) On the other hand, if T = Tot is a tableau, then T is uniquely determined by
T, because the sequence (T(1),..., T(n» is the weight of T (Chapter I, 1). In
general the standard tableau t is not uniquely determined by T: for example, if

8. THE POLYNOMIAL REPRESENTATIONS OF GLm(k) 165
r:;: 2 then t can be either  2 or  3. However, if T -= Tot == T 0 I] with t, t]
standard tableaux, then t 1 == vt for some v e S n such that T = T 0 v, and hence
Xt1,T == E W(!'l XT)
we S"
= E WV(!, XT) =Xt,T
we S"
and we may therefore unambiguously define XT == xt, T when T is a column-strict
tableau.
(e) It follows from (b) and (c) above that FA(X) is spanned by the elements XT'
where T: A --+ [1, q] is a column-strict tableau. But now the dimension of FA(X) is
by (7.2) equal to the Schur function SA evaluated at (lq) == (1,...,1), which by
Chapter I, (5.12) is equal to the number of tableaux T as above. Hence the XT
from a k-basis of FA(X).
2. In continuation of Example 1, let Y be a k-vector space with basis Yl'...' Yp'
and let a: X --+ Y be a k-Iinear mapping, say
p
aXj == E aijY;'
i-1
so that a is represented by the p X q matrix A = (a;j) over k, relative to the given
bases of X and Y. Then a induces Tn( a): Tn(x) --+ Tn(y), given by
(5)
Tn( a )X'T = E auTYu
CT
where atTf' == aU(l)T(l) . . . aU(n)T(n) and in the sum u runs through all mappings
[1, n] --+ [1, p].
If now A is a partition of n, we have a k-Iinear mapping FA(a): FA(X) --+ (y),
say
(6)
FA(a)XT = E aSTYs
S
summed over column-strict tableaux S: A  [1, p]. We shall now compute the
matrix coefficients aST.
If T = Tot as in Example 1, then
FA(a)xT = E weft)  Tn(a)xTW-l
we s"
== E we!,)  E aO',TW-1 Yu
weS" CT
by (3) and (5) above. By replacing u by u W -1 in the inner sum above and
obseIVing that auw-l,TW-1 =aU,T' we obtain
(7)
FA( a)xT == E auTYt,u
CT

166
APPENDIX A: POLYNOMIAL FUNCTORS
summed over 0': [1, n] -+ [1,p]. For each such 0', choose v E Sn such that a :r::
0' V-I is an increasing mapping; then by (4) of Example 1 we have 0
(8)
Y,. U == E Ps(v) Ys. eTo
S
== E Ps(v)Ys
s
summed over standard tableaux s such that S == u 0 0 s is a tableau. From (7) and
(8) we obtain
F).(a)XT= E ps(v)aUOv,TYs
summed over all triples (uo, s, v) such that uo: [1, n]  [1, p] is increasing, s is a
standard tableau of shape A, S == 0'00 S is a column-strict tableau, and v E Sn runs
through a set of representatives of the cosets Gv of the stabilizer G of 0'0 in S".
Hence finally
aST == E Ps(v )aUOV,T
s,v
summed over s, v as above. In particular, this formula shows that aST is a
polynomial in the aij with integer coefficients.
3. As in Chapter I, 7, Example 15, let A be a partition of n and let T be any
numbering of the diagram of A with the numbers 1,2,..., n. Let R (resp. C) be the
subgroup of S n that stabilizes each row (resp. column) of T, and let
e == eT == E e(u)uv E k[Sn],
u,v
summed over (u, v) E ex R. Then (loc. cit.) k[Sn]eT is a minimal left ideal of the
group algebra k[Sn], isomorphic to MA. Deduce from (5.5') that
F).(V) :: eT Tn(v)
as G-modules.
4. Let el'...' em be the standard basis of V == km, so that
m
gej == E gjje;
i-I
if g == (gij) E G. Then T == Tn(v) has a basis consisting of the tensors e., =
eT(l)  ...  eT(n)' where T runs through all mappings of [1, n] into [1, m]. Let (u,v)
be the scalar product on T defined by (eu, eT) = BUT' and for each u E T let
cp == 'Pu: T  pn be the linear mapping defined by
cp(v )(g) == (u, gv)
for vET and g E G. Verify that 'P is a G-module homomorphism and deduce that
(in the notation of Example 3) cp(eT T) is either zero or is a G-submodule of pn
isomorphic to F).(V).

8. THE POLYNOMIAL REPRESENTATIONS OF GLm(k) 167
5. If g = (gij) E G let
g(r) == (g;j)l <i,J<r
for 1  r  m (so that g(m) == g), and for any partition A of length  m let
A(g) = fl det(g(Ai».
i;. 1
Then A is a polynomial function on G. If b == (b;j) EGis upper triangular, we
have
 A ( b) = b tl . . . b  ,
and
 A (gb) = ,\ ( b ' g) ==  A (g ) ,\ ( b)
for all g E G.
Show that the G-submodule of P generated by A is irreducible and isomorphic
to FA(V). (This is a particular case of the construction of Example 4. Take T to be
the standard tableau of shape A in which the numbers 1,2,..., n = I AI occur in
order in successive rows, and let u = e.,. where (T(l),..., T(n» is the weakly
increasing sequence in which i occurs Ai times, for each i = 1,2,..., m. The
subgroup of Sn that fixes u is the row-stabilizer R = S,\ X ... X S,\ of T, so that
1 m
EWER W(U) == ru, where r = IRI = A)! . . . Am!. Hence cp(eT u) is the function on G
whose value at g EGis
r. E s(w)(e,.,gweT) =rA(g)
weC
where C is the column stabilizer of T. Hence A E cp(eT T); since A  0, it follows
that cp(eT T) is irreducible and isomorphic to F,\ (V).)
6. The Schur algebra of degree n of G (or V) is
6n == Endk[s"] Tn(v).
For each g E G we have Tn(g) E 6n, and G acts on E)n by the rule ga = Tn(g)o a
for g E G and a E 6 n.
If a E 6 n, let P a be the function on G defined by
(1)
P a (g) == trace (g a) ·
(a) Show that Pa E pn and that a...... Pa: 6n  pn is an isomorphism of G-mod-
ules. (Relative to the basis (e,.) of Tn(v) (Example 4), each a E 6n is represented
by a matrix (aUT) such that aUW,TW == aUT for all w E S,.o We have trace(gu) ==
EO',T aTU gUT and hence
Pa = E aTUxUT
CT , .,.
where XUT == 0i-l XU(i),T(i) == XUW,TW for w E S,.. It follows that a...... Pa is a linear
isomorphism of 6n onto pn, and it is clear from (1) that hPa = Ph a for h E G.)

168
APPENDIX A: POLYNOMIAL FUNCTORS
(b) The Tn(g), g E G, span (;n as a k-vector space. (Suppose not, then there exists
a ::1= 0 in 6 n such that trace(Tn(g)a) = 0 for all g E G, since the bilinear form
(a, (3)....... trace(af3) on n is nondegenerate. But then Po(g) = 0 for all g E G, so
that Po = 0 and therefore a=- 0.)
Hence the G-submodules of 6 n are precisely the left ideals.
(c) We have
6n == n Endk(F,\(V»)
A
as k-algebras, where the product is taken over all partitions A of n of length  m.
(Use (5.6).)
Notes and references
Practically everything in this Appendix is contained, explicitly or implicitly,
in Schur's thesis [S4] and his subsequent 1927 paper [S6]. We have
assumed throughout that the ground field k has characteristic zero; for a
study of the polynomial representations of GLn(k) when k has positive
characteristic, see [G14], and for polynomial functors over k (or indeed
over any commutative ring) see [,A2].

APPENDIX B:
Characters of wreath products
1. Notation
If G is any finite group, let G* denote the set of conjugacy classes in G,
and let G* denote the set of irreducible complex characters of G. If
c E G * and l' E G* we shall denote by ')'(c) the value of ')' at an element
x E c, and by 'c the order of the centralizer of x in G, so that the number
of elements in c is Icl = IGI/ 'C.
Let R(G) denote the complex vector space spanned by G*, or equiva-
lently the space of complex-valued class functions on G. We have
dim R(G) = IG*I = IG* I. On R(G) we have a hermitian scalar product
1
(1.1) (u,v)G = IGI E u(x)v(x)
xeG
relative to which G* is an orthonormal basis of R( G), i.e. ( {3, 1')0 == 813'"1
for f3, y E G * ·
Later we shall encounter families of partitions indexed by G* and by
G*. In general, if X is a finite set and p = (p(x))xex a family of partitions
indexed by X, or equivalently a mapping p: X --.gD (where gD is the set of
-all partitions), we denote by II pI! the sum
II pI! = E I p(x)1
xeX
and by 9'(X) (resp. gDn(X)) the set of all p: X -+ (resp. the set of all
p: X 9' such that II pI! == n). Finally, if p, u E9'(X), then p U u is the
function x  p(x) U u(x) for x EX.
2. The wreath product G ,..., Sn
::' Let Gn = G x .. . x G be the direct product of n copies of G. The
,.'symmetric group Sn acts on Gn by permuting the factors: s(g],...,gn)=
.jgs-l(l)'" . , gs-l(n»). The wreath product Gn == G f"V Sn is the semidirect prod-
Fuct of Gn with S n defined by this action, that is to say it is the group
\whose underlying set is Gn X Sn, with multiplication defined by (g, s)(h, t)
-_.:;= (g. s(h), st), where g, h E Gn and s, t E Sn' More concretely, the ele-
.'.ments of Gn may be thought of as permutation matrices with entries in G,
.:the matrix corresponding to (g, s) having (i,j) element g; B;,s(j)' where
.. g == (g l' · · · , g n ).

170 APPENDIX B: CHARACTERS OF WREATH PRODUCTS
When n = 1, G1 is just G. When n = 0, Go is the group of one element
The order of Gn is IGln . n! for al n  O. 0
An embedding of Sm X Sn in Sm+n gives rise to an embedding of
Gm X Gn in Gm +n' and any two such embeddings are conjugate in G
'"+110
3. Conjugacy classes and types
Let x = (g, s) E Gn, where g = (gl"'" gn) E on and S E S". The permuta.
tion s can be written as a product of disjoint cycles: if z = (iti2... i,) is one
of these cycles, the element gj,g;,-l'" g;1 EGis determined up to conju.
gacy in G by g and z, and is called the cycle-product of  corresponding to
the cycle z. For each conjuga(,j' class c E G * and each integer r  1, let
m,(c) denote the number of r-cycles in s whose cycle-product lies in c. In
this way each element xEGn detennines an array (m,(c)),;>I,CEG of
non-negative integers such that E"c rm,(c) = n. Equivalently, if p(c). de.
notes the partition having m,(c) parts equal to r, for each r  1, then
p = (p(c))c E G. is a partition-valued function on G* such that II pI! = n, i.e.
p EfJlJn(G *) in the notation introduced in 1. This function p is called the
type of x = (g, s) E Gn. Note that the cycle-type of s in S" is u=
UCEG. p(c).
We shall show that two elements of Gn are conjugate if and only if they
have the same type.
(a) If WE S", let w = (1, w) E Gn. If x = (g, s) E Gn we have wxW-1 =
(wg, wsw-1). If z = (i1 ... i,) is a cycle in s, as above, then wzw-1 =
(w(i1)...w(i,)) is a cycle of WSW-I; moreover (wg)w(io)=gio' so that the
cycle-product is unaltered, and therefore x and wxw-1 have the same type.
(b) Let h = (hI'.'" hn) E Gn. Then hxh-1 = (hgs(h-1), s), and hgs(h-1)
has ith component h;g;h;-\(i)' Hence the cycle-product of hxh-1 corre.
sponding to the cycle z in s is
(h. g.h-:-t )(h. g. h-:-l )...(h. g. h-:-l) =h.(g. .. g. )h-:-t
" I, ',-I ',-I ',-1 ',-2 '1 '1 I, ., I, · 'I I,
which is conjugate in G to gi ... gi . It follows that x and hxh -1 have the
, 1
same type.
(c) From (a) and (b) it follows that conjugate elements in Gn have the same
type. Conversely, suppose that x = (g, s) and y = (h, t) in Gn have the
same type. Then s, t E Sn have the same (,j'cle-type u = (Ut, U2"") and
are therefore conjugate in Sn' Hence by conjugating y by a suitable__
permutation w E Sn we may assume that t = s; then both x and y lie in
the same subgroup Gu = GU1 X GU2 X ... of Gn, and it is enough to show
that they are conjugate in Gu' This effectively reduces the question to the

4. THE ALGEBRA R(G)
171
ase wbere s  S n is an n-cycle, say s == (12. . . n), and the products g =
lIgll-1 · · · g!? h !!-1I r -1 · · · hI re conjugate in G. Now ch?ose ull E G
such that g == unhun , and define "1'...' Un-1 e G successively by the
equations
- h -1 - h -1 - h -1
gl - U1 lUn ,g2 - U2 2Ul ,...,gn-l - Un-1 n-1Un-2.
!\ simple calculation now shows that uyu-1 ==X, where U == (Ut,..., un);
hence x and yare conjugate in Gn, and tbe proof is complete.
(d) We shall now compute the order of the centralizer in Gn of an element
x(g,s) of type pEfPn(G*). First, the number of possibilities for seSn
is
(1)
n!/n rm'.m,!
r;.l
where m, == Ee m,(c). Next, for each such s and each r  1 there are
m,!/f}. m,(c)!
ways of distributing the m, r-cycles among the conjugacy classes of G.
Finally, for each cycle z == (i1 . .. i,) in s such that g;,... git E e, there are
(2)
'(3)
IGI,-l .Iel == IGI' / 'e
choices for (gi , . . · ,-g; ). From (1), (2), and (3) it follows that the number of
1 ,
,elements in Gn of type p is
n ! IGln
n n J'el( p(e»
Z p( e) · !,,4
C C
nd hence that the order of the centralizer in Gn of an element of type p
'is
(3.1)
z == n z r1(p(e»
p p(e)!=.e.
ceO.
..' The algebra R( G)
let
R(G) = e R(Gn).
n;.O
!e define a multiplication on R(G) as follows. Let u e R(Gm), v e R(Gn),
d embed Gm X Gn in Gm+n. Then U X v is an element of R(Gm X Gn),
!;d we define
.(4J)
uv == indGGm+IIG (u X v)
mX "

172 APPENDIX B: CHARACTERS OF WREATH PRODUCTS
which is an element of R(Gm+n). Thus we have defined a bilinear multipli..
cation R(Gm) X R(Gn) -+ R(Gm.+n), and just as in Chapter I, 7 (which
deals with the case G = (1}) it is not difficult to verify that with this
multiplication R(G) is a commutative, associative, graded C-algebra with
identity element.
In addition, R( G) carries a hermitian scalar product; if u, U E R( G), say
u = EUn and v = LUn with un' Un E R(Gn), we define
(4.2)
(U,v)= E (un,vn)Gn
n>O
where the scalar product on the right is that defined by (1.1), with G
replacing G. 11
5. The algebra A(G)
Let Pr(e) (r  1, e E G *) be independent indeterminates over C, and let
A(G) = C[p,(e): r  1, e E G*].
For each e E G* we may think of Pr(e) as the rth power sum in a
sequence of variables Xc = (Xic)i> 1. We assign degree r to Pr(e), and then
A( G) is a graded C-algebra.
If u = (UI, U2'.") is any partition, let pu(e) = PU1(C)PU2(C)... . If now p
is any partition-valued function on G *, we define
(5.1)
Pp = n pp(C)(c).
ceG.
Clearly the Pp form a C-basis of A(G). If fE A(G), say f= LpfpPp (where
all but a finite number of the coefficients fp E C are zero), let
(5.2)
1= Eippp
p
so that in particular Pp = Pp'
Next, we define a hermitian scalar product on A(G) as follows: if
f = Lp fpPp' g = Ep gpPp then
(5.3) (f,g)=EfpgpZp
p
with Zp given by (3.1). Equivalently,
(5.3')
( Pp, Pu) = 8pu Z p ·
Finally, let '¥: Gm --. A(G) be the mapping defined by 'I'(x) = Pp if

6. THE CHARACTERISTIC MAP
173
X E Gm has type p. If also y E Gn has type u, then x Xy E Gm X Gn is
well-defined up to conjugacy in Gm+n, and has type p U u, so that
(5.4)
'I'(x X y) = 'I'(x )'I'(y).
6. The characteristic map
This is a C-linear mapping
ch: R(G)  A(G)
defined as follows: if IE R( Gn) then
ch(/) = <I, '1')0"
1
= IG I E f(x)'I'(x).
n xE G,.
(6.1)
If fp is the value of I at elements of type p, then
(6.2) ch(/) = E Z;llpPp'
p
In particular, if 'Pp is the characteristic function of the set of elements
x E G n of type p, we have ch( 'Pp) = z; 1 Pp' from which it follows that ch is
a linear isomorphism.
Let I, g E Gn. Then from (5.3) and (6.2)
<ch(/),ch(g» = E Z;l/pKp
p
= (/,g)G"
from which it follows that ch is an isometry for the scalar products on
R(G) and A(G) defined by (4.2) and (5.3).
If U E R(Gm), v E R(Gn), then by (4.2) and (6.1)
ch(uv) = (indg:-+x"G,,(u X v), 'I')Gm+"
= (u X v, 'l'IGmxG")c;",XG,,
by Frobenius reciprocity,
= (u, 'I')G (v, 'I')G
m "
= ch(u)ch(v).
by (5.4)

174 APPENDIX B: CHARACTERS OF WREATH PRODUCTS
Hence we have proved that
(6.3) ch: R(G)  A{G) is an isometric isomorphism of graded C-algebras.
7. Change of variables
For each irreducible character 'Y E G* and each r  1 let
(7.1)
p,('Y)= E ,;ly(e)p,(c)
CEO.
so that (by the orthogonality of the characters of G) we have
(7.1') p,(e) = E 'Y(e)p,( 'Y) = E 'Y(e)p,( 'Y).
-yE G* "IE G*
The Pre 'Y) are algebraically independent and generate A(G) as C-algebra.
From (7.1') we have
E ,;lp,{e)Pr(e)= E E ,;l'Y(e)f3(e)Pr('Y)Pr(f3)
C c P,"I
= E (f3, 'Y )G Pr ( 'Y)  Pr ( f3 )
(3,"I
(7.2)
= EPr('Y) p,{'Y)
-y
in A{G)  A(G).
We may regard Pre y) as the rth power sum of a new sequence of
variables Y'Y = (Yi'Y)i h and we may then define, for example, Schur
functions Sp.( y) = Sp.(Y'Y) for any partition J.L, and more generally
(7.3)
SA = n SA('Y)( Y)
-yE 0*
for any A E9'(G*).
(7.4) (SA)'\E .9'(G.) is an orthonormal basis of A(G).
Proof. In view of the definition (5.3') of the scalar product on A(G), it is
enough to show that
(1)
ES).SA= EZ;lppPp,
,\ p

8. THE CHARACTERS '1",)'
175
where the sum on the left is over all A E[P(G*), and that on the right is
over all p E9'(G*). Now the left-hand side of (1) is equal to
n (ES#L(i')S#L(i'»)= n (exp E P'('Y)p,('Y»)
-yE G. p. 'YE G. r> 1 r
(2)
=exp( E  -E P'('Y)p,('Y»).
r> 1 r 'YEO.
On the other hand, the right-hand side of (1) is by virtue of (3.1) equal
to
n (E C;I(O')Z;lpO'(C)  Pcr(C») = n (exp E  l;lp,(c)  p,(C»)
ceO. (1' CEO. r> 1 r
(3) = exp( E  E l;lp,(C)  P,(C»)
r> 1 r cEG.
Now (7.2) shows that (2) and (3) are equal. I
8. The characters 71n,-y
Let E1 be a G-module with character i' E G*. The group Gn acts on the
nth tensor power Tn(Ey) = Ey  ... :y as follows: if "1""'"n E Ey and
(g, s) E Gn, where g - tg1,..., gn) E G and S E Sn, then
(g, s)(ut  ...  un) = g]Us-l(l)  ...  gnus-1(n).
We wish to compute t'character 71n = 71n( i') of this representation of
Gn. First, if x E G, and y E Gn-" so that x acts on the first r factors of
Tn(E'Y) and y on the last n - r factors, it is clear that
(8.1)
71n (x X y ) = 71, ( x ) 71n - r (y ) ·
Hence it is enough to compute 71n(g, s) when g E G" and S is an n-cycle,
say s = (12... n). For this purpose let e1,..., ed be a basis of £y and let
gej = E aij(g )ej
i
so that g  a(g) = (aij(g» is the matrix representation of G defined by
this basis. Then we have
(g,s)(ej.  ... ej.) =gtej. g2ej.  ... gnej.
1 II II 1 ,.-1

176 APPENDIX B: CHARACTERS OF WREATH PRODUCTS
in which the coefficient of eh  ...  ej" is
(1) a jd" (g 1 )aj2j.(g 2) · · · aj"jn _.(gn).
To obtain the trace we must sum (1) over all jl'...' jn' which gives
l1n(g, s) = trace a{gn)a(gn-I).'. a(gl)
= trace a{ g n g n - 1 · · · g 1) = Y ( c)
if the cycle-product gngn-l ...81 lies in the conjugacy class c. From (8.1) it
now follows that if x E Gn has type p, then
(8.2)
l1n( Y )(x) = n Y{ c )/( p(c».
c
We now calculate:
E ch(l1n(Y» = E Z;1 n Y(C)/(p(c»PP(C)(c)
n;;.O p ceG.
- n (E (,; ly{C)/(CT) z; IpCT(C))
ceG. (1'
= n exp( E 2.c;lY(C)Pr(C»)
cEG. r;;.l r
= exp( E 2. E C;lY(c)Pr(c»)
r;;.l r cEG.
= exp( E 2. pr< Y ») by (7.1)
r;;. 1 r
= E hn(y).
n;;.O
Hence
(8.3)
ch{ l1n{ y») = hn{ Y)
for all /' E G* and n  o.
9. The irreducible characters of Gn
For each partition J.L of m and each y E G*, let
(9.1)
x II- ( y) = de t ( 111L; - i + j ( Y ) ) .

9. THE IRREDUCIBLE CHARACTERS OF Gn 177
This is an alternating sum of induction products of characters, hence is a
(perhaps virtual) character of Gm. From (8.3) and (6.3) we have
(9.2) ch( X(y)) = det(hi-i+j(Y)) ==s}£(y).
Next, for each A E9'n(G*), define
(9.3)
XA= n XA(-Y)(y)
-yE G.
which for the same reason is a character of Gne From (9.2) it follows that
(9.4)
ch(XA) = n SA(-Y)( y) = SA;
-yE G.
and since ch is an isometry (6.3) it follows from (7.4) that
(XA, X,." )G" = (SA'S,.,,) = 6AJ.L
for all A, J.L E9'n(G*). Hence the XA are, up to sign, distinct irreducible
characters of Gn; and since L9'n(G*) 1= L9'n(G*)1 = I(Gn) *1, there are as
many of them as there are conjugacy classes in Gn, so that they are all the
irreducible characters of G n. It remains to settle the question of sign,
which we shall do by computing the degree of each character XA.
Let X: denote the value of XA at elements of Gn of type pE9'n(G*).
From (9.4) and (6.2) we have
(9.5)
SA = E Z;lXpApp
p
or equivalently
(9.5')
(9.5" )
XpA == (SA' Pp),
Pp== EXpASA = Ex:sA.
A A
Let Co E G * be the class consisting of the identity element. The type of the
identity element of Gn is p where p(co) = (In), p(c) = 0 if c :;: Co. Hence
Pp = Pl(CO)n, and since PI(CO) = E-y d-yPI( y), where d-y = y(co) is the degree
of 'Y, it follows from (9.5') that the degree of X A is equal to the coefficient
of SA in (L-y d.."PI( Y ))n, i.e. to the coefficient of O-y SA(..,,)( y) in
n ! 1 A( .." )1
n IA(y)lt !J(dyP1(Y»)
-y

178 APPENDIX B: CHARACTERS .OF WREATH PRODUCTS
which by Chapter I, (7.6) is equal to
(9.6)
n! n ( dA(Y)I/h('\( ;' )))
'Y
where h( A( ;,)) is the product of the hook-lengths of the partition A( 1').
Since this number is positive, it follows that XA (and not -XA) is an
irreducible character. So, finally,
(9.7) The i"educible complex characters of Gn = G,.,. Sn are the XA (A E
gtJn(G*)) defined by (9.3), and the value of xA at elements of type p E9'n(G *)
IS
X:=(SA'Pp).
Moreover the degree of XA is given by (9.6).
Example
The simplest nontrivial case of this theory is that in which the group G has two
elements :i: 1. Then Gn is the group of signed permutation matrices, of order 2nn!,
or equivalently the hyperoctahedral group of rank n. The conjugacy classes and the
irreducible characters of Gn are each indexed by pairs of partitions (A, IL) such that
IAI + I ILl = n.
If the power sums corresponding to the identity (resp. non-identity) conjugacy
class are denoted by p,(a) (resp. p,(b », and those corresponding to the trivia]
(resp. nontrivial) character by p,(x) (resp. p,(y», the change of variables formula
(7.1') reads
p,(a) = p,(x) + p,(y),
p,(b) == p,(x) - p,{y)
and the formula (9.5") reads
pp(a)pu(b) == E X:':SA(X)Sp,(Y)
A, JJ.
where X': is the value of the character indexed by (A, IL) at the class indexed by
(p, u) (where IAI + I ILl = I pi + 10"1 == n).
Notes and references
The characters of the wreath products G fIOcJ Sn' where G is any finite
group, were first worked out by W. Specht in his dissertation [S17], and our
account does not differ materially from his. See also [M3].

II
HALL POLYNOMIALS
1. Finite a-modules
Let 0 be a (commutative) discrete valuation ring, tJ its maximal ideal,
k = o/tJ the residue field. Later we shall require k to be a finite field, but
for the present this restriction is unnecessary. We shall be concerned with
finite o-modules M, that is to say, modules M which possess a finite
composition series, or equivalently finitely-generated o-modules M such
that tJ'M = 0 for some r  O. If k is finite, the finite o-modules are
precisely those which have a finite number of elements.
Two examples to bear in mind are
(1.1) Example. Let p be a prime number, M a finite abelian p-group.
Then p'M = 0 for large r, so that M may be regarded as a module over
the ring Z/p'Z for all large r, and hence as a module over the ring 0 = Zp
of p-adic integers. The residue field is k = Fp.
(1.2) Example. Let k be a field, M a finite-dimensional vector space over
k, and let T be a nilpotent endomorphism of M. Then M may be regarded
as a k[t]-module, where t is an indeterminate, by defining tx = Tx for all
x EM. Since T is nilpotent we have t'M = 0 for all large r, and hence M
may be regarded as a module over the power series ring 0 = k[[t]], which is
a discrete valuation ring with residue field k.
Remarks. 1. In both these examples the ring 0 is a complete discrete
valuation ring. In general, if M is a finite o-module as at the beginning, we
have tJ'M = 0 for all sufficiently large r, so that M is an o/tJ'-module and
hence a module over the -adic completion a of 0, which has the same
residue field k as o. Hence there would be no loss of generality in
assuming at the outset that 0 is complete.
2. Suppose now that k is finite. The complete discrete valuation rings with
finite residue field are precisely the rings of integers of p-adic fields
(Bourbaki, Alg. Comm., Chapter VI, 9), and a tJ-adic field K is either a
finite extension of the field Qp of p-adic numbers (if char. K = 0) or is a
field of formal power series k«t» over a finite field (if char. K> 0). The
two examples (1.1) and (1.2) (with k finite) are therefore typical.
3. The results of this chapter will all be valid under the wider hypothesis
that 0 is the ring of integers (i.e. the unique maximal order) of a division
algebra of finite rank over a tJ-adic field (Deuring, Algebren, Ch. VI,  11).

180
II HALL POLYNOMIALS
Since 0 is a principal ideal domain, every finitely generated o-module is
a direct sum of cyclic o-modules. -Por a finite o-module M, this means that
M has a direct sum decomposition of the form
(1.3)
r
M =:. EB o/P A,
i-I
where the Ai are positive integers, which we may assume are arranged in
descending order: Al  A2  ...  A, > O. In other words, A = (AI' · · · , Ar) is
a partition.
(1.4) Let ILi = dimk(i-lM/iM). Then IL = (ILl' IL2'.") is the conjugate
of the partition A.
Proof. Let xj be a generaor of the summand 0 I Ai in (1.3), ad let 1T' be
a generator of . Then tJ,-lM is generated by those of the 7r,-lXj which
do not vanish, i.e. those for which Aj  i. Hence ILj is equal to the number
of indices j such that Aj  i, and therefore ILi = Xi. I
From (1.4) it follows that the- partition A is determined uniquely by the
module M, and we call A the type of M. Clearly two finite o-modules are
isomorphic if and only if they have the same type, and every partition A
occurs as a type. If A is the type of M, then I AI = E Ai is the length I(M) of
M, i.e. the length of a composition series of M. The length is an additive
function of M: this means that if
o --+ M' --+ M --+ M" --+ 0
is a short exact sequence of finite o-modules, then
I(M') -1(M) + I(M") = O.
If N is a submodule of M, the cotype of N in M is defined to be the
type of MIN.
Cyclic and elementary modules
A finite o-module M is cyclic (i.e. generated by one element) if and only if
its type is a partition (r) consisting of a single part r = I(M), and M is
elementary (i.e.  M = 0) if and only if the type of M is (1'). If M is
elementary of type (1'), then M is a vector space over k, and I(M) =
dimkM = r.
Duality
Let 7r be a generator of the maximal ideal . If m  n, multiplication by

1. FINITE o-MODULES
181
11',,-m is an injective a-homomorphism of a/vm into a/n. Let E denote
the direct limit:
E= lim a/n.
-.
Then E is an injective a-module containing 0 Iv = k, and is 'the' injective
envelope of k, i.e. the smallest injective a-module which contains k as a
submodule.
If now M is any finite a-module, the dual of M is defined to be
"
M=Homo(M,E).
M is a finite a-module isomohic to M, hence of the same type as M. (To
see this, observe that M.-+ M commutes with direct sums; hence it is
"
enough to check that M == M when M is cyclic (and finite), which is easy.)
Since E is injective, an exact sequence
o N -+M -+MIN -+ 0
(where N is a submodule of M) gives rise to an exact sequence
" "
D +- N +- M +- (MINT+- D
" " "
and (M IN) is the annihilator NO of N in M, i.e. the set of all gEM

such that g(N) = O. The natural mapping M -+ M is an isomorphism for
all finite a-modules M, and identifies N with Noo. Hence
"
(1.5) N  NO is a one-one con-espondence between the submodules of M, M
respectively, which maps the set of all N c M of type v and cotype J..L onto the
set of all NO eM of type J.L and cotype v. I
Automorphisms
Suppose that the residue field k is finite, with q elements. If M is a finite
o-module and x is a non-zero element of M, we shall say that x has height
r if .p'x = D and '-lX =1= O. The zero element of M is assigned height O.
We denote by M, the submodule of M consisting of elements of height
r, so that M, is the annihilator of V' in M.
(1.6) The number of automorphisms of a finite a-module M of type A is
aA (q) = qlAI+ 2n(.\) n 'Pml(.\)(q-l),
i 1
where as usual 'Pm(t) = (1 - tX1 - t2)... (1 - tm).
The number of automorphisms of M is equal to the number of se-
qUences (Xl"'" X,) of elements of M such that Xi has height Ai(l  i <; r)

182
II HALL POLYNOMIALS
and M is the direct sum of the cyclic submodules ox;. To enumerate such
sequences we shall use the folowing lemma:
(1.7) Let N be a submodule of M, generated by elements of height  r, and
let x EM. Then the following conditions on x are equivalent:
(i) x has height r and ox n N = 0;
(ii) x E Mr - (Mr- t + N,J.
Moreover the number of x E M satisfying these equivalent conditions is
(1.8)
if v is the type of N.
Proof. If x satisfies (i), clearly x E Mr. If X E Mr-I + N" then 0 =1= p'-lX C
Nr eN, so that 0 x n N =1= 0, contrary to assumption. Conversely, if x
satisfies (ii), it is clear that height (x) = r. If 0 x n N =1= 0, then for Some
m < r we shall have pm X eN, and therefore p r- 1 X is contained in the
socle Nt of N. Since N is generated by elements of height  r, it follows
that '- t X =  ,-1 Y for some yEN; hence x - y E Mr-1 and therefore
x E (M,_ 1 + N) n M, = M'-1 + N,.
We have M/Mr = p'M, so that
q A'l + ... + A (1 _ q v - A', )
r
l(Mr) = l(M) -l(prM) = E l(pi-1M/piM) == Xl + ... + A
i-I
by (1.4). Also
I(Mr-t +Nr) = l(M,_1) + l«M,_l +N,)/M,-1)
= l(M,_I) + l(Nr/N,-1)
= X. + ... + X,_ 1 + v:
which proves (1.8). I
The number of automorphisms of M is therefore the product of the
numbers (1.8) for r = A., A2,..., where v = (A.,..., Ak-1) if r = Ak. It is
not hard to see that this product is equal to aA(q) as defined in (1.6).
Example
Let M, N be finite o-modules of types A, I-L respectively. Then AI (lj N has type
A U p., and M (lj N, Hom(M, N), Tort(M, N), and Ext1(M, N) all have type A X IL
(Chapter I,  1).
2. The Hall algebra
In this section the residue field k of 0 is assumed to be finite.

2. THE HALL ALGEBRA
183
Let A, J.L(1),..., J.L(r) be partitions, and let M be a finite o-module of type
,\. We define
Gj1)...(,)(o)
to be the number of chains of submodules of M:
M = M ° -=> M 1 -=> ... -=> Mr = 0
such that Mi-1/Mi has type J.L(i), for 1  i  r. In particular, G:v(o) is the
number of submodules N of M which have type v and cotype J.L. Since
Z(M) = l(M/N) + I(N), it is clear that
(2.1) G;v(o) = 0 unless IAI = I J.LI + I vi.
Philip Hall had the idea of using the numbers G:v(o) as the multiplica-
tion constants of a ring, as follows. Let H = H(o) be a free Z-module on a
basis (uA) indexed by all partitions A. Define a product in H by the rule
uuv = E G:v( 0 )u)..
,\
By (2.1) the sum on the right has only finitely many non-zero terms.
(2.2) H(o) is a commutative and associative ring with identity element.
Proof. The identity element is uo, where 0 is the empty partition. Associa-
tivity follows from the fact that the coefficient of u). in either u(uvup) or
(uru,Jup is just G;"P" Commutativity follows from (1.5). which shows that
Gp.v == Gv. I
The ring H( 0) is the Hall algebra of o.
(2.3) The ring H( 0) is generated (as Z-algebra) by the elements u(1') (r  1),
and they are algebraically independent over Z.
Proof. For convenience let us write vr in place of U(l')' and for any
partition A consider the product
VA' = V)., VA' ... VA'
1 2 ,
where X = (Xl'...' Xs) is as usual the conjugate of A. This product VA' will
be a linear combination of the u, say
(1)
VA' = E aAu
JL

184
II HALL POLYNOMIALS
in which the coefficient a>..p. is by definition equal to the number of chains
(2)
M=Mo M1::> ... Ms = 0
in a fixed finite o-module M of type J..L, such that Mi-1/Mi is of type (1 Ai),
i.e. elementary of length Xi' for 1 < i < s. If such a chain (2) exists (that is
if a>..p. =1= 0) we must have VMi-1 eM; (1 <: i <: s) and therefore ViM cM.
for 1 < i < s. Hence I
I(M/ViM)  I(M/Mi)
which by virtue of (1.4) gives the inequality
JL1 + ... + p!;  Xl + ... + Xi
for 1 <: i <: s. Hence Jl  X and therefore by Chapter I, (1.11), J..L  A.
Moreover, the same reasoning shows that if J..L =,\ there is only one
possible chain (2), namely Mi = ViM.
Consequently we have a>..p. = 0 unless J..L <:'\, and aAA = 1. In other
words, the matrix (a >..p.) is strictly upper unitriangular (Chapter I, 6), and
so the equations (1) can be solved to give the u JI, as integral linear
combinations of the VA" Hence the VA' form a Z-basis of H( 0), which
proves (2.3). I
From (2.3) it follows that the Hall algebra H( 0) is isomorphic to the
ring A of symmetric functions (Chapter I). The obvious choice of isomor-
phism would be that which takes each U(l') to the rth elementary symmet-
ric function er; however, as we shall see in the next chapter, a more
intelligent choice is to map U(l') to Q-r(r-1)/2er, where q is the number of
elements in the residue field of o. Thus each generator u,\ of H(o) is
mapped to a symmetric function. We shall identify and study these
symmetric functions in the next chapter; the remainder of the present
chapter will be devoted to computing the structure constants G;v(o).
3. The LR-sequence of a submodule
Let T be a tableau (Chapter I, 1) of shape ,\ - J..L and weight v =
(vI"'" Vr). Then T determines (and is determined by) a sequence of
partitions
s = (,\(0), ,\(1) t..., ,\(r»)
such that ,\(0) = J..L, A(T) = '\, and A(i)::> A(i-1) for 1  i <: r, by the condition
that ,\(i) - ,\(i-1) is the skew diagram consisting of the squars occupied by
the symbol i in T (and hence is a horizontal strip, because T is a tableau).

3. THE LR-SEQUENCE OF A SUBMODULE 185
A sequence of partitions S as above will be called a LR-sequence of type
(J.L, v; ,\) if . .
(LR1) ,\(0) = J..t, A(r) == '\, and A(')::> A(,-I) for 1  i  r;
(LR2) A(i) - ,\(i-l) is a horizontal strip of length Vi' for 1  i <: r. (These
twO conditions ensure that S determines a tableau T.)
(LR3) The word w(T) obtained by reading T from right to left in
successive rows, starting at the top, is a lattice permutation (Chapter I, 9).
For (LR3) to be satisfied it is necessary and sufficient that, for i  1 and
k  0, the number of symbols i in the first k rows of T should be not less
than the number of symbols i + 1 in the first k + 1 rows of T. In other
words, a condition equivalent to (LR3) is
k k+l
(LR3') E ('\Ji) - AY-I»)  E (AY+ 1) - Ai»)
j-l j-l
for all i  1 and k  o.
We shall show in this section that every submodule N of a finite
o-module M gives rise to a LR-sequence of type ( p:, v'; X), where '\, J.L, II
are the types of M, M I N, and N respectively. Before we come to the
proof, a few lemmas are required. We do not need to assume in this
section that the residue field of 0 is finite.
(3.1) Let M be a finite o-module of type A, and let N be a submodule of type
v and cotype J..L in M. Then J.L C A and V C A.
Proof. Since
'pi-l(MIN) pi-1M + N , pi-1M
trJi(MIN) ::: piM + N = pi-1M n (trJiM + N)
and since also '!fJi-1M n (piM + N)::> piM, it follows that
l(trJi-I(M/N)/'!fJi(M/N)) <: 1(pi-1M/piM)
and hence that JJ1  Xi by (1.4). Consequently J.L c A. By duality (1.5), it
follows that v c,\ also. I
Let M be an o-module of type '\, N an elementary submodule of M.
Then t,J N = 0, so that N c S where
S = {x EM: t,Jx = O}
is the socle of M, i.e. the unique largest elementary submodule of M.
(3.2) The type of MIS is ,\ = (AI - 1, A2 - 1,...).

186
II HALL POLYNOMIALS
Proof. If M= E90/A" then clearly S= EBA,-I/AI, whence M/S
E9 0 /  A,- 1 . I
(3.3) Let M be a finite o-module of type A, and N an elementary submodule
of M, of cotype J.L. Then A - J.L is a vertical strip (i.e. Ai - J.Li = 0 or 1 for
each i).
Proof. We have NcS, hence M/S:::.(M/N)/(S/N), and therefore Xc
J.L C A by (3.1) and (3.2). Hence
o  A. - II..  A. - ,\. = 1
, r-, , I
and therefore A - J.L is a vertical strip. I
Notice that if J.L c A then A - J.L is a vertical strip if and only if A c J.L.
(3.4) Let M be a finite o-module of type A, and let N be a submodule of M
of type v and cotype J.L. For each i  0, let A(i) be the cotype of iN. Then th
sequence
S(N) = (A(O), , A(I), ,..., A(r)/)
(where prN= 0) is an LR-sequence of type (Ii, v'; X).
Proof. Clearly A(O) = J.L and A(r) = A, and A(i)::> A(i-l) by (3.1) applied to the
module M/piN and the submodule i-lN/iN. Hence (LR1) is satisfied.
Since p;- 1 N / iN is an elementary o-module, it follows from (3.3) that
A(i) - A(i-l) is a vertical strip, and hence that AO)' - A(i-1)1 is a horizontal
strip, of length equal to l(i-IN /piN) = vI (by (1.4». Hence (LR2) is
satisfied.
As to (LR3'), we have
Ai)' = l(pj-l(M/iN) /j(M/piN»
(again by (1.4), so that
k
E Ai)' = 1«M/iN)/k(M/iN» = l(M/(pkM + piN»
j-l
and therefore
k
E (>..Y)' - >..i-l)') = I(Vki)
j-l
where Vki = (pkM + i-lN)/(pkM + 'piN). Likewise
k+l
E (Ai+l)1 - Ai)/) = 1(Vk+1,i+l).
j-l

4. THE HALL POLYNOMIAL
187
Since multiplication by a generator of p induces a homomorphism of Vki
onto Vk+ 1,i+ l' it follows that l(Vki)  l(Vk+ 1,;+ 1)' and hence (LR3') is
satisfied. I
Example
Let V be an n-dimensional vector space over an algebraically closed field k. A flag
in V is a sequence F == (Vo, VI'...' Jt;.) of subspaces of V, such that 0 = Vo e V1 e
. .. c v;. == V and dim  == i for 0  i  n. Let X denote the set of all flags in V. The
group G = GL(V) acts transitively on X, so that X may be identified with G IB,
where B is the subgroup which fixes a given flag, and therefore X is a (non-singu-
lar projective) algebraic variety, the flag manifold of V.
Now let u e G be a unipotent endomorphism of V. Then as in (1.2) V becomes
a k[t ]-module of finite length, with t acting on V as the nilpotent endomorphism
u - 1. Let ,\ be the type of V, so that ,\ is a partition of n which describes the
Jordan canonical form of u, and let X).. eX be the set of all flags F eX fixed by u.
These flags F are the composition series of the k[t]-module V. The set X).. is a
closed subvariety of X.
For each F == (VO, VI'. . . , Jt;.) e X),., let ,\(i) be the cotype of the submodule Jt;. _;
of V. Then by (3.1) we have
o == ,\(0) e ,\(1) e ... e A (n) := ,\
and I,\(i) - ,\(i-11 == 1 for 1  i  n, so that F determines in this way a standard
tableau of T of shape '\. Hence we have a partition of X).. into subsets XT indexed
by the standard taJ>leaux T of shape '\.
These subsets X T have the following properties (Spaltenstein [S16]):
(a) XT is a smooth irreducible locally closed subvariety of XA.
(b) dim XT = n('\).
(c) XT is a disjoint union U j..1XT,j such that each XT.j is isomorphic to an
affine space and Uk... jXT,k is closed in XT, for j == 1,,2,..., m.
From these results it follows that the closures XT of the XT are the irreducible
components of X).., and all have the same dimension n( ,\). The number of
irreducible components is therefore equal to the degree of the irreducible charac-
ter XA of Sn (Chapter I, 7).
If k contains the finite field Fq of q elements, the number XA(q) of Fq-rational
points of X),. is equal to Q(1n)(q) (Chapter III, 7).
More general results concerning the partial flags may be found in [H8], [S14].
4. The Hall polynomial
In this section we shall compute the structure constants G:v( 0) of the Hall
algebra. (The residue field of 0 is assumed to be finite, as in 2.) Let S be
an LR -sequence of type (Ii, v'; A.'), and let M be a finite o-module of
type A. Denote by Gs(o) the number of submodules N of M whose

188
II HALL POLYNOMIALS
associated LR-sequence S(N) is S. By (3.4), each such N has type v and
cotype JL.
Let q denote the number of elements in the residue field of 0, and
recall that n( A) = E(i - 1) Ai' for any partition A. Then:
(4.1) For each LR-sequence S of type (Ii, v'; X), there exists a monic
polynomial gs(t) E Z[t] of degree n(A) - n( JL) - n(v), independent of 0,
such that
gs(q) = Gs(o)
(In other words, Gs(o) is a 'polynomial in q'.)
Now define, for any three partitions A, JL, v
(4.2)
g;JI(t) = E gs(t)
s
summed over all LR -sequences S of type (Ii, v'; X). This polynomial is ';
the Hall polynomial corresponding to A, JL, v. Recall from Chapter I (5.
and 9) that C;JI denotes the coefficient of the Schur function SA in the
h A A' A' ·
product sp.sJl; t at cJLJI = cp.'JI'; and that cp.'JI' IS the number of LR-sequences
of type ( P:, v'; X). Then from (4.1) it follows that
(4.3) (i) If C;JI = 0, the Hall polynomial gJI(t) is identically zero. (In parlicu-c,
lar, g;JI(t) = 0 unless IAI = I JLI + Ivl and J.L, v CA.) .il
(ii) If C JI =1= 0, then g ;JI(t) has degree n( A) - n( JL) - n( v) and leading coeffi-::
. A c.'
Clent c . 1
p.JI .C
(iii) In either case, G:JI(o) = g:JI(q).
(iv) g;JI(t) = g(t).
The only point that requires comment is (iv). From (2.2) we have.
G:JI( 0) = GJI( 0) for all 0, hence g;JI(q) = g;p.(q) for all prime-powers q':
and so gJI(t) = g:p.(t). I
The starting point of the proof of (4.1) is the following proposition:
(4.4) Let M be a finite o-module of type ,\ and let N be an elementary:_
submodule of cotype a in M (so that ,\ - a is a vertical strip, by (3.3)). Let'PrI
be a partition such that a c (3 C A and let Ha/3A(O) denote the number oP1
submodules PeN of cotype (3 in M. Then :,:j'
Hal3A (0) = hal3A (q)
;;'J

4. THE HALL POLYNOMIAL
189
where ha/3A(t) E Z[t] is the polynomial
(4.4.1)
ha13A(t) =td(a,I3,A)n[A- /3i, /3! - a;](t-1),
i;. 1
in which
(4.4.2)
d( a, /3, A) = E ( /3, - a,)( As - /3s)
r<s
[r+s]
and [r, s] is the Gaussian polynomial r (Chapter I, 2, Example 1) if
r,s  0, and is zero othe1Wise.
Proof. Let (J = A - /3, 'P = /3 - a. Also let N; = N n ;M. Since
iM/N; = (iM +N)/N= i(M/N)
we have
1(N;) = 1(t,JiM) -I(i(M/N»
=  (A'. - a)
i..J J J
j>i
by (1.4), or equivalently
(1)
ni = 1(N;) = E (OJ + 'Pj).
j>i
Now let P be a submodule of N, of cotype /3 in M, and let Pi =
Pn 'piM = P n N;. Then
P /Pi =- (P + t,JiM)/iM
and therefore
I(Pi-I) -I(Pi) = I(P /Pi) -1(P /Pi-1)
= 1«P + t,JiM)/t,JiM) -J«P + i-lM)/'pi-lM)
= l(t,Ji-lM/iM) -[«P + pi-1M)/(P + iM»
= X. - fJ! = O
I 1J, ,
by (1.4). Conversely, if P is a submodule of N such that
(2)
I(Pi-I/Pi) = 8;
(i  1)

190
II HALL POLY-NOMIALS
then the preceding calculation shows that P has cotype {3 in M.
Suppose that i  1 and Pi re given. We ask for the number of
submodules Pi _ I of N; _ I which satisfy (2) and
(3)
Pi-InN;=Pi.
The number of sequences x = (Xl'...' X9j) in N;-l which are linearly
independent modulo N; is
(4) (qn'-l _ qnt)(qnt-l _ qn,+l)... (qn,-t _ qnt+9i-I).
For each such sequence x, the submodule Pi-1 generated by Pi and .r
satisfies (2) and (3). Conversely, any such Pi-1 can be obtained in this way
from any sequence x of 8; elements of Pi-1 which are linearly indepen-
dent modulo Pi. The number of such sequences is
(5) (qPI-l - qP')(qPI-l - qPI+l)... (qP'-l _ qPt+9:-1)
where
(6)
Pi = l(Pi) = E 8;
j>;
by virtue of (2). So the number of submodules Pi _ 1 of N; _ 1 which satisfy
(2) and (3) is the quotient of (4) by (5), namely
q 8;(n,-1-p'-1)[ 6; t 'Pi ](q-l ).
Taking the product of these for all i;> 1, and obseIVing that n; - 1 - Pi -1 =
Lj> i'P; (from (1) and (6» we obtain
Hal3A ( 0) = qd n [ Xi - PI , fJi - a;]( q -1 )
i> 1
where
d = E 6;'P;.
i<j
Now 8; is the number of squares of the skew diagram 0 = A - P in the
i th column, and L j> i 'P; is the number of squares of cp = fJ - a in the same
or later colums, hence in higher rows. It follows that
d = E 'PrOs =d(a, Pt A)
r<s
which completes the proof of (4.4). I

4. THE HALL POLYNOMIAL
191
Suppose in particular that N in (4.4) is the socle S of M, so that a == A
by (3.2). Then HXA (0) is the number of elementary submodules P of
cotype f3 in M, so that
HAA(O) = O;(1m)( 0)
where m = I A - f31 = 191. In this case we have 'P = f3 - A, so that 'Pi =
Pi - Ai + 1 = 1 - 9;, and the exponent d = d(A, f3, A) is therefore
d = E (1 - 9,)9$
r<s
= E(l-O,)Os= E 0s- E 9,Os.
r<s
r<$
r<s
because 0, = 6,2 for all r. Since EO, = m we have
(4.5)
'" (J 6 = 1m2 - 1 '" (J2 = !m(m - 1)
I..J 's 2 2I..J, 2
r<s
and hence
d = E (8 -1)9s - m(m - 1) = n(A) - n( f3) - n(lm).
So from (4.4) we have the formula
Gt(1m)(O) = qn(A)-n()-n(1m)n [Ai - (3i, f3! - Ai+1](q-1)
i> 1
(because Xi = )(i+ 1 for all i  1). This is valid for any partitions A, {3, where
m = I AI - I {31. (If A - (3 is not a vertical strip, both sides are zero.)
Equivalently, if we define '
(4.6)
g A (t) = tn(A)- n( )- n(lm) n [A'. - tJ.! Q! -)(. ](t-1)
(1m) , tJ" tJ, ,+1
i> 1
then we have g:(1m)(q) = GI3lm)(O).
The next stage in the proof of (4.1) is
(4.7) Let R = (a', f3', X) be a three-term LR-sequence. Then there exists a
monic polynomial FR(t) E Z[t], depending only on R, of degree n( (3)-
n(a) - (;) where n = I f3 - al. with the following property: if M is a finite
o-module of type A, and P is an elementary submodule of cotype (3 in M, then
the number of submodules N of cotype a in M such that tJ N = P is equal to
FR(q ).

192
II HALL POLYNOMIALS
(Observe that when {3 = A (so that A - a is a vertical strip) we have
P = 0, so that N is elementary nd therefore FR(t) = g ;(1n)(t) in this case.)
Proof. Let Q be a submodule of P, and let 'Y be the cotype of Q in M, so
that we have a C f3 C 'Y C A.
Let f(P, Q) (resp. g(P, Q» denote the number of submodules N of
cotype a in M such that N:) P :) Q :) P N (resp. N:) P ::> Q =  N). The
number we want to calculate is g(P, P). Now f(P, Q) is easily obtained, as
we shall see in a moment, and then g(P, Q) can be obtained by MObius
inversion [R8].
First of all let us calculate f(P, Q). We have
N::> P :) Q :) P N  N:) P and N /Q is elementary
 S :)N:)P, where S/Q is the socle of M/Q
N/PcS/P.
By (3.2) S has cotype .y in M, and so by (4.4) (applied to the module MjP
and its elementary submodule SIP) we have
(i)
f(P,Q) =H:Yal3(o)
if .y C a, i.e. if l' - a is a vertical strip, and f(P, Q) = 0 otherwise.
Now it is clear that
(ii)
f(P,Q) = E g(P,R)
RcQ
summed over all submodules (i.e. vector subspaces) R of Q. The equations
(ii) for fixed P and varying Q C P can be solved by Mobius inversion: the
solu tion is
g(P, Q) = E f(P, R)J.L(R, Q)
RcQ
where J.L is the Mobius function on the lattice of subspaces of the vector
space P, which (loc. cit.) is given by
J.L(R, Q) = ( _l)d qd(d-l)/2
where d = dimk(Q/R). Hence in particular we have
(iii)
g(P,P)= E (_1)mqm(m-l)/2f(P,R)
RcP
where m = dimk(P /R) and the summation is over all subspaces R of p,

4. THE HALL POLYNOMIAL
193
Now for each partition 8 such that f3 C SeA, the number of R c P of
cotype 8 in M is H138A (0). Hence from (i) and (Hi) it follows that
(iv) g(P, P) = r, ( _1)m qm(m -1)/2 H138A (0 ) HSal3 (0)
8
where the sum is over all 8 such that f3 C 8 C A and such that 8 - a is a
vertical strip, and m = 18 - ,B I. SO if we define the polynomial FR(t) by
(4.8) FR(t) = r,( -1)mtm(m-l)/2hI3SA(t)hSal3(t)
8
summed over partitions 8 as above, then it follows from (iv) and (4.4) that
g(P, P) = FR(q).
To complete the proof it remains to be shown that this polynomial FR(t)
is monic of degree n( M - n(a) - (;). where n = I,B - a I.
The degree of the summand corresponding to 8 in (4.8) is by (4.4.2)
equal to
d = tm(m - 1) + r, (cp, t/1s + (1 - 8, - cp,) 8s)'
r<as
where 8 =,B - a, cp = 8 - f3 and t/1= A - 8, and m = 1 cpl. Each of 8, cp, t/1
is a vertical strip, so that each 0" cp" and t/1, is 0 or 1: in particular we have
!m(m - 1) = L, < s cp, 'Ps - m by (4.5), so that
(v)
d = r, 'P,( CPs + t/1s - 8s) + r, (1 - 8,) 8s -I cpl.
r:(s
r:(s
Now (a I, (3 I, A') is an LR-sequence. This implies that, for each r  1, the
number of squares of ,B I - a I = 0' in the columns with indices  r is not
less than the number of squares of A' - {3' in the same columns: that is to
say, we have
(vi)
r, Os  r, (CPs + «/1s)
sr s,>,
for each r  1. From (v) and (vi) it follows that
d  r, (1 - 8,) 8s
r<s
with equality if and only if cp = 0, i.e. if and only if 8 = f3. Hence the
dominant term of the sum (4.8) is hal3(t) = g!(1n)(t), which by (4.6) is
monic of degree n( M - n( a) - ( ; ). This completes the proof. I
The proof of (4.1) can now be rapidly completed. Let M be an
o...module of type A, and let S = (A(O)"..., A(')/) be an LR-sequence of type
(J.L', JI '; A'). Let N be a submodule of M such that S(N) = S, and let

194
II HALL POLYNOMIALS
NI == 'p N. Then clearly S(N1) = (>..(1)',..., >..(r),) == S. say. Conversely, if We
are given a submodule N1 of M such that S(N1) == St, the number of
submodules N such that S(N)"': Sand tJ N = Nt is equal to PRt(q), where
RI == (>..(0)', >..(1)', >..(2),), as we see by (4.7) applied to the module M/VN1
and its elementary submodule Nt/tJNt. Consequently
Gs(o) == Gs.(o) .PRt(q)
and therefore
r
Gs(o) = nFR(q)
. 1 I
,-
where Ri == (>..(i-l)" >..(i)', >..0+1),). (When i == r we take >..(r+l)' = A(r)" so
that as remarked earlier FR,(q) == g;(]tft)(q), where a == >..(r-l), and m ==
I>.. - al.)
Hence if we define
(4.9)
r
gs(t) == n FR(t)
. 1 I
,-
we have gs(q) = Gs(o), and by (4.7) the polynomial gs(t) is monic of
degree
i ( n( ,\(i)) - nC\(i-l)) - ( i)) = n('\) - n( IL) - n( v). Q.E.D.
The proof just given provides an explicit (if complicated) expression for
g;v(t), via the formulas (4.2), (4.4), (4.8), and (4.9). If a, b, c, N are
non-negative integers such that b  C, let us define
tP(a, h, c; N; t) = ro ( _l)r tNr+r(r+ l)/2[ ][: =]
where [] and [: = ] are Gaussian polynomials in t. (The sum on the
right of (4.10) is finite, since the term written is zero as soon as r>
min{a, b).)
With this notation we have
(4.10)
(4.11) Let S == (a(O), a (1), ..., a(r» be an LR-sequence of type (JL', v'; A').
Then
g (t) == tn(A)-n(p.)-n(v) n <l>(aoo b.. COO' N.' t-I)
S 'J ' '1' I) ' I) '
i-;.j-;. 1

4. THE HALL POLYNOMIAL
195
where
a.. == aj+ 1) - aj) b == a(j-l) - a(j)
IJ ,+1 1+1' ;j; i+I'
C..==aj)-aj) N (a a )
'J , 1+1' ij= I...J h-l,j-l- h,j ·
h<i
(Thus aij is the number of symbols j + 1 in the (i + l)t row of the tableau
defined by S; cij is the number of columns of a(J) of length i, and
c.. - b;j = a;-I,j-l is the number of j's in the ith row of S. Finally, N;j is
the excess of the number of j's in the first i rows of S over the number of
°i/ + 1)'s in the first i + 1 rows, and hence is  0 by virtue of the lattice
permutation property.)
Proof. With the notation of (4.7) let R == (a', 13', '\') be a three-term
LR-sequence, and let
- ,\' n'
a; - ; + 1 - #-Ii + l'
b - , n' IJ.' IJ.'
; - a; - #-Ii + l' c; == /Ji - /Ji+ 1.
As in (4.8) let 8 be a partition such that 8 cae f3 C 8 C '\, where 8; == 8:+ 1
for all i  1, and let 'i == 8:+ 1 - f3:+ l' so that m == 18 - 131 == E'i. (Since R is
an LR-sequence we have f3 == 81 == '\'1' so that ao == '0 == 0; also 0  'i  ai
(Ci and 0 'i  bi  ci.)
We shall use (4:8) to calculate FR(t). From (4.4) we have
(1) tm(m-l)/2h{38A(t)hSa13(t) == tA n [](t-l)[: =](t-I),
i;;.1 I , I
in which the exponent of t is
(2) A = H Erj)2 - t E rj + E {(OJ - rj)rj + (cj - b)(bj - rj»)
i<j
== -! E 'j('j + 1) - E 'j + E (Ci - bi)bj
i<j
where N. = E.< .(c. - a. - b.). Moreover,
J 1 J I , I
E (cj - bj)bj == E (p: - ai)(aj - 13J+l)
i<j i<j
which reduces to
(3)
n( {3) - n(a) - !s(s -1)
where s == I f3 - a I.

196
II HALL POLYNOMIALS
From (1), (2), (3), and (4.8) we obtain
(4)
F (t) = tn( 13)- n(a)-s(s ....1)/2 n <I>(a. b. C.' N.' t-])
R " " "" ·
i> 1
Since by (4.9)
r
gS(t) = nFR(t),
. 1 J
J-
where Rj = (a(j-l), a(j), aU+1)), the formula (4) leads directly to (4.11).
(Since (j + l)'s appear for the first time in the (j + l)th row of S, it follows
that aij = 0 if i <j, whence the restriction i  j in the product on the
right-hand side of (4.11). I
As an example (which we shall make use of later) we shall compute
g ;v(t) when v = (r) is a partition with only one part r = I A - J.LI. (The case
when v = (lr) is given by (4.6).) First we have
(4.12) g;(r)(t) = 0 unless A - J.L is a horizontal strip of length r.
Proof. Let M be a finite o-module of type A, and let N be a cyclic
submodule of cotype J.L in M. Let Ni = N n iM. Then since iMIN; 
(iM + N)/N::: i(MIN) we have
Z(N 11N) = A'. - II.
 - , , r-,
and since P-l = peN n pi-1M) eN n p;M = Ni, it follows that
o  Xi - JLi  I{Ni-l/pNi-l)'
Since N is cyclic, so is Ni, and hence l(Ni-l/ Ni-l)  1. Hence Xi -Ili == 0
or 1 for each i  1, and therefore A - J.L is a horizontal strip. So unless
A - J.L is a horizontal strip we have g;(r)(q) = 0 for all prime-powers q, and
therefore g;(r)(t) = O. I
Remark. Alternatively (and more rapidly) (4.12) follows from (4.3): we
have g;(r)(t) = 0 unless c:(r) =1= 0, i.e. unless SA occurs in the product s",h"
which by Chapter I, (5.16) requires A - J.L to be a horizontal strip.
Assume then that A - J.L is a horizontal r-strip. Then A' - Ji is a vertical
r-strip, and there is only one LR-sequence S = (a(O), a(1),..., a(r») of type
( Ji, (Ir); A'), namely that obtained by filling in the squares of the vertical
strip X - J.L' consecutively, starting at the top. For this S we have aij = 0 or
1 for each pair (i, j), and also cij - b;j = ai-1,j- J = 0 or.1, since there is at
most one square of A' - Ji in any row. Moreover, if aij = 1 (so that there is

4. THE HALL POLYNOMIAL
197
a square labelled j + 1 in the (i + 1)th row) we have N;j == O. Hence, from
the definition (4.10) of cI> we find that <I>(aij, bij, cij; N;j; t) is equal to
1
1- t
(1 - tCi}) /(1 - t)
if ai-1,j-l == aij == 0 or 1;
if a i-I, j - 1 == 0, a i j == 1;
if a i-I, j - 1 = 1, a i j == O.
Moreover, in the latter case (i.e. when there is a square labelled j in the
ith row, but none labelled j in the row below), Cij is the number of
columns of A' of length i, that is Cij = m;('\). Putting these facts together,
we obtain
(4.13) Let u == ,\ - J.L be a horizontal strip of total length r, and let I be the set
of integers i  1 such that u;' = 1 and u/+ 1 == o. Then
tn(A)- n( p,)
gA (t) = n (1 - t-mi(A»
#L( r) 1 t - 1
- iel
where m;( A) is the multiplicity of i in '\.
Examples
1. Let {x; t)r = (1 - xXI - xt)... (1 - xtr-1) for all r  O. Then <I>(a, b, c; N; t) is
equal to the coefficient of xb in {xtN+1;t)a/(X;t)c_b+l. (Use Chapter I, 2,
Example 3.) Deduce that if N = c - a - b we have
[c - a]
4>(a, b, c; N; t) = b
This applies in particular to the terms corresponding to i = j in the product (4.11),
since N. = c.. - a.. - b...
II II " "
2. Let
 (a; t )r( /3; t)r
2({)t(a,/3;'Y;t,x)==  ( .t)(t./) xr
r> 0 Y, r , r
in the standard notation for basic hypergeometric series (see e.g. [05]). In this
notation we have
(a,b,c;N;t) = [:]2fPt(ta,tb;tC;rt,tN)
so that gs(t) is a product of Gaussian polynomials and terminating 2 ({)l'S.
Notes and references
The contents of 1 and 2, together with the theorem (4.3), are due to
Philip Hall, who did not publish anything more than a summary of his

198
II HALL POLYNOMIALS
theory [H3]. The contents of 3, in particular (3.4), are due to J. A. Green
[G13]. Theorem (4.1) was first. proved by T. Klein [KID], a student of
Green. OUf proof is different from hers.
It should be pointed out that Hall was in fact anticipated by more than
half a century by E. Steinitz, who in 1900 defined what we have called the
Hall polynomials and the Hall algebra, recognized their connection with
Schur functions, and conjectured Hall's theorem (4.3). Steinitz's note [S26]
is a summary of a lecture given at the annual meeting of the Deutsche
Mathematiker-Vereinigung in Aachen in 1900; it gives neither proofs Dor
indications of method, and remained forgotten until brought to light by K.
Johnsen in 1982 [J13].
For generalizations of the notion of the Hall algebra, see the papers by
C. M. Ringel [R2], [R3], [R4].

APPENDIX*: Another proof of
Hall's theorem
This Appendix is devoted to a simple proof of the following slightly
weakened form of (4.3) (we shall freely use the terminology and notation
of Chapter II):
(AZ.1) For any three partitions A, p." v there exists a polynomial g;1I (t) E Z[ t ]
such that Gp.>;'(o) = g;v(q). Moreover, g:v(t) has degree  n('\) - n( p.,)-
n(v), and the coefficient of tn(A)-n(p.)-n(v) is equal to c;v.
Our proof does not use the Littlewood-Richardson rule (Chapter I,
(9.2)). Therefore, combining it with the arguments of 3 and 4 of Chapter
II we shall obtain a new proof of the Littlewood-Richardson rule which
makes the appearance of the lattice permutations more natural.
Our proof is based on a combinatorial interpretation of the coefficients
aAJI- from the formula (1) of Chapter II, 2. We need some definitions.
A composition is a sequence a = (at, a2'. . .) of non-negative integers
with only a finite number of non-zero terms. So a partition is a composi-
tion such that 'at  a2  ... . The group Sot) of finite permutations of
N+ = {I, 2,3,.. .} acts on compositions by wa = (aw-1(1>' aw-l(2)' ...). We
shall write a  f3 if a and f3 are conjugate under this action: clearly, each
S-orbit contains exactly one partition.
As well as partitions the compositions have diagrams: the diagram of a
is formally defined as the set {(i, j) E Z2: 1 j  ail and is graphically
displayed as the set of squares containing ai squares in the ith row.
If a and /3 are two compositions, an an-oy of shape a and weight f3 is a
numbering of the squares of the diagram of a by positive integers such
that for any i  1 there are /3; squares numbered by i (more formally, an
array is a function A: a  N + such that Card A - 1 (i) = /3; for all i  1; it
will be convenient for us to assume that A is defined on all of N+ X N+ so
that A(i, j) = + 00 when j > ai)' For any x = (i, j) E N+ X N+ let x -+ =
(i,j + 1). An array a will be called row-ordered (resp. row-strict) if
A(x -+) A(x) (resp. A(x -+) > A(x)) for any x E a. Likewise we define
column-ordered and column-strict arrays.
· This Appendix was written by A Zelevinsky for the Russian version of the first edition of
this book, and is reproduced here (in English, for the reader's convenience) with his
permission.

200 APPENDIX: ANOTHER PROOF OF HALL'S THEOREM
We define a total ordering on N+X N+ by
(i, j)  (i' , j')  ei tQer j < j' , or j = j' and i > i' .
Finally, for each row-strict array A of shape a we let
d(A) = Card{(x, y) E a X a: Y <L x, A(x) <A(y) <A(x -t)}.
(AZ.2) Let A, IL be partitions and a, {3 compositions such that a '""-J J.L and
f3  A'. Then the coefficient a p. (Chapter II, 2) is equal to
a p. = E qd(A)
A
summed over all row-strict a"ays A of shape a and weight f3.
Before proving (AZ.2) we shall derive from it (AZ.I). From (AZ.2) the
polynomial
E td(A) E Z[t],
A
where the sum is the same as in (AZ.2), depends only on A and J.L; we
denote it by ap.(t).
(AZ.3) (a) Q).p.(t) has coefficients  o.
(b) a).p.(I) is equal to the number of (0, I)-matrices with row sums !-Lt, J.L2,."
and column sums A'l' X2, . .. .
(c) ap.(t) = 0 unless !-L  A. Moreover, a(t) = 1.
Proof. (a) is evident. To prove (b) it is enough to establish a bijection
between row-strict arrays of shape !-L and weight A', and (0, I)-matrices
with row sums J.LI, !-L2,... and column sums Xl' A'2'... . To do this we assign
to an array A the matrix (c;j)' where C;j = 1 if the ith row of A contains j,
and cij = 0 othelWise; this is the required bijection. Finally, (c) follows at
once from (a), (b), and the Gale-Ryser theorem (Chapter I, 7, Example
9). I
From (AZ.3Xc), (aAp.(t)) is a strictly upper unitriangular matrix over Z[t].
Hence it is invertible, and its inverse has the same form. Therefore, the
entries in the transition matrices between the bases (v x) and (u) in R(o)
are integer polynomials in q. Since the multiplication law in H(o) with
respect to the basis (v,) does not depend on q, it follows that the structure
constants in the basis (u) are integer polynomials in q. This proves the
existence of the Hall polynomials g;.,(t) E Z[t]. It remains to find their
degrees and leading coefficients.
(AZ.4) ap.(t) has degree  n( J.L) - n(A), and the coefficient of tn(p.)-n() is
equal to Kp!' (Chapter I, 5).

APPENDIX: ANOTHER PROOF OF HALL'S THEOREM 201
This follows at once from the next combinatorial lemma:
(p;l.5) For any row-strict array A of shape JJ., and weight A' let d(A) denote
the number of pairs (x, y) E JJ., X I-L such that y lies above x (in the same
column) anA(x) <A(y) <A(x -'). Then
(a) (A) + d(A) == n( J-L) - n( A);
(b) d(A) = 0 if and only if A is column-ordered.
Proof. (a) Let
D(A) = ((x,y) E I-L X I-L: y <L x, A(x) A(y) <A(x)},
N( I-L) = {(x, y) E J.L X I-L: y lies above x},
D(A) == {(x, y) E N( I-L): A(x) <A(y) <A(x )};
then we have
Card D(A) == d(A) + E (A';) = d(A) + n(A),
i> 1 2
Card N( I-L) == E ( I-L) == n( I-L),
i> 1 2
and
Card D(A) = d(A).
We shall construct a mapping cp: D(A) --+ N( I-L). Let (x, y) E D(A) and
suppose that x == (it, jl)' Y = (i2, j2)' Clearly it:;: i2; let i = max(it, i2),
i' = min(it, i2), j == min(jl' j2) and finally cp(x, y) == «i, j), (i', j». The _ defi-
nitions readily imply that cp is a bijection of D(A) onto N( /-L) - D(A),
whence our assertion.
(b) The 'if part is evident. Now suppose that A is not column-ordered:
that is. to say, there are x = (i, j), Y == (i', j) E I-L such that i > i' and
A(x) <A(y). Choose such a pair with least possible j; clearly it belongs to
D(A), hence d(A):;: O. I
Now let o').p.(t) == tn(P.)-n(A)aAp.(t-t). From (AZ.4) and (AZ.5) we have
(AZ.6) o').p,(t) = E td(A)
A
summed over all row-strict arrays of shape I-L and weight A'; in particular,
QAp.(t) E Z[t]. Moreover, o'Ap,(O) = Kp.')." I
Now consider the ring A[t] = AZ[t] of polynomials in t with coefficients
from A; we shall write down the elements of A[t] as P(x; t). Clearly, A[t]

202 APPENDIX: ANOTHER PROOF OF HALL'S THEOREM
is a free Z[t]-module with basis (eA). From (AZ.3Xc), the matrix (aAJ.t(t)) is
strictly upper unitriangular. Therefore, the equations
(1)
eA, = E QAP.(t)PP.(x; t)
JL
uniquely determine the elements Pp.(x; t) E A[t], and they form a Z[t].
basis of A[t]. Let f;v(t) be the structure constants of A[t] with respect to
this basis, i.e.
Pp. (x; t) PI' (x; t) = E f;v (t ) PA (x; t);
A
it is clear that I;v(t) E Z[t] for all A, JL, v.
(AZ.7Xa) g;JJ(t) = tn(A)-n(p.)-n(v)f;lI(t-1).
(b) PA(x; 0) = SA(X). In particular, 1;11(0) = c;JJ are the structure constants of
A in the basis (SA)'
Proof (a) follows at once from the definitions. To prove (b) it is enough to
observe that
M(e, s)Xp. = KP!A' = aAp.(O)
by (AZ.6) and the results of Chapter I, 6. I
From (AZ.7), g;v(t) has degree  n(A) - n( JL) - n(v), and the coeffi-
cient of this power of t is equal to f;II(O) = c;v' This completes the proof
of (AZ.l).
It remains to prove (AZ.2). For this we reformulate the definitions of an
array and of d(A) in terms of sequences of compositions. If a and f3 are
two compositions, we shall write 13  a if a; - 1  13i  ai for any i  1. If
f3  a then we define d(a, 13) to be the number of pairs (i,j) such that
f3i = ai' 13j = Uj - 1 and (j, aj) <L (i, a;).
(AZ.8) Let a and (3 be two compositions and suppose that 13; = 0 lor i > r.
There is a natural one-to-one co"espondence between row-strict a"ays A of
shape a and weight 13, and sequences of compositions (a(O), a(1),..., a(r»)
such that 0 = a(O)  a(l)  ...  a(r) = a and I a(i)I-1 a(i-1)1 = Pi for i  1.
Moreover, this co"espondence transforms d(A) into Li> 1d(a(i), a(i-l»).
Proof. We attach to an array A the sequence (a(i»), where aU) =
A -1 ({I, 2,. . ., i»). All our assertions are verified directly. I
Remembering the definition of aAp' (Chapter II, 2) we see that an
evident induction reduces the proof of (AZ.2) to the next statement:

APPENDIX: ANOTHER PROOF OF HALL'S THEOREM 203
(AZ.9) Let A, J.L be partitions with IAI = I J.LI + r, and a a composition such
that a "" A. Then
G:(1')(O) = E qd(a,!3)
f3
summed over all compositions 13 such that 13 -i a and f3  JL.
Proof. Let M be a finite o-module of type A. Recall that G;(1')( 0) is the
number of submodules N eM of type (1') and cotype JL. The condition
that N has type (1') means that N is an r-dimensional vector k-subspace
of the socle S of M. Let G,(S) denote the set of these subspaces. We shall
use the decomposition of G,(S) into Schubert cells. Recall that if a basis
(V)i e I of S is given, where I is a totally ordered set, then the correspond-
ing Schubert cells C J in G,(S) are parametrized by ,-subsets J e I. The
elements of CJ have coordinates (c;jEk:jEJ,iEI-J,j<i); the sub-
space corresponding to (c;j) has the basis (vj + EiCijVi)j e J. It is known
(and easy to prove) that G,(S) is the disjoint union of the CJ. Moreover,
we have Card (C/) = qd(J), where d(J) is the number of pairs (i,j) such
that j E J, i E I - J and j < i.
Now we express M in the form M= EB;>10Xi' where Ann(xi) = al. Let
I denote the set of indices i  1 such that ai > 0, and for each i E I let
Vi == 7T' aj-l Xi' where 'IT is a generator of . It is clear that the vj(i E I) form
a k-basis of S. We order I by requiring that j precedes i if and only if
(j, aj) <L (i, a;). Consider the corresponding decomposition of G,(S) into
Schubert cells. The subsets J c I are in natural one-to-one correspondence
with compositions 13 -i a: to a subset J there corresponds the composition
f3 such that Pi = ai - 1 for i E J, and Pi = ai for i E I - J. It is clear that
this correspondence transforms d(J) into d( a, f3). Finally, it is easy to
prove that all submodules N E CJ have the same cotype JL, where JL is the
partition such that JL /'OW p. This completes the 'proof of (AZ.9). I
Remarks. 1. The polynomials P).(x; t) defined by (1) are called the Hall-
Littlewood functions. They will be studied in detail in Chapter III.
2. It is easy to see that (AZ.9) is equivalent to Chapter II, (4.6): each of
these statements follows at once from the other by means of the following
well-known expression for Gaussian polynomials:
(2) [ ;] = ;: t d(J)
where the sum is over all r-subsets of a totally ordered n-set /, and d(J) is
defined in the proof of (AZ.9) above. The formula (2) follows e.g. from
Chapter I, 2, Example 3. On the other hand, one of the standard ways to
prove (2) is to count the number of points on a Grassmannian over a finite
field in two different ways, which was essentially done above.

III
HALL- LITTLEWOOD SYMMETRIC
FUNCTIONS
1. The symmetric polynomials RA
Let x,..., xn and t be independent indeterminates over Z, and let A be a
partition of length  n. Define
( X.-tx.)
R,\(x1,...,xn;t)= E W Xfl...X;"O 1_ } .
wES 1<) Xi Xj
n
The denominator in each tenn of the sum on the right-hand side is, up to
sign, the Vandermonde polynomial
a= n(xi-xj)
i<j
(Chapter I, 3), and therefore we have
(1.1) RA(X1'''.' xn; t) = ai1 E e(w). w( X;l... x:" 0 (Xi - aj»)
we S I <J
n
where as usual B( w) is the sign of the permutation w. The sum on the
right-hand side of (1.1) is skew-symmetric in Xl"'" xn' hence is divisible
by as in the ring Z[x1,..., xn' t], and consequently R,\ is a homogeneous
symmetric polynomial in Xl"'" X n' of degree I '\1, wi th coefficien ts in Z[ t ].
Hence R,\ can be expressed as a linear combination of the Schur functions
S(Xl"'" xn), with coefficients in Z[t]. In fact we have
(1.2)
R,\(x1,...,Xn;t) = E u'\JL(t)sJL(x1,.",xn)
J.L
where u'\JL(t) E Z[t], and u'\JL(t) = 0 unless 1,\1 = I J.LI and ,\  J.L.
Moreover, the polynomial u,\,\(t) can be explicitly computed. For each
integer m  0 let
m 1 - ti
vm(t} = n 1 = 'Pm(t}/(l - t}m
i-I - t

1. THE SYMMETRIC POLYNOMIALS R).
205
. . and for a partition A = (At,..., An) of length  n (in which some of the Ai
may be zero) we define
u..\(t) = n urn (t)
. 0 I
,
where mi is the number of Aj equal to i, for each i  O. Then we have
(1.3)
u..\,\(t) =: u,\(t).
Proof of (1.2) and (1.3). The product Il; < j(x; - txj), when multiplied out, is
a sum of terms of the form
n xilJ( - tx. )rJI
. . )
, <)
where (rij) is any n X n matrix of O's and l's such that
0)
r.. = 0
u ,
r.. + r.. = 1 if i =1= Jo .
I) }I
For such a matrix (rij)' let
(ii)
a.=A.+"r..,
I I  I)
j
d = E rji.
i<j
(Hi)
Then from (1.1) it is clear that R,\ is a sum of terms
( )d - 1
-t aaas
where as in Chapter I, 3, aa is the skew-symmetric polynomial generated
by xa =xjl ° 0 ° x:". Since aa = 0 if any two of the a; are equal, we may
assume that al,..., an are all distinct. We rearrange them in descending
order, say
(iv)
awO) = ILi + n - i
(1  i  n)
for some permutation W E Sn and some partition IL = (ILl'"'' ILII)' Then
aaail is equal to e(w)s/oL' and to prove (1.2) it is enough to show that
Ii <: A.
Let S ij = r w(i), w(j). The matrix (s ij) satisfies the same conditions (i) as the
matrix (rij), and from (ii) and (iv) we have
ILi + n - i = AW(i) + E Sij.
j

206 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
Hence for 1  k  n
k n k
(v) ILl + ... + ILk = Aw(l) + ... + Aw(k) + E E Sij - L (n - i).
i-1 j-1 i-1
Now
k n k k n
E E Sij = E Sij + E E Sij
i-I j-I i,j-1 i-1 j-k+1
k
= !k(k - 1) + E L Sij
i-1 j>k
 !k(k - 1) + k(n - k)
k
= E (n - i).
i-1
Hence it follows from (v) that
ILl + ... + ILk <; Aw(l) + .. · + Aw(k)
 A1 + ... + Ak
and therefore IL <; A. This proves (1.2).
Furthermore, these calculations show that A = IL if and only if Aw(i) = Ai
for 1 <; i <; n, and Sij = 1 for all pairs i <j. It follows that
(vi)
UAA(t) = E s(w)( _t)d
w
summed over all w E Sn which fix A, where
d = '" roo = '" S -I( 0) -1(0)
'-}f '- w J t W I
i <j i <j
is equal to the number n( w) of pairs i < j in {I, 2, . . . , n} such that
w-1(j) < w-1(i); this number n(w) is also the number of pairs 1< k such
that w(k) < w(l), and the signature s( w) is equal to ( - l)n(w). Hence from
(vi) we have
(vii)
u).). (t) = E tn(w)
weS"
n
where S; is the subgroup of permutations w E'Sn such that Aw(i) = Ai for
1  i <; n. Clearly S; = fIi;. 0 Sm; where as before mi is the number of Aj

1. THE SYMMETRIC POLYNOMIALS RA
207
equal to i, and hence to prove (1.3) it is enough to show that
(viii) E tn(w) = urn (t).
weSm
We prove (viii) by induction on m. Let wi denote the transposition
(i, m), for 1 <; i <; m (so that wm is the identity). The wi are coset represen-
tatives of S m _ 1 in 8m, and we have
n(w'wi) = n(w') + m - i
for w' E 8m-I' because in the sequence (w'wi(1),..., w'wi(m)) the number
m occurs in the ith place and is therefore followed by m - i numbers less
than m. Consequently
E tn(w) = ( E tn(w'»)O + t + ... +tm-l)
weSm w'eSm-1
: from which (viii) follows immediately. This completes the proof of (1.3).
(1.4)
By taking A = 0 in (1.2) and (1.3) we have
( X.-txo)
E w Il' J =un(t),
weSn i<j Xi - Xj
which is independent of Xl'.'" Xn.
Next we shall show that
,;.,(1.5) RA(X1,..., Xn; t) is divisible by v).(t) (i.e. all the coefficients of R). are
'.:divisible by U A (t) in Z[ t ]).
:;,-.Proof. Suppose for example that Al = ... = Am > Am + l' Then any w E 8 n
,;which permutes only the digits 1,2,..., m will fix the monomial X;I ... x;n,
and by (1.4) we can extract a factor um(t) from R).. It follows that
( X.-txo)
RA(X1,...,xn;t)=uA(t) E w xt1...x:n Il ' _ J
wesn/s; Ai> Aj Xi Xj
=UA(t)PA(X1,...,xn;t), say.
.?:Since R A is a polynomial in Xl'...' X n' so also is PA; and since t occurs only
:(.in the numerators of the terms in the sum PA, it follows that P). is a
?symmetric polynomial in Xl'...' xn with coefficients in Z[t]. I
:f'::;'It is these polynomials PA, rather than the R)., which are the subject of
;.this chapter.

208 III HALL-LITTLEWOOD. SYMMETRIC FUNCTIONS
2. Hall-Littlewood functions
Th polynomials PA(X1,..., xn; t) just defined are called the Hall-
Littlewood polynomials. They were first defined indirectly by Philip Hall in
terms of the Hall algebra (Chapter II) and then directly by D. E. Litle.
wood [LI2], essentially as we have defined them. From the proof of (1.5)
we have two equivalent definitions:
(2.1) P,,(Xl'"'' xn; t) = ) 1: W (xtl... x:" n Xi  txj ),
VA t weSn 1<) Xi Xj
( X.-tx,)
(2.2) PA (X 1 , . . . , X n; t) = 1: w x t 1 .. . X: n n I _ J .
wESnIS; A;>Aj Xi Xj
The PA serve to interpolate between the Schur functions SA and the
monomial symmetric functions m A' because
(2.3)
PA ( XI' · · · , X n ; 0) = SA ( X ] , · · · , X n )
as is clear from (2.1), and
(2.4)
PA(xl,..., Xn; 1) = mA(x1,..., Xn)
as is clear from (2.2).
As with the other types of symmetric functions studied in Chapter I, the
number of variables Xl'''.' Xn is immaterial, provided only that it is not
less than the length of the partition A. For we have
(2.5) Let A be a partition of length =s:; n. Then
PA ( XI' · · · , X n , 0; t) = PA (x 1 , · · · , X n ; t ) ·
Proof. From (2.2) we have
PA(X1"", xn+l; t) = 1:
we S,.+ II S;+ 1
( X.-tx.)
AI An + 1 I }
W Xl ."Xn+1 n _ ·
A;> Aj Xi Xj
When we set Xn+ 1 equal to 0, the only terms on the right-hand side which
survive will be those which correspond to permutations W E S n + 1 which
send n + 1 to some r such that A, = 0; modulo S; + l' such a permutation
fixes n + 1, so that the summation is effectively over Sn/S;, I
Remark. The polynomials RA(x],..., xn; t) defined in 1 do not have this
stability property, and are therefore of little interest.

2. HALL-LITTLEWOOD FUNCTIONS
209
By virtue of (2.5) we may pass to the limit and define P>..(x; t) to be the
element of A[t] whose image in An[t] for each n  1(>..) is P,\(x1,..., xn; t).
The symmetric function P.\(x; t) is the Hal/-Littlewood function corre-
sponding to the partition A. It is homogeneous of degree IAI.
. From (1.2), (1.3), and (1.5) it follows that
P,\(x1,...,xn; t) = E wA}£(t)S,u(x1,...,xn)
J.'
with coefficients W,\,u(t)EZ[t] such that W,\,u(t) = 0 unless I>..I=IJ.LI and
,\  JL, and w,\,\ (t) = 1. Hence
(2.6) The transition matrix M(P, s) which expresses the PA in terms of the S,u IS
strictly upper unitriangular (Chapter I, 6). I
Since the sp. form a Z-basis of the ring A of symmetric functions, and
therefore also a Z[t ]-basis of A[t], it follows from (2.6) that the same is
true of the PA:
(2.7) The symmetric functions P,\(x; t) form a Z[t]-basis of A[t]. I
Next we consider P,\ when >.. = (lr) and when >.. = (r). In the first case
we have
(2.8)
P(1r)(X; t) = er(x),
the rth elementary symmetric function of the Xi.
Proof. By stability (2.5) P(1r) is uniquely determined by its image in Ar[t]:
in other words, we may assume that the number of variables is r. But then
it is clear from (2.2) that P(1r)(X l' · · · , xr; t) = Xl. · · X r = ere I
We now define
qr = q,(X; t) = (1 - t)p(r)(X; t)
qo = qO(X; t) = 1.
From (2.2) we have, for r  1,
(r  1),
n x.-tx.
(2.9) qr(x1, .. ., xn; t) = (1 - t) E xi n I ) .
i-I j+i Xi-Xj
The generating function for the q, is
(2.10)
00 1 - Xi tu
Q(u) = E qr(X; t)ur = n
r-O i 1 -Xiu
= H(u)/H(tu)

210 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
in the notation of Chapter I.
Proof. Suppose first that the "number of variables Xi is finite, and put
z = u -1. By the usual rule for partial fractions we have
Ii z - txj = 1 + t (1 - OXj n Xj - txj ,
i-I Z-Xi i-1 Z-X; j..i X;-Xj
so that
nn 1-tuxi n x.u n Xi-txj
=l+(l-t)E I
i-I l-uxi i-I l-xiu j..i Xi-Xj
in which the coefficient of u', for r  0, is equal to q,(x1,..., xn; t) by (2.9).
Now let n -+ 00 as usual. I
It will be convenient to introduce another family of symmetric functions
Q,\(x; t), which are scalar multiples of the P).(x; t). They are defined as
follows:
(2.11)
Q,\(X; t) = b).(t)P).(x; t)
where
(2.12)
b,\ (t) = n 'Pm (,\)(t)
. 1 I
,
Here, as usual, mi(A) denotes the number of times i occurs as a part of A,
and 'Pr(t) = (1 - t Xl - t 2) . . . (1 - t '). In particular,
(2.13)
Q(r)(X; t) = q,(x; t).
We shall refer to the Q,\ as well as the P,\ as Hall-Littlewood functions.
They may also be defined inductively, as follows. If f is any polynomial or
formal power series in XI'.. . , X n (and possibly other variables), and 1  i 
n, let f(i) denote the result of setting Xi = 0 in f. Then if we write QA for
Q,\(xt, .. ., xn; t), we have
(2.14 )
n
Q,\ = E Xi'\t giQ),
i-I
where IL = (A2, A3' . . . ) is the partition obtained by deleting the largest part
of A, and
X. - tx.
gi = (1- t) n I } .
j..i Xi - Xj

2. HALL-LITTLEWOOD FUNCTIONS
211
proof. Let I be the length of A. From the definitions of bA(t) and vA(t) we
bave
vA(t) = vll_1(t)bA(t)/(1 - t)l,
and therefore
(1 - t)l
QA.(Xt,...,x,,;t) = ( ) RA.(xt,...,x,,;t)
vn-I t
( I X.-tx.)
= (1-1)' L W X;I...X/A./n n 1_ )
wE S,.I S,. _I l - 1 J > I X; X j
by (1.4), where Sll-1 acts on xl+l'...' XII. It follows that
QA(Xt,...,xll;t) = L W(Xt1g1QI£(X2,...,xlI;t»)
wE S,.ISn-l
which is equivalent to (2.14).
Let
l-u
F(u) = 1
-tu
=: 1 + L (t' - t,-1 )u'
r> 1
as a power series in u. Then we have
(2.15) Let ut' u2, ... be independent indetermin{ltes. Then QA(X; t) is the
coefficient of u A =: utI U2 . .. in
Q(U1,U2, ...) = n Q(ui) n F(U;1Uj).
,  1 , <J
Proof. We proceed by induction on the length I of A. When 1 = 1, (2.15)
follows from (2.13). So let I> 1 and assume the result true for JL =:
('\2' "-3' ...).
As before, let Q(i)(u) (resp. Q(i)(u1, u2' ...» denote the result of setting
t; = 0 in Q(u) (resp. Q(u1, u2' ... ». From (2.10) we have
Q(i)(u) = F(xiu)Q(u)
and hence
(1)
Q(i)(ut, U2' ...) = Q(ut, U2' ...) n F(xiuj).
j>l
h

212 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
From (2.14), QA(X1,..., Xn; t) is the coefficient of uA in
n
E ut1 E xt1 giQ(i)(U2, u3,...)
A1> 0 i - 1
(2)
n
= Q(U2' u3, ...) E ut1 E xt1g; n F(x;uj)
A1;;. 0 i-I j;;. 2
by (1) above. Let us expand the product rIj__ 2 F(xiuj) as a power series in
Xi' say
n F(x;uj) = E Imx't
t2 m;;'O
where the coefficients 1m are polynomials in t, U2' U3' .... Then the
expression (2) is equal to Q(U2' U3' ...) multiplied by
n
 UA1  of  xl+mg.= """ uA1q of
'-' t '-' J m '-' I I i..J t A1 +mJ m
A1;> 0 m > 0 i-I A l' m
= Q(u1) E Imu1m
m;;.O
=Q(ut)OF(ul1uj)
J;;.2
and hence finally QA is the coefficient of u A in
Q(Ut) 0 F(uIIUj).Q(U2,U3, ...) = Q(Ut,U2'U3, ...)
1;;.2
For any finite sequence a = (at, lr2' ...) of integers let
qa = qa(x; t) = n qa (x; t)
. 1 l
,
with the convention that qr = 0 if r < 0 (so that qa = 0 if any ex; is
negative). The raising operators Rij, i <j (Chapter I, 1) act as follows:
Rijqa:: qR;Ja. Then the coefficient of ua in
F( U;l uj) n Q(Uk) = (1 + E (tr - tr-1 )UirUj) E qu
k;>l rl p
IS
I-R..
1 + " (tr - t,-1 )R.q = 'i q
i..J 'i a 1 R Q ·
r> 1 - t ij

2. HALL-LITTLEWOOD FUNCTIONS
213
It follows that (2.15) may be restated in terms of raising operators:
(2.15')
l-R..
Q" = n I-tR'. q"
i <j IJ
for all partitions A.
Hence
QA= n<.(1+(t-1)R;j+(t2-t)R7j+...)q,\
I J
and it follows from Chapter I, (1.14) that
Q,\ = E aAJL(t)qJL
JJ.
where the polynomials aAJL(t) E Z[t] are such that a'\JL(t) = 0 unless ,\  JL,
and a,\A(t) = 1. By (2.7), the Q,\ form a Q(t)-basis of A z Q(t); and
therefore
(2.16) The symmetric functions qA form a Q(t)-basis of A z Q(t), and the
transition matrix (Chapter I,  6) M(Q, q) is strictly lower unitriangular.
Examples
1. If we set Xi = t;-l(l  i  n) we have
R A ( 1 , ( , . . . , (n - 1 ; () = (n( A) V n ( ( )
directly from the definition (1.1), because the only term in the sum which does not
vanish is that corresponding to w = 1. Hence
Q.\ (1, (,. . ., tn -1; t) = tn(A)'Pn(t) / 'Pmo(t)
where mo = n -1(A). Let n -.. 00, then also mo --+ 00 and so when Xi = (;-1 for all
i) 1 we have
Q,\ ( 1 , ( , (2 , . . . , ; t) = t n( A) .
2. We can use the inductive definition (2.14) to define Q,\ for any finite sequence
(Ah '\2" · · , A) of non-negative integers, not necessarily in descending order. The
formula (2.15') will still be valid and can be used to extend the definition of Q,\ to
. .qny sequence (At, A2' ... AI) of integers, positive or negative. If Al < 0, clearly
Q.\ = O.
To reduce QA when the A; are not in descending order to a linear combination

214 III HALL-LITTLEWOOD. SYMMETRIC FUNCTIONS
of the Q where J.L is a partition, we may proceed as follows: since interchange o:-
x 1 and x 2 transforms f..
x1'xi(X2 - U1)(XI - U2)
xl -X2
into
XfX2(X2 - UI)(XI - tt2)
xl - X2
it follows that
Q(r.s+1) - tQ(r+l,s) = - (Q(s,r+l) - tQ(s+l,r»
or, replacing r by r - 1, that
Q(s,r) = tQ(r,s) - Q(r-l,s+l) + tQ(s+l,,-l).
Assuming S < r, this relation enables us to express Q(s r) in terms of the Q( .
, , - " oS + i)
where 0  i  [i(r - s)] == m say. The result is
m
Q(s,r) = tQ(r,s) + E (ti+ 1 - ti-1 )Q(r-i,s+O
i-1
if r - S = 2m + 1, whereas if r - S == 2m the last term in the sum must be replaced
by (tm - tm-1) Q(r-m,s+m).
For simplicity we have stated these formulas for a two-term sequence (s, r): but
the same holds for any two consecutive terms of a sequence A..
3. The definitions (2.1) and (2.2) of PA can be written in terms of lowering
operations R ji(i < j):
PA = vA(t) -1 fl (1 - tRji)s).
i<j
== fl (1 - tRjj)s).
AI> AJ
Hence, for example,
n
Pen) = fl (1 - tRj1 )s(n)
j-2
= E(-t)IJlflRjlS(n)
J jeJ
summed over all subsets J of {2, 3, . . . ,n}. The only J for which Il j e J R j1 s(n) =1= 0
are J = {2, 3" .. . ,i} (2  i  n), and therefore
n-1
p(n):=I E (-t)' s(n-r,1').
r-O

F:" 3. THE HALL ALGEBRA AGAIN 215
r[&,
IJ':,..'The Hall algebra again
g;;),. .
..;r':_
;/':'
1 (2.7), th: poduct PJLP" of two Hall-Littlewood functions ill be .a
J@ear mbinatIn. of th P>.., where II = I JLI + Ivl ('\, JL,. v beIng patl-
.ITQPs) WIth coeffIcIents In Z[t]: that IS to say, there eXist polynomials
n."(t) E Z[ t] such that
,p.JI
. ..I.
<- PJL{x; t)P,,{x; t) = Et;,,(t)PA{x; t).
c. A
'.When t = 0 we have by (2.3)
f,'(b) l;v(O) = c;v.
\'.: '';'
iJb coefficient of sA in sJLs", Likewise, from (2.4), t;,,(l) is the coefficient
;.6f" m>.. in the product mJLm" when expressed as a sum of monomial
srnmetric functions.
.:. '-
iE.In order to connect the symmetric functions P>.. with the Hall algebra,
1;e need to compute the polynomial t;v(t) in the case v= (1m).
.{;.:lRecall that
[;] = ft'n(t)/ft'r(t}ft'n-r(t)
if 0  r  n and that [;] = 0 otherwise, where
CPr (t) = (1 - t ) (1 - t 2) . . . (1 - t r) .
:(3.2) We have
[X, - X.+l-]
1;(1..)(t} = n . _ I I.
I  1 I J..L,
(and therefore flm){t) = 0 unless A - JL is a vertical m-strip).
Proof. We shall work with a finite set of variables Xl"..' xn' where
n  I( JL) + m. We have to multiply PJL, given by the formula (2.2), by .l(1m),
which by (2.7) is equal to the mth elementary symmetric function em. Let
k = ILl be the largest part of JL, and split up the set {x l' . . . , xn} into subsets'
Ko,.. " Xk (some possibly empty) such that Xj E Xi if and only if JLj = i, so
that IXil = mi, the multiplicity of i as a part of IL. Let e,{Xi) denote the
elementary symmetric functions of the set of variables Xi. Since
n k
n (1 +xit) = n n (1 +xit)
i-I J'-o x.EX.
. 1

216 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
it follows that
(1)
em(xt,..., xn) = E ero(XO)." erk(Xk)
r
summed over all r = ('0'''.' 'k) E Nk+ 1 such that ri  mi for each i and
E 'i = m.
To each such r there corresponds a partition ,\ defined by
x. = Il. + r. 1
I r-, I -
(1  i  k + 1).
Clearly ,\ - J.L is a vertical m-strip, and conversely every partition A such
that ,\ - J.L is a vertical m-strip arises in this way.
Each eri(Xi) on the right-hand side of (1) is equal to P(1ri)(Xi; t) by (2.7),
and hence by (2.1) can be written as a sum over the symmetric group S
acting on the set Xi. In this way (1) takes the form ntj
(2) em (x 1 , .. · , X n) = E c,\,,u ( t ) - 1 cI>,\, ,u (x 1 , .. · , X n ; t)
A
summed over all partitions ,\::) 'J.L such that A - J.L is a vertical m-strip,
where
( x.-tx.)
<PA,,u = E w X;l - ,ul . . . x:" -,un n I _ J
weS: I <} Xi Xj
J.L; -=s IL j
and
k
CA,,u(t) = Il Vr{t)Vm.-r{t).
i=O' "
If we now multiply both sides of (2) by P,u(X1,..., Xn; t) as given by (2.2)
we shall obtain
P,uem = E c,\, ,u(t) -1 RA
A
from which it follows that the coefficient of PA in P,uem is
f;(1 m) (t) = v,\ ( t ) c,\, ,u (t ) - 1
which easily reduces to the expression given. I
If we now compare (3.2) with Chapter II, (4.6) we see that
(3.3)
GA (0) = qn(A)-n(,u)-n(lm).fA (q-l)
,u(lm) J ,u(lm) ·

IA:r 3. THE HALL ALGEBRA AGAIN 217
!::.==.I.' '_.
f()111 this we deduce the following structure theorem for the Hall algebra
!J(o ):
fJ.4) Let 1/1: B(o)  Q -+ AQ be the Q-linear mapping defined by
!: 1/1 (u ) = q -n(A)p (x' q-l ).
if. A A ,
hen f/J is an isomorphism of rings.
-p,oof. By (2.7), «/I is a linear isomorphism. Since H(o) is freely generated
[(s Z-algebra) by the U(1r) (Chapter II, (2.3)), we may define a ring
lphqmomorphism «/I': H(o)  Q --+ AQ by
!  '.: .
,,:'.:.: 1/1'(U(lr») = q-r(r-l)/2er.
I:-; 
(We shall show that «/I' = «/I, which will prove (3.4). To do this we shall
i;pf9ve by induction on the partition A that .p'(uA) = tjJ(u>..). When A = 0,
Hhis is clear from the definitions. Now assume that A -+ 0, and let J.L be the
,'p.artition obtained from A by deleting the last column. Suppose that this
::iast column has m elements. Then from Chapter II, (4.6) we have Gp.F")(O)
{J, and G;'(lm)(O) = 0 unless v:) J.L and v - J.L is a vertical m-strip.
'Jdoreover, if v - J.L is a vertical m-strip, it is easy to see that v  A. Hence
-:'J)
;.;'r  - . .
Up.U(1m) = uA + E G;(1m)( 0 )uv
v<A
and likewise
'(2) Pp.P(F") = PJ.. + E f;(1m)(q-t )Pv
v<A
:."::r
'where Pp. stands for Pp.(x; q-l), and so on. If we now apply tjI' to both
'sides of (1) and recall that P(1m) = em by (2.7), ana compare the result with
(2), taking as inductive hypothesis that «/I'(uv) = q-n(v)pv for all v < A, then
it follows from (3.3) that .p'(uA) = q-n(A)PA = .p(uJ..). I
Remark. This proof depends on the identity (3.3), which appears as merely
a happy accident. It is possible to give a proof of (3.4) which does not
depend on an apparent miracle: see the notes at the end of Chapter V.
; From (3.4) it follows that
(3.5)
G;v(o) = qn(A)-n(p-)-n(v)f:v(q-l)
for any three partitions A, JL, v, and therefore by Chapter II, (4.3) the
polynomials f;v are related to the Hall polynomials g:v of Chapter II as
follows:
(3.6)
g;v(t) = tn(A)-n(IL)-n(v)f:v(t-1).

218 III HALL-LITTLEWOOD-SYMMETRIC FUNCTIONS
In particular:
(3.7) (i) If f;v(O) == 0, then f;v(t) is identically zero.
(ii) I;v(t) == 0 unless I AI == I J.LI + I vi and J.L, v C A.
The results of Chapter II, 4 (especially (4.2) and (4.11)) provide an
explicit formula for l;v(/), namely
(3.8) fAJL(t) = E fs(t)
s
summed over LR-sequences of type (Ii, Vi; X), where in the notation of
Chapter II, (4.10) and (4.11)
(3.9)
fs(t) = n CP(a;j' b;j' cij; N;j; t).
i'>i
In particular:
(3.10) If A :::> #-t' and 0 = A - #-t is a horizontal r-strip, then
I:(r)(t) = (1 - t) -1 n (1 - tm;(A»)
je/
where I is the set of positive integers i such that 0; > (J;+ 1 (so that 8; = 1 and
0;+ 1 == 0) and mi(A) is the multiplicity of i as a part of A. In all other cases
I;(r)(t) == O.
In view of (3.6), this is simply a transcription of Chapter II, (4.13).
Remark. We have arrived at this result (which we shall later make use of)
by an indirect route, via the Hall algebra. It is also possible to derive (3.10)
directly from the definition of the PA (x; t): see Morris [M13].
Examples
1. Let A be a partition. Then for each m  0,
E Gl"')( 0)
p.
is the number of elementary submodules E of type (1m) in a fIXed o-module M of
type A. All these submodules E lie in the socl S of M, which is a k-vector space
of dimension I = I( A), and the sum above is therefore equal to the number of
m-dimensional subspaces of an I-dimensional vector space over a finite field with q
elements, which is [J (q). Hence by (3.3) we obtain
E tn(p.)fI'")(t) = t"(.I.)-m(m -1)/2 [] (r 1).
JL

f:/\: 3. THE HALL ALGEBRA AGAIN
i::.:";:'.
I';';'
r,,>:
QW let y be an indeterminate. Then
.. -
:,':' ( E tn(P.)Pp.) ( E emym ) = E ymt,,(p.)flm)(t)PA
,s;" p. m A, p., m
r:; .:.'::
['; -:.' '..
219
', :; ,"" .
{.:'
I:"",
" . 
= E tn(.\)p" E ym t- m(m -1)/2 [ 1(A) ] (t-1 )
A m m
'. ';
;';:.= :'
;.:;(i)
I(A)
= E t n(.\) P" n (1 + t 1- j Y )
,\ j-I
_.'# the identity of Chapter I, 2, Example 3. In particular, when y = -1 we obtain
. i,  
( E t"( p. )Pp. ) ( E ( -1) m em) = 1
p. m
I   :
;:d therefore
[(2)
E tn(p,)Pp. = hll.
1p.I-n
:Hence the identity (1) takes the form
(3)
1(,\)
n (1 + Xi Y ) / (1 - Xi) = E tn(.\) n (1 + t 1- j Y ) . p.\ (x; t).
i__l A j-I
,2 (a) Let
P" = q-n(.\>p.\ (x; q-l),
Q.\ = ql.\l+n(.\)Q.\ (x; q-l)
SO' that Q" = a,,(q)A., where by Chapter II, (1.6) a.\(q) is the order of the
:automorphism group Aut M" of an o-module of type' A. From 2, Example 1 it
follows that the specialization xi.-. q -i(i  1) maps each Q.\ to 1 and hence P.\ to
Q).(q)-I. Hence from (3.4) the mapping u.\ ..... tAut M.\I-1 of the Hall algebra H(o)
into Q is multiplicative, i.e. is a character of H( 0 ).
(b) From Example 1 above we have
E P" = hn
IA/-n
ind therefore, specializing as above,
E JAutM.\I-t =hll(q-t,q-2, ...).
IAI-n
fience
E lAutM"I-1 = n (l_q-i)-t = E q-I.\I
A i> 1 A

220 III HALL-LITTLEWOOD- SYMMETRIC FUNCTIONS
or equivalently
E !AutMAI-1 = E IM,\I-1
A A
as convergent infinite series.
(c) More generally, the identity (3) of Example 1 above leads to
E (X;q)/(A) = n l-xq-i
A JAutMAI i>1 1-q-i
where (x; q)n = (1 - xXI - qx)... (1 - qn-1x).
3. (a) Let J.L = ( J.L(1), . . ., I-L(r» be any sequence of partitions, and let
v(M» = g;(l)... (r)(q)
be the number of chains of submodules
MA = Mo Ml ::> ... ::>M, = 0
such that Mi-1/M; has type I-L(i), for 1  i  r. Show that
(1)
v (M ), .
E  ,\ = n IAut M')1-1 .
A JAut MAl i-I
In particular, if v(MA) is the number of composition series of M,\, then
E v(MA) _
1.\1=, tAut MAl
1
, .
(q - 1)
(Specialize the identity
E g;(1)...JL(r)(q )u,\ = UJL(l)... UJL(r)
A
as in Example 2 above.)
(b) Let H = (H1,..., H,) be any sequence of finite abelian groups, and for each
finite abelian group G let v H( G) denote the number of chains of subgroups
G = Go  G1  ...  G, = 0
such that Gi-1/Gi = Hi for 1  i  r. Show that
vH(G)' -1
E = n lAutHil
(G) JAut GI i= 1
where the sum on the left is over all isomorphism classes of finite abelian groups.
(Split G and the Hi into their p-primary components, and use (a) above.)

3. THE HALL ALGEBRA AGAIN
221
;.4. The fact (Example 2 (a)) that uA ...... a,\(q)-l is a character of the Hall algebra
1:'I/( 0) is equivalent to the following relation:
\ (1) E a).(q)-lap,(q)av(q)g;v(q) = 1
A
,: for all partitions J.L, v. This may be proved directly as follows.
Fix o-modules L, M, N of respective types A, J.L, v, and letE1:tN denote the set
of all exact sequences
a f3
E( a, {3): 0 --+ N --+ L --+ M --+ O.
In such an exact sequence, N' = a (N) is a submodule of L of type v, with
LIN' = M of type J.L; hence the number of choices for N' is g;v(q). Given N',
there are aJl(q) choices for a, and ap,(q) choices for {3, and therefore
ektN = IE1tNI = ap.(q)av(q)g;v(q).
Hence (1) is equivalent to
E ai,lektN= 1
(L)
where aL = (Aut LI and the sum is over isomorphism classes of finite D-modules L.
The group Aut L acts on E1:tN by <pE(a, 13) =E(<pa, {3q;-1), and it is not
difficult to verify that cp E Aut L fixes E(a,.{3) if and only if <p = 1 + a8f3, where
8 E Hom(M, N). Hence the orbits are all of size aLlh, where h = IHom(M, N)I,
and the number of them is therefore ai1ettNh.
On the other hand, the orbits of Aut L in E1tN' for all L, are the equivalence
classes of extensions of M by N (see e.g. [Cl], Chapter XIV) and are in one-one
correspondence with the elements of Ext1(M, N). Now Ext1(M, N) and
Hom(M, N) are finite o-modules of the same type J.L X v (Chapter II, 1, Example)
and in particular have the same order h( = ql p,X vi). Hence we have
(2)
E aL1ektNh = h
(L)
which proves (2) and therefore also (1).
5. Let J.L = ( J.Ll'. . . , J.Lr) be a partition of length r, and for a finite o-module M let
wp,(M) denote the number of chains of submodules
M = Mo => M1 => ... => Mr = 0
such that IMi-11 M;I = q p.j for 1  i  r. Thus
wp.(M,\) = Eg1)...v(r)(q)
summed over all sequences (v(l),..., vCr») of partitions such that I v(i)1 = J.Li for
1  i  r. From Example 2 it follows that
E wp.(M).)P,\ = hp,
A

222 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
and hence that
(1)
 wjA.(MA) _ -1-2
l.,., lA M I -hjA.(q ,q ,...).
A ut A
Since (Chapter I, 2, Example 4)
h (-1 - 2 ) -'/ ( -1)
, q , q ,. .. = q cp, q
= q,(r-1)/2 /(q _ 1)... (q' - 1),
we may rewrite (1) in the form
 wlI(MA) I . - II!.
l.,., r- z:: qn( p;) n (q' - 1) r-..
A tAut MAl i__1
(2)
In particular, when r == 2 and J.L == (n - k, k), wjA.(MA) is the number of
ules of M).. of order qk. More particularly still, when J.L == (n - 1, 1), W(MA) is
number of sub modules of MA of order q. These all lie in the socle, which
k-vector space of dimension l(A), so that wjA.(M») = (ql(A) - 1)/(q - 1) when IJ.
(n - 1, 1). Deduce that ..
ql(A)tIAI 1 + t
 !Aut MAl = n (1- q-it)
i> 1
where the sum on the left is over all partitions A.
Notes and References
The identities of Examples 2(b) and 3(a) are due to P. Hall [HI], [H2].
also [M 4].
4. Orthogonality
We shall now generalize the developments of Chapter I, 4 by giving three;
series expansions for the product
n (1- txiYj)/(I-xiYj)'
i, j
The first of these is
(4.1) n (1 - txiYi) /(1 - xiYj) = E z,\ (t) -1 PA (x) PA (y)
i,j A
summed over all partitions A, where
n A-I
ZA(t)=Z)... (1-t I) .
i;;.1

,
1f We have
U1f1-v  ·
t{-:lOg n (1 - ttiYj) /(1 - xiYj) = E (log(l - txiYj) - log(l - XiYj))
If i,j i,j 00 1 _ tm m
J:- = E E (x Y )
IV i, ml_ tmm i j
it, = ml m Pm(X)Pm(Y),
Il*d therefore
ij_r_,_! 0(1- tt.y.)/(1-x.y.) = n°o exp( 1- tm P (x)p (y))
:::L: - , J , ) m m m
If ;,j m-1
L3;;'jj:
Ii'"
tV: .
t
1-i!t.
,'--'. ;..
:'I"'"
F'
t':;:" ,
:'\;.
 . .
r:::..
:
4. ORTHOGONALITY
223
00 00 (1 _ tm)'m
- n E , , Pm(x)'m Pm(y)'m
ma=:l r -0 m m.'m.
m
= E Z,\(t)-l PA(X)PA(Y).
A
j:_ Next we have
n (1 - tXiYj) /(1 -xiYj) = E qA(X; t)mA(y)
i,j A
(4.2)
i":., .
'Yi '
 :.'. ':
= E mA(x)qA(Y; t)
A
:;
:;!ummed over all partitions A.
,".
L- .
\p,oofi From (2.10) it follows that
I'
;:... :
00
n (1- ttiYj)/(1-xiYj) = IJ E q,lx; t)yJj
, ,j } 'j - 0
nd the product on the right, when multiplied out, is equal to
E q,\(x; t)mA(y).
A
ikewise with the x's and y's interchanged.
-- From (4.2) it follows that
:4.3) The transition matrix M(q, m) is symmetric.

224 III HALL-LITTLEWOOD. SYMMETRIC FUNCTIONS
Next, in gneralization of Chapter I, (4.3) we have
(4.4) n (1 - tx;Yj) /(1 .:- xiYj) = L PA (x; t )QA (y;t)
i,) A
= L b). (t ) PA ( X ; t) PA (Y; t) .
A
Proof Consider the transition matrices
A = M(q, Q),
B =M(m,Q),
C=M(q,m)
and let D = B'A. By (2.16), A -1 is lower unitriangular, hence so is A. Also
B is upper triangular, because
B-1 =M(Q,P)M(P,s)M(s,m)
and M(Q, P) is diagonal by (2.11), M(P, s) is upper triangular by (2.6), and
M(s, m) is upper triangular by Chapter I, (6.5). It follows that D is lower
triangular. On the other hand, D = B'CB, and C is symmetric by (4.3);
hence D is symmetric as well as triangular, and is therefore a diagonal
matrix, with diagonal elements equal to those of B, so that D = M(P, Q)
and therefore DA). = b).(t )-1.
Hence
L q).(x; t)m).(y) = L AAP.BAJlQp.(x; t)Qv(Y; t)
A At,JI
= L bp.(t)-1 Q,u(X; t)Qj.L(Y; t)
J.L
= L P(x; t)Q,u(Y; t)
J.L
and so (4.4) follows from (4.2). I
Remark. There is yet another expansion for 0(1 - tXiYj)/(l -XiYj)' Let us
define
(4.5)
S).(X; t) = det( q).l-i+j(X; t))
for any partition '\, so that
(4.6) S).(X;t)= n(l-Rij)q).= n(l-tRjj)Q).
i<j i<j
by Chapter I, (3.4") and (2.14'). If we introduce a set of (fictitious) variables
i by means of
n (1- fxiy)/(l-xiy) = n (1- {iy)-1
i i

4. ORTHOGONALITY
225
':"then qr(x;t)=hr(g) and therefore S>..(X;t)=SA(g) by Chapter I, (3.4).
"Bence from Chapter I, (4.3) we have
(4.7) Il (1 - tx;Yj) /(1 - XiYi) = E SA (x; t )SA (Y)
i,j A
= E SA(X)SA(Y; t).
A
We now define a scalar product on A[t] (with values in Q(t)) by
requiring that the bases (q).(x; t)) and (mA(x)) be dual to each other:
--(4.8) (qA(X; t), m,u(x) = 8A,u.
The same considerations as in Chapter I, 4, applied to the identities (4.1),
(4.4), and (4.7) show that
(4.9) ( PA ( X ; t), Q JL ( x; t ) ) = 8A JL '
(4.10)
(4.11)
(SA (X; t), S,u(X) = 5,\,u'
(PA(X),P,u(X)= zA(t)8A,u.
(4.12) The bilinear lorm (u, v) on A[t] is symmetric.
When t = 0, this scalar product specializes to that of Chapter I, because
P,\(x; 0) = QA(X; 0) = s,\(x). When t = 1 it collapses, because bA(l) = 0 and
therefore Q,\(x; 1) = 0 for all partitions A =1= O.
Remark. Since z,\(t) = zAP>..(l, t, t2, ...), it follows from (4.11) and Chapter
I, (7.12) that the scalar product (4.8) is given by
(4.13)
(I, g) = (/* g)(1,t,t2: ...)
for all I, g E A[t], where 1 * g is the internal product of 1 and g defined in
Chapter I, 7.
Examples
1. By taking Yi = ti-1 for all i  1 in (4.4) and making use of 2, Example 1, we
obtain
E tn(A)PA(X; t) = f1 (l-X)-l,
,\
so that
E tn(A)p,\(X; t) = hn(x).
IAI-n
(See also 3, Example 1.)

226 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
2. By applying the involution Cd (Chapter I, 2) to the y-variables in (4.7) we obtain
E S.\(x; t)SA'(Y) = [J (1 +xiYj)/(l + txiYj).
A i,j
Now take Yi = ti-1 for all i  1; then (Chapter I, 3, Example 2)
s.\,(y) = tn().')H). (t)-1
and therefore
E SA (X; t ) t n().') H). (t ) - 1 = [J (1 + Xi) .
A
3. For each integer n  0, under the specialization
p, ...... (1 - tn,) /(1 - t')
(r  1)
Q). (x; t) specializes to
I(A)
tn().) [J (1 - tn -i+ 1)
i-I
(2, Example 1). Since this is true for all n  0, we may replace tn by an
indeterminate u: under the specialization
p, (1- u')/(1- t')
(r  1)
Q A (x; t) specializes to
I(A)
t n().) [J (1 - t 1 - i U ) .
i-I
By applying this specialization to the y-variables in (4.4), we obtain another proof
of the identity (3) of 3, Example 1.
Notes and references
The identity (4.4) is equivalent to the orthogonality relations for Green's
polynomials (7). The proof given here is due to Littlewood [L12].
5. Skew Hall-Littlewood functions
Since the symmetric functions PA form a basis of A[t], any symmetric
function u is uniquely determined by its scalar products with the PA: for by
(4.9) we have
u = E (u, PA)QA.
A

5. SKEW HALL-LITTLEWOOD FUNCTIONS 227
In particular, for each pair of partitions '\, J.L we may define a symmetric
function QA/P. by
(5.1)
( Q A/ p. , Pv) = (Q,\, Pp. Pv) = fv (t )
or equivalently
(5.2)
Q,\/p. = Ef;v(t)Qv.
v
Since f;v(t) = 0 unless J.L C ,\ (3.7), it follows that
(5.3) .QA/P. = 0 unless J.L C A.
From (5.1) we have (Q,\/p.' u) = (Q,\, Pp.u) for any symmetric function u.
In particular, when J.L = 0 it follows that QA/O = Q,\o When t = 0, Q,\/p.
reduces to the skew Schur function s,\/p..
Likewise we define P,\/p. by interchanging the P's and Q's in (5.1):
(5.1')
(P,\/p., Qv) = (P,\, Qp.Qv)
for all partitions v. Since Q,\ = b,\ (I )P,\ it follows that
(5.4) Q,\/p. = b,\/p.(t)P,\/P.
where b,\/p. (t) = bA (t) / bp. (t).
From (5.2) it follows that
E QA/P.(X; t )PA (y; t) = E t;v(t)Qv(x; t)PA (y; t)
A A,v
= E Qv(x; t)Pp. (y; t )Pv(Y; t)
v
and therefore by (4.4)
1 - tx.y.
E Q,\/p. (x; t ) P,\ (y ; t) = Pp. (y , I) 0 I ) .
I J I-x.y.
A " . I }
From this we have
E Q A/ p. (x ; t ) P,\ (y ; t) Q p. ( z; t )
A,p.
1 - /x.y.
= E PI'(y; t)QI'(Z; t) 0 1 I J
J-l i,j -xiYj
1 - a.y. 1 - tyJ.Zk
=0 IJO
i,j l-xiYj j,k l-Yjzk

228 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
which by (4.4) again is equal to
 Q A ( X ,-z; t) PA ( y ; t ) .
A
Consequently,
(5.5)
QA(X, z; t) =  QA/(X; t)Q(Z; t)
J.l.
where by (5.3) the summation is over partitions J..L C '\. Likewise,
(5.5' )
PA (x, z; t) =  PA/,u (x; t) P,u (z; t) .
J.l.
Just as in Chapter I, 5, these formulas enable us to express the
symmetric functions PA/ and QA/,u as sums of monomials. Since (m})) and
(qv) are dual bases for the scalar product (4.8), it follows that
(5.6)
Q)./ =  (QA/' qv )mv
v
=  (QA' P,uqv)mv.
v
Now from (3.10) we have
(5.7)
Pp.qr =  'PA/P.(t )PA
).
summed over all partitions ,\ such that 8 = A - J.L is a horizontal r-strip,
where
(5.8)
'P)./,u (t) = n (1 - tm;(A»)
iE/
in which I is the set of integers i  1 such that 8! > e!+ 1 (i.e. e; = 1 and
8I+ 1 = 0).
We shall use (5.7) to express Pp.qv, where J.L and v are any partitions, as
a linear combination of the PA. Let T be a tableau (Chapter I, 1) of shape
A - J.L and weight v. Then T determines (and is determined by) a sequence
of partitions (,\(0),..., ,\(r») such that J..L = ,\(0) C ,1(1) C ... C A(r) = A
and such that each ,\(i) - ,\(i-l) is a horizontal strip. Let
(5.9)
r
'P T ( t) = n 'PA (i) / A (i - 1) ( t ) ,
i-1
then we have
(5.10)
Pp.qll =  (  'PT(t)) P,\
,\ T

t.
",;.=.O
5. SKEW HALL-LITTLEWOOD FUNCTIONS
229
'sUIIllI1ed over all partitions A :J # such that IA - #1 = lvi, the inner sum
ffrbeing over all tableaux T of shape A - J..L and weight v. This is a direct
::.,.;'consequence of (5.7) and induction on the length of v.
:_:. From (5.6) and (5.10) it follows that
QA/p. = E 'PT(t)XT
T
.,::'summed over all tableaux T of shape A - J..L, where as in Chapter I, (5.13) xT
L_is the monomial defined by the tableau T.
..:..( 5 .11 )
(:.. There is an analogous result for PA/p.' First, when A - J.L = () IS a
';':hhorizon tal strip, we define
., (5.8')
II, (t) = n (1 - tmj( J.L))
o/A/ p.
jeJ
m.where J is the set of integers j  1 such that OJ < 0j+ 1 (i.e. OJ = 0 and
f"OJ+ 1 = 1), and then we define
:(5.9')
r
f/lT (t) = n f/lA(i) / A(i-1)(t)
i-1
for a tableau T as above. It is easily verified that
(5.12)
'PA/p. (t) / f/lA/p. (t) = bA (t) /bJ.L (t)
. and hence that
(5.13)
'PT(t)/f/lT(t) = bA(t)/bp.(t)
if T has shape A - J..L. From (5.4), (5.11), and (5.13) it follows that
(5.11') PA/p. = E t/lT (t )xT
T
summed over all tableaux T of shape A - J..L.
Also, in place of (5.7) we have
(5.7' )
Qp.q, = E f/lA/p.(t )QA
,\
summed over all A::> J..L such that A - J..L is a horizon tal r-strip, by virtue of
(5.7) and (5.12).
Remarks. 1. Whereas the skew Schur function SA/J.L depends only on the
difference A - JL, the symmetric functions PA/p. and QA/J.L depend on both
,\ and J.L; this is clear from (5.11) and (5.11').

230 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
2. In the case where there is only one variable x, we have
(5.14 )
QA/(X; t) = lpA/(t)xl.\-1
if A - IL is a horizontal strip, and QA/(X; t) = 0 otherwise. Similarly
(5.14') PA/(X; t) = t/1.\/p,(t)XIA-1
if A - IL is a horizontal strip, and PA/(X; t) = 0 otherwise.
These are special cases of (5.11) and (5.11').
Examples
Several of the formal identities given in the Examples in Chapter I, 5 can he
generalized.
1.
E PA(x; t) = TI (1 _X;)-l TI (1- tX;xj)/(I-xixj),
,\ i i <j
summed over all partitions A.
It is enough to prove this identity when x = {Xl'...' Xn}, the sum on the left
being over all partitions of length  n. By induction on n, it is therefore enough to
prove that
1 n 1-txiy
EpA(x,y;t)= TI1 Epv(x;t)
.\ l-y i-I -XiY v
where we have written Y in place of xn+l. Now by (5.5') and (5.14') we have
PA(x, Y; t) = E PA/Jt(Y; t)PJt(x; t)
C.\
= E tJlA/Jt(t)yIA- JtIPJt(x; t),
CA
summed over partitions J.t C A such that A - J.t is a horizontal strip. On the other
hand, by (2.9),
n 1 - IXiy
TI 1 _ E Pv(x; t) = E Pv{x; t)q,{x; t)y',
i-I xiY v v"
and
Pv{X; t )q,(x; t) y' = E lpJt/v(t )PJt(X; t) yl Jt- vi,
p,-::Jv
by (5.7), the summation being over all J.L:J II such that J.L - II is a horizontal r-strip.
Hence we are reduced to proving that
(I)
E tJlA/Jt(t)yIA-JtI = {1- y)-l E c,oJt/v(t)yIJt-vl
A vc
for any partition J.L, where A - J.L and J.L - II are horizontal strips.

1ft 5. SKEW HALL-LITTLEWOOD FUNCTIONS 231
f-'
:::
I For each subset I of [1, p.d consider the partitions A::::> p. and II C P. such that
.-'.
;gE::.'
I) I/JA/,..{ t) = r,Dp./,,( t) = }J (1 - t m i( p.») ·
iLet iI' it + i2,..., i1 + ... +i, be the elements of I in ascending order. Then it is
easi1y seen that the contribution to the left-hand side of (1) from the partitions A
Itisfying (2) is
:{!3) (1 +y+ ... +yil-1)(y+y2+... +yi2-1)...(y+ ... +yi,-1).y(l_y)-t
ind likewise that the contribution to the right-hand side of (1) from the partitions
.' satisfying (2) is
Ii' (1_y)-l(y + y2 + ... +yil)(y + y2 +... +yi2-1)... (y + ... +yi,-l)
J'Which is visibly equal to (3). Hence the two sides of (1) are indeed equal, and the
:"proof is complete.
t:.: When t = 0, this identity reduces to that of Chapter I, 5, Example 4.
;'2
C. ·
Ep(x;t)= n(l-x;)-I n(l-tx;xj)/(I-x;xj)
 i i<j
:.j'summed over all partitions J.L with all parts even.
In view of Example 1 it is enough to show that
E P(x; t) E em (x) = E PA(x; t)
#l- mO A
..
'::where the sum on the right is over all partitions A. Now each partition A
":determines uniquely an even partition J.L by decreasing each odd part of A by 1, so
:,that A'; - J.L = 0 if i is even and = mj( A) if i is odd. It follows from (3.2) that
,!lm)(t) = 1, where m = IA - J.LI, and hence that the coefficient of PA in Pp,em is 1.
.3.
E clI(t)PV(x; t) = n (1 - txjXj) /(1 - xixj)
II i<j
here the sum is over all partitions v with all columns of even length, and
.ev(t) = 0;;.1(1- tXl- t3).. .(1- tm/-I), where mi = m;(v).
From Example 1 it follows that
n (1 - ttjxj)/(1 - XiXj) = ( E P,\(x; t») ( E (_I)m em(x»)
I <} A m
= E (-I)mEf1(tm)(t)Pp,(x;t),
A,m p,
from which and (3.2) the coefficient of P(x; t) (J.L any partition) In
Oi<j(l- tt;xj)/(l-xjxj) is equal to
m ;( J.I.) [ () ]
n E (_1)'; mir .
,;.1 ,.-0 .
J

232 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
Now (see [A2], Theorem 3.4) the inner sum is 0 if mi( JL) is odd, and is equal t
(1 - t Xl - t 3) . .. (1 - tm/( 1£)- 1) if m j( JL) is even. Hence the result. 0
Since cv(t) = bv(t)/bp/2(t2), where v/2 is the partition defined by mj(v/2)
= tmi( v) for all i, the identity just proved can be restated in the form
" 2 -1 . n
i..Jbv/2(t) Qp(x,t)= (l-ttixj)/(I-xjxj)
v i<j
again summed over partitions v with all columns of even length, i.e. partitions in
which each part occurs an even number of times.
4.
1 - tx. 1 - IX.x.
I} I-X; Q l-X;X; =  d,,(t)-lQ,,(X; t)
summed over all partitions A, where
[m;/2]
dA(t) = n n (1 - t2j)
i;.1 j=l
in which mi == mj( A).
From Example 3 and (2.10) we have
n 1 - txi n 1 - txiXj (  ) ( 2 -1 )
. 1 _ .. 1 _ = i..J q,(x; t) E bv/2(t) Qv(x; t)
t Xi t<} XiXj r-O v
.pA/v (I)
= E b (2) Q,,(X; t)
At v v/2 1
by (5.7'), where the summation is over all v with all columns even and all A::) v
such that A - v is a horizontal strip. Each partition A determines a unique such JI,
by removing the bottom square from each column of odd length in A. If A - v = (),
this means that 0; = 1 or 0 according as  is odd or even. We have m;(v) =
m;( A) - 0; + 0;+ l' and .pAlv(t) is the product n(l - tmi(A)- 8; + 8+ 1) taken over
those i  1 for which 0; == 0 and 0;+ 1 == 1. It follows that bvI2(t2) / .pAlv(t) == dA(t) as
defined above.
5. Let
n
<I>(Xt,...,xn;t) = n (l-xJ-1 n(l-tx;xj)/(I-x;xj)
i-I i <j
= E P). ( Xl' .. . , X n ; t)
A
(Example 1). Let u be another indeterminate, then the formal power series
s (u) = E PA ( Xl' . .. , X n ; t) u >"0

:\:-$:/'.:
tWI'.'
-.: ..
t;here the sum is over all Ao  Al  ...  An  0, and A = (AI' . . . , An), is equal to
'," E <I>(xf',...,x:";t)
!,:::;'.:'. 1 - UOXP- &;)/2
,..<-", E I
5. SKEW HALL-LITTLEWOOD FUNCTIONS
233
."",,:
....;.,.;
";.JunlIned over all B = (B l' . . . , Bn) where each Bi is :I: 1 (hence a sum of 2 n terms).
tJ
r;-;,';' Let X denote the set of variables Xl"..' xn' and for any subset E of X, let peE)
t:'denote the product of the elements of E. (In particular, p(0) = 1.)
.;:>isuppose A = (AI' . . · , An) is of the form ( ILll, 11}"i, · · · , /.L'i/), where /.Ll > /.L2 > ... >
i;ffk;;;' 0, and the 'i are positive inegers whose sum is n. Then from (2.2) we have
.;
p.\ (x 1 , .. . , X n ; t) = E p (r 1 (1) ) 1£, .. . P (r 1 (k » p.k n xi - Ix j
/ /(Xj) </(Xj) Xi - Xj
i.:....t1.)
':.:summed over all surjective mappings f: X  {1,2,..., k} such that Ir1(i)1 = ri for
:-:)1' i  k.
:L.... Each such f determines a filtration of X:
7: 0 =Fo eF1 e ... cFk =X
-:- :"n:.
;
(all inclusions strict) according to the rule
 .. ".:
xEFrf(x)r.
-,'Conversely, such a filtration g- of length k determines a surjection f: X 
'{1,2,..., k} uniquely. From (1) it follows that
k
S(U) = E 1Tg-- E U 1-'0 n p( - F;-1 )J.tj,
!F i-1
;.:'(2)
(:where the outer sum is over all filtrations !T= (Fo,..., Fk) of X (with k arbitrary)
'.:and the inner sum is over all integers /.Lo, /.Ll'" · ,/.Lk such that /.Lo  /.Ll > /.L2 >
:;:.. > ILk  0, and finally
n Xi-txj
'IT' g-- =
f(x,) <f(xj) Xi - Xj
: where f is the function X -+ {I, 2,. . . , k} defined by g:
Now let Vi = /.Li - ILi+ 1 (0  i  k - 1), Vk = ILk' so that Vo  0, JJk  0 and Vi > 0
for 1  i  k - 1. Then the inner sum in (2) is
E
k 1 k-1 p(F)u 1
uVo+." + Vie n p(fi) Vi = n
;_ 1 1 - U ;-1 1 - p(Fj)u 1 - p(X)u
Vo, .. . , v Ie
= (1 - U) -1 Ag--(u) , say,
and hence from (2) we have
(3)
S(u) = (1 - u) -] E 'IT'yAg-(u),
!F

234 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
summed over all filtrations g- of X as before.
ObseIVe that
s ( u ) (1 - u) == E p..\ ( Xl' . .. , X n ; t) u Al
,\
and hence that S(uXl - u)lu-t is equal to <I>(Xt,..., xn; t), so that
(4)
<I>(Xt,...,Xn;t)== E 1TyAy(l).

The formula (3) shows that S(u) is a rational function of u (with coefficients in
Q(Xl'.'" xn' t». The denominators on the right-hand side of (3) are products of
factors of the form (1 - p(Y)u), where YeX. We may therefore express S(u) as a
sum of partial fractions: say
(5)
a(Y)
S(u) == E
YcX 1 - p(Y)u
To compute a(Y), we have to decompose (1 - u)-lAu) into a sum of partial
fractions in the usual way, for each filtration Sf which contains Y, and then add
together the results. The reader will not find it difficult to verify, using (4), that for
each YcX we have
(6)
a(Y) == <P(xi1,..., x:"; t)
where 8i == -lor + 1 according as Xi E Y or xi  Y. Since
n
p(y)= n xi= nxp-c;)/2,
xjeY i-I
this gives the stated result.
6. Show that
n(A) 1 + q-;
"q == (l-q-t)n
'-: lAut M..\I i 1 1 - q-i '
qn(p) 1
E IA t M I - n 1 -j I
#Leven U JJ. i 2 - q
where (as in Chapter II) M..\ is a finite o-module of type A, and q is the cardinality
of the residue field of o. (Apply the specialization Xi t-+ q-i(i  1) to the identities
of Examples 1 and 2 above, bearing in mind 3, Example 2(a).)
7. (a) Let Q(x; t) = QA(X'; t) where the x' variables are the products xitj-1(i,j 
1). (In A-ring notation, Q'A(X; t) = Q,\(x/(I- t); t). From (4.4) it follows that
(1)
E PA(x; t)Q(y; t) = n (1 - X;Yj)-1
A ;,j

r:'
5. SKEW HALL-LITTLEWOOD FUNCTIONS
235
i8r2'quivalently that (Q'A) is the basis of A(t] dual to (P.\), relative to the scalar
;t;t6duct of Chapter I, 4. .
l;ffJNOW let n be a positive integer and ,= exp2'TT'i/n, and set xj = '1-1 (1 j 
I#):x. = 0 for j > n. From 2, Example 1, if A is a partition of length  n we have
Irr:: J
;{]rf
R.Jt:;;i; :,
l;'C I .
fWhbre mj = mj(A) for j '> 1, and mo = n -1(A): The right-hand side of (2) will
riain 1 - ,n in the numerator, and hence will vanish, unless m s = n for some s,
111it is to say unless A = (sn); and in that case, since n( A) = tn(n - 1)s, we have
;7;;":':':-:  .
!::.; ">:.
. L.;..;.:.._.:::.":" ')'
'''''(3
PA ( 1, , , . . . , , n - 1 ; , ) = , n( A )'Pn ( , ) / Il 'Pm'< ( ) ,
. OJ
,;.
PA (1, , , . . . , ,n -1; () = ,"(A) = ( _1)s(n -1).
'iiF(OD1 (1), (2), and (3) it follows that
\';'
-I-:::: - - .
:'.{t..
E (_1)s(n- t)Q(s")(Y; ,) = Il (1- yp)-1
$O j;.l
i: ..:,,::: i..., -
:::'''_::! . 
rplhat
r:=:f(4 )
Q(s")(Y; ,) = (_l)s(n-1)hs(yn)
:.{here yn = (Yi, yi, . . .).
:1:() From (5.7') we have
:',(5)
Q(x; ,)q;(x; ,) == E .pA/(' )Q(x; ,)
A
'.(\Vhere q = Q(r» summed over partitions '\::> JL such that A - JL is a horizontal
t'strip, where .pA/ is given by (5.8'). It follows that, for each s  1,
:':' ..
;:;(6)
Q'P.l.I(S")(X; ')q(x; ,) = E .pA/(')Q'AU(Sn)(X; ,)
A
:.ith the same coefficients .pA/ as before.
, Let 'Ps: Ac --+ Ac denote the linear mapping defined by
'Ps(Q' (x; ,» -= Q' u (S")(x; ,).
....Prom (4) we have lI's(1) = ( _l)s(n -l)hs(xn). The equations (5) and (6) show that 'Ps
:.,commutes with multiplication by q, for each r '> 1, and hence that 'Ps is Ac-linear.
:-It follows that 'Ps(f) =: fCfJs(1) for all f E Ac, i.e. that
(7)
Q'p.u (s")(x; ,) == ( _1)s(1I-1)Q' (x; ,)hs(xn).
(c) Deduce that the symmetric functions Q(x; ,) enjoy the following factorization

236 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
property: let mi(A) = nq; + 'i where 0  ';  n - 1, and let p, = (lQ12Q2...) ,.."
(1'12'2 .. .). Then ' P--
Q'..\ (x; , ) = ( _1)(n -1)I/-'IQ (x; , )h/-,(xn).
In particular, when n = 1 (so that (= 1) we have PA(x; () = m)., and p = 0, J.L::::,\.
so we recover the fact that (h..\) is the basis of A dual to (mA). .
8. As in Chapter I, 5, Example 25, let  denote the diagonal map (f == f(x, y))
Also, for each f E A[t], let f- denote the adjoint of multiplication of f, relative t
the scalar product (4.8), so that (f-g, h) = (g,fh) for all g,h E A[t].
(a) Show that  is the adjoint of the product map, i.e. that
(1)
(tJ.f, g  h) = (f, gh)
for all f, g, h E A[t]. (Use (5.5) and (5.1).)
(b) We have
(2)
qll = E qr ti!J qs
r+s-,.
and hence by (a) above
(qn, gh) = E (qr' g )(qs, h)
r+s-,.
for all g, h E A[ t]. By taking g = q m deduce that
.1 _ { (1 - t)qn-m
qmq,. - q,.
if m  1,
if m = O.
If we define
QJ.(u) = E q,; um ,
mO
Q(v) = E qnun
,.;;.0
where u, v are indeterminates, this last relation may be written in the form
(3)
Q.L(u)(Q(v» = Q(v )F(uV)-1
where F(uv) = (1- uv)/(1- tuv) as in 2.
(c) Show that Q.l(u): A[t] -) A[t, u] is a ring homomorphism. (We have
(qll.L(fg), h) = (fg, qnh)
= (fti!Jg,(q,)(h»

':-."
}.
:::. ,
i:::':.-. ..
....
5. SKEW HALL-LITTLEWOOD FUNCTIONS
237
il>Y (1) and the fact that .Ii is a ring homomorphism,
Fr:::
i:' .
'!::..',
r:<:.,
>
= E «q/f) 0 (q/-g), h)
t;;;.;. :
r+s-n
:.:..
= E «q/-!) (q/-g ), h)
1::.
r+sz:n
I:."
£;by (2) and (1) again.
r:r Hence
qn.l.(fg) = E (q,.l.f)(qs.l.g).)
t:_, .
r+s-n
:,,>
i  . .
r. It now follows from (3) that
..
J4)
(t(u) 0 Q(v) = F(uD) -lQ(V) 0 Q.l(u)
i"as operators on A[t], where Q(v) is regarded as a multiplication operator, i.e.
[;'Q( v ) f = En > 0 q n tv n.
[!d) Let Q(u)-t = E'nun =R(u) and let
R.l.(u) = E 'n.l. un.
From (4) we have
R(v) 0 Q.l(u) = F(uv) -lQ.l(u) 0 R(v)
and hence, on taking adjoints,
(5)
R.l.(v) 0 Q(u) = F(uv)Q(u) 0 Rl.(v).
(e) Now define operators Bn (n E Z) by
Bn = E qn+i'i.l.
i> 0
or collectively
(6)
B(u) = E Bnun = Q(u) 0 R.l.(u-1).
n
Since
( 1 - tn )
Q(u) = exp E n Pnun
n>l
it follows that
B(U)=exp( E 1 tnpnun)oexp(_ E ltn pnLu-n).
n>l n n>l

238 III HALL-LITTLEWOOD SY.MMETRIC FUNCTIONS
Also define
(7)
B(Ul'.'., un) = Q(U1)-..' Q(un)Rl.(ult) ... Rl.(u; 1)
where Ut'..., un are independent indeterminates. Deduce from (5) that
(8)
B(u)B(v) =B(u,v)F(u-1v)
and hence by iteration that
(9)
B(ut)... B(un) = B(Ut,..., un) [J F( UitUj).
i<j
From (9) it follows that
B(Ut)... B(un)(l) = Q(ut)... Q(un) [J F( ui1Uj)
i<j
= Q(Ul'''.' un)
in the notation of (2.15). Hence for all partitions A = (At,..., An) of length  n we
have
Q,\(x;t) =B,\ ...B,\ (1).
1 n
(0 Deduce from (8) that
B (u) B ( v) (u - tv) = B ( v) B (u) (tu - v)
and hence that
Bm-tBn - tBm Bn-t = tBnBm -1 - Bn-1 Bm
for all m, n E Z.
Notes and references
Example 7 is due to Lascoux, Leclerc, and Thibon [L6].
Example 8. The expression B(u) is a 'vertex operator': see Jing [J12].
6. Transition matrices
We have met various (integral or rational) bases for the ring A[t] of
symmetric functions with coefficients in Z[t]: in particular (4)
(PA), (Q,\) are dual bases,
(qA)' (m,\) are dual bases,
(8,\), (s,\) are dual bases,
with respect to the scalar product of 4.
(6.1)

i.,:,.
j;;.;  ..
(..:".>,'
T.....;.: .
:i::.....;.
:'T.;':'>
6. TRANSITION MATRICES
239
?;BY (2.6) the transition matrix
';J  .
K(t) = M(s, P)
!fbn1 the Schur functions SA(X) to the Hall-Littlewood functions p...<x; t) is
rctlY upper unitriangular. For any t, u we have
t.2) M(P(x; t), P(x; u» =K(t)-l K(u);
lso K(O) is the identity matrix, and K(l) is the Kostka matrix K of
fhapter I, 6. Also from (6.1), (6.2), and Chapter I, (6.1) we have
M(Q,q) =M(P,m)* = (K(t)-l K)* =K(t)'K*
here * denotes transposed inverse. Since (Chapter I, 6) K* is the matrix
pf the operator fl; < j(l - R;j)' it follows from (2.15) that
6.3) K(t)' is the matrix of the operator TIj<j(l- tR;j)-l.
Moreover, the rule (5.11') for expressing PA(x; t) as a sum of monomials
shows that
.(6.4)
(K(t)-l K)Ap. = E «PT(t)
T
summed over all tableaux T of shape A and weight JL.
The table of transition matrices between the six bases listed in (6.1) is
now easily calculated, and is as shown on p. 241 (b(t) denotes the diagonal
matrix whose entries are the b).. (t )).
, For n  6 the matrices K(t) are given below.
4 31 22 212 14
3 21 13
4
31
22
212
14
1 t t2 (3 (6
1 t t+ t2 t3 + t4 + (5
1 ( t2 + (4
1 ( + (2 + (3
1
2 12
2
3
21
13
1 ( t
1 (+ t2
1
5 41 32 312 221
213
15
5
41
32
312
221
213
15
1 t t2 (3 (4 t6 t10
1 ( t+ t2 t2 + t3 (3 + (4 + (5 (6 + t 7 + t8 + t9
1 t t+ (2 t2 + (3 + t4 t4 + (5 + t6 + t 7 + t8
1 t ( + t2 + (3 t3 + (4 + 2 (5 + (6 + t 7
1 t + t2 t2 + t3 + t-t + (, + ('
1 t + t2 + t3 + t4
1

240
III
HALL-LITTLEWOOD SYMMETRIC FUNCTIONS

....
....
-
rf')  +
... ,..... 0\_
- - .... 00
+ + - + -
N +
N  N + "L
- - ... I"'-
 -  + QI ....
.- + + + - - + II'!
- 2 N I"'- -
+ .... c - + '<>
- + N +
- - ,-j ..... -
rf') N N - :c + - N 't
....
- + - + + -
+ +  .c_ +
+ ,. QI \C II'!
0 - + N - - rf') ,.....01
VI N .... --
'oD ,-j .... - N + I"'- + + + -
,.....c .... .... - +
+ + + 00 ('f) , VI 't
- - - - N
01  + + N N -
 - +
,-j N N + +
- \C 'oD + tf'\
+ + - - - -
+ N r:r_ tf'\
-
0 OCI I' +
.... .... - + +
- + + VI I N
- - -
.....  N
-
+

-
\0
-
 r-. +
- -
+ + II'!
 - 
- r-- f.o_ C"'
-
+ I"'- + + rf')
+ + -
oc -
0 foC_ I'! +   -:t
,.....c .... - - - +
-
N - + N N VI N + + N ,...-4
- -
I"'- + + + + - N
- - +
!v\_  tf'\
+ -:t - + -
-
 + + N
- + -
 1...
- N
-
\0 r,  (f'\
- ....
-
+' + +  +
-
I"'- VI   + N - - '1""""'4
-
- - + N
+ + N
-
rr'I - +
 - -
- N

-
II'! + N
\0 - rf') -
N + ('t'} ('t'} + 0 ,.-..4
- - - -
 + -
-N
-
\f)  jrr) ....
- ....
+ + + N
-
'1""""'4 'oD  rf')  ('t'} + ,.-..4
('f') - - - - -
+ + + -
tf'\ N -
- ....
tf'\ N
,.....c - -
N  + + - - ,.....c
-
('f') N -
-
rf') N 0 ,.....c
('f') - .... ....
N
-
--c rf') + 
 .... -
-
N N
 - - 
,.....c - ,.....01
V')
,.....01
N
,.....c
N
N
(f'\
tf'\
N
N
\0
,.....01 N
N N N(f'\ rf') 't -c
\C ,...-4 N   ('f') --c N N ,.....oI,.....c
V') ('f') NN

6. TRANSITION MATRICES
241

:::-
-
--- :::-
 -
:::- ---
 
I -
--
...-  :::- 
- I
--  -
..Q .... --- -- 'I""""C
I  -
...- --
- ...- ..Q
-- -
 -- ...-
..Q -
--
 
I


I ....
-- I
-
--- .... ...-
 -
I --
--  ...- 
... - I -
I --- ....- --- ...-
..Q -  'I""""C -
 * -- --
 ...- ,.Q
--:;:- - *
--
-- ..Q ...-
 -
--
 ::<

I
-- ...
- * I
---
,.Q - .... -- *
- I -
...- --- -- ...-
-  -- 'I""""C ..0 -
--- - --
 -- ...- 
..Q -
 --
 
I

'i:' 
...- ---- -
-  ----
-- 
 * ...- --
 - - *
 - 'I""""C -- -- 
---- ..Q 
I  ----
 E< 
*
 *
:::- 
- *
--  :::-
 -
* "-'"
 ....-  
I -
-- *
...-  :::- -
- 'I""""C I 
-- -
,.Q - -- --
I  -
...- --
- -- ..Q
--- -
 -- --
..0 -
--
 
I

 

I -
...-  I
-
--   ....-
 -
I --
...-  ...- 
- I - 
'I""""C -- -- --- ...-
,.Q -  -
-- --
*  ...- ..Q
- - *
- ---
-- ..Q ...-
 -
--
 

o


E
E
o

V)
o..c


242 III HALL-LITTLEWOOD SYM-METRIC FUNCTIONS
These tables suggest that the polynomials K). (t) have all their coeffi-
cients  O. Since KAi1) = KAiJ. is «;hapter I, (6.4)) the number of tableaux
of shape A and weight , Foulkes [F7] conjectured that it should be
possible to attach to each such tableau a positive integer c(T) such that
K).p. (t) = L tc(T),
T
the sum being over all tableaux T of shape A and weight J.L. Such a rule
has been found by Lascoux and Schutzenberger ([L4]; see also [BI0]), and
goes as follows.
We consider words (or sequences) w = a1 ... an in which each ai is a
positive integer. The weight of w is the sequence J.L = ( J.Lt, J.L2' · . .), where
i is the number of a { equal to i. Assume that J.Ll  J.L2  ... , i.e. that J.L is
a partition. If J.L = (1 n) (so that w is a derangement of 12. . . n) we call w a
standard word.
(i) Suppose first that w is a standard word. Attach an index to each
element of w as follows: the number 1 has index 0, and if r has index i,
then r + 1 has index i or i + 1 according as it lies to the right or left of r.
The charge c( w) of w is defined to be the sum of the indices.
(ii) Now let w be any word, subject only to the condition that its weight
should be a partition J.L. We extract a standard word from w as follows.
Reading from the left, choose the first 1 that occurs in w, then the first 2
to the right of the 1 chosen, and so on. If at any stage there is no s + 1 to
the right of the s chosen, go back to the beginning. This procedure selects
a standard subword wt of w, of length ,It. Now erase wt from wand
repeat the procedure to obtain a standard subword W2 (of length Ji2)' and
so on. (For example, if w = 32214113, the subword w1 is 2143, consisting of
the underlined symbols in w: 32111. When w1 is erased, we are left with
3211, so that W2 = 321 and W3 = 1.)
The charge of w is defined to be the sum of the charges of the standard
subwords: c(w) = E c(wi). (In the example above, the indices (attached as
subscripts) are 2t104231 for Wh so that c(w1) = 4; 322110 for W2, so that
c(w2) = 3; and 10 for w3, so that c(w3) = 0: hence c(w) = 4 + 3 + 0 = 7.)
(iii) Finally, if T is a tableau, by reading T from right to left in
consecutive rows, starting from the top (as in Chapter I, 9), we obtain a
word w(T), and the charge c(T) of T is defined to be c(w(T)).
The theorem of Lascoux and Schutzenberger now states that
(6.5) (i) We have
K).p.(t) = E tc(T)
T
summed over all tableaux T of shape A and weight J.L;

6. TRANSITION MATRICES
243
o:(ii) If ,\  JL, KJ,.p,(t) is monic of degree n( JL) - n('\). (If ,\  J.L, we know
\already (2.6) that KAp,(t) == 0.)
For the proof of (6.5) we refer to [IA], [BID]. (6.5Xii) is an easy
c9nsequence of (6.3) (see Example 4).
Examples
1. K(n)I£(t) = tn(l£) for any partition J.L of n. This fol1ows from the identity
hn = E tn( I£)PI£
1p,I-n
(3, Example 1; 4, Example 1). Alternatively, use (6.5).
2. KA(l")(t) = tn(A')cp,,(t)HA(t)-l for any partition A of n, where
HA(t) = n (1 - th(x»
xEA
is the hook-length polynomial. This follows from 4, Example 2, since K)'(111)(t) is
the coefficient of S),.(x; t) in Q(111)(X; t) = CPn(t)e,,(x).
From Chapter I, 5, Example 14, it follows that
K).(l")(t) = E tC'(T)
T
summed over all standard tableaux T of shape A, where e'(T) is the sum of the
positive integers i < n such that i + 1 lies in a column to the right of i in T.
3. Tables of the polynomials KAI£(t) suggest the following conjecture: if the lowest
power of t which is present in KAI£(t) is tD(A,JJ-), then a(A, J..L) = a( J..L', A').
4. Let 81"'" 8" be the standard basis of Z n and let R + denote the set of vectors
&i - Sj such that 1 <:. i <j <:. n. For any  = (1'''.' n) E zn such that E i = 0, let
P(;t) = E tEma,
(mQ)
summed over all families (ma)a e R+ of non-negative integers such that g = E mQ a.
The polynomial P( f; t) is non-zero if and only if  = Ei - 1 'YJ;( 8; - 8; + 1), where the
11; are ;>0, and then it is monic of degree L'TJ;. Since "1; = 1 + ... + i (1  i  n -
1), the degree of P( f; t) is
E (n - i) i = <  ,  )
where as usual  = (n - 1, n - 2, . . . , 1,0) and the scalar product is the standard
one.
Now by (6.3), K(t»,.1£ is the coefficient of SA in
n (1 + tR ij + t 2 R  + ... ) S JJ- '
1 < , <} < n

244 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
hence is the coefficient of a A + 8 in
E tEmaa,.,.+8+r.maD
(ma)
or equivalently the coefficient of x A + 8 in
E B ( W) E t E m a X w( JJ. + 8 + r. m a a) .
we Sn (ma)
It follows from this that
(1)
K (t ) AP. = E B (W ) P ( W -1 (,\ + 8) - ( J.L + 8); t) .
weSn
Now
(w -1 (,\ + 5) - ( J.L + 8), 5) = (,\ + 5, w5) - ( J.L + 5, 5)
 (,\+ 5,5) - (J.L+ 5,5)
= n( p.) - n('\)
and equality holds if and only if w = 1. Hence the dominant term in the sum (1) is
that corresponding to w = 1, and therefore K(t)AP. is monic of degree n( J.L) - n(A).
From (1) it follows that K A,.,.(t) is equal to the coefficient of x p. in
det(x/l-i+j)/ni<j(l-txilXj) (P. Hoffman).
5. From the tables of this section and Chapter I, 6, we have
M(e,P) =M(e,s)M(s,P) =K'JK(t)
or equivalently
(1)
M(e, P) Ap. = E K(t )v,.,.Kv' A.
II
On the other hand, the formula (1) in the Appendix to Chapter II shows that
M(e P) =" td(A)
, A P. /.,.., ,
A
where the sum is over- all row-strict arrays of shape p. and weight '\, and d(A) is
defined in Chapter II, Appendix (AZ.5).
From (1) and (2) we conclude that Foulkes's conjecture is equivalent to the
following combinatorial statement: for any two partitions '\, J.L there exists a
bijection cp between pairs of tableaux (T, T') of conjugate shapes and respective
weights p. and '\, and row-strict arrays of shape J.L and weight '\, such that
d( cp(T, T'» does not depend on T'.
Indeed, letting c(T) = d(cp(T, T'», we have
(2)
K(t) =" tc(T)
AIL I....J
T

i:
6. TRANSITION MATRICES
245
sUJ11med over all tableaux T of shape A and weight J.L. It would be interesting to
;\'e an explicit construction of such a bijection cpo
6. The polynomials KAI£(t) have the following geometrical interpretation, due to G.
:tusztig [L1S]. Let V be an n-dimensional vector space over an algebraically closed
i'field k. The conjugacy classes of unipotent operators on V are parametrized by the
partitions of n: the class cA consists of operators with Jordan blocks of sizes
-,'AI' '\2'''' · The closure cA of C.\. is an algebraic variety (singular, in general) of
dimension 2n( A'), and we have
CA = U cIJ-.
#J-"A
Let x E cp, C CA' let Bi(CA) denote the Deligne-Goreski-MacPherson cohomology
sheaves on cA (see, for example, [L1S]), and let Hi(c).)x be the stalk of Hi(C).) at x.
Then ni(CA) = 0 for i odd, and
E ti dim H2i(C.\.) x = tn(I£)-n().)K).I£(t-1).
iO
, This gives another proof of (6.5) (in and of the fact that aU coefficients of K).I£(t)
:are  O. It would be interesting to give a geometrical interpretation of the
'Lascoux-Schlitzenberger theorem (6.5) (n.
,7. If a = (a(O), a(1), a (2), ...) is any finite sequence of partitions, let
n
Pnk( a) = E (ai(k-1) - 2 ai(k) + a;(k+ 1»
i-I
::for k, n  1, and
( a(k-1) - a(k»)
C(a) = E n n.
k,nl 2
[a+b]
Also let [a, b] denote the Gaussian polynomial b . Then
KAI£(t) = E tC(er) n [Pnk( a), ak) - akH)]
a k,n>l
where a runs through .all sequences of partitions such that a(O) = J..L' and I a(k)1 =
'Xk+ 1 + Ak+ 2 + ... for k  1. (This is a consequence of (6.5Xi); for the proof, see
;_[_9].)
.. Let NA( 0) denote the number of submodules of a finite a-module of type A,
"here (as in Chapter II) 0 is a discrete valuation ring with finite residue field k.
then
NA( 0) = qn(.\.) E (J.L1 - J.L2 + 1)KI£A(q-l)
J.L
summed over partitions J.L  A of length  2, where q is the cardinality of k.

246 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
(We have
M.(o) = E g;v(o)
p., II
= qll(A) E q-n( p,)-n(v)f;v(q-1)
p., v
= qll().) E q-n( p,)- n(v)( Q). (q-1), Pp,(q -1 )Pv(q-1)
p., v
= qn(A)(QA(q-l), (E hn )2).
Now
(E hn)2 = I1 (l-X;)-2 = E s/xh/1, 1)
p.
z= E (J.Ll - J.L2 + l)sp,(x),
p.
summed over partitions J.L of length  2. Hence
NA(o) =qn().)E(J.L1 - J.L2 + 1)(Q).(q-1),s)
p.
= qll().) E (J.Ll - J.L2 + 1)Kp,).(q-1 ).)
p.
Notes and references
Examples 5 and 6 were contributed by A. Zelevinsky. Example 7 is due to
Kirillov and Reshetikhin [K9].
7. Green's polynomials
Let X(t) denote the transition matrix M(p, P) between the power-sum
products Pp and the Hall-Littlewood functions PA:
(7.1)
pp(x) = E XpA(t)P).(x; t)
A
(so that p is the row-index and A the column-index). By (2.7), X;(t) E Z[t],
and is zero unless IAI == I pl. When t = 0 we have PA(x; 0) = s..\(x), so that by
Chapter I, (7.8)
(7.2) XpA(O) = XpA,
the value of the irreducible character X A of Sn at elements of cycle-type p.

7. GREEN'S POLYNOMIALS
247
:;grom (4.1) and (4.4) we have
. "'::
I;t.;, E Zp(t)-l pp(x)pp(y) = E bA(t)PA(x; t)PA(y; t).
(r;; I pl-n IAI-n
Itbstituting (7.1) in this relation, we obtain
i);:..:
t1:.3) X'(t)z(t) -1 X(t) = bet)
;?'':/""'
here X'(t) is the transpose of X(t), and bet) (resp. z(t» is the diagonal
Hnatrix with entries bACt) (resp. z).(t». An equivalent form of (7.3) is
::,,,,,'".""
",":"':'.: .
[t(7.4) X(t )b(t) -1 X' (t) = z(t).
t#. other words we have the orthogonality relations
::i ' .".
 .' i .. .'.
f 3' ')
.sf7 ·
':;'l;:';;
E Zp(t) -1 X;(t )X:(t) = S).p,b). (t),
Ip/-n
-d):.:..
r!.41)
 ;".:; ':
:i' .
E b).(t)-1 X;(t)X;(t) = Spuzp(t)
I>.I-n
iWich when t = 0 reduce to the orthogonality relations for the characters
.d(the symmetric group Sn'
'.:,::.Since X(t)-l = b(t)-1X'(t)Z(t)-1 by (7.3), we can solve the equations
)(7.1) to give
J7.5)
Q).(x; t) = E Zp(t)-l X:(t)pp(x).
p
:(i':'.Next, since M(s,P)=M(p,S)-lM(p,P), we have K(t)=X(O)-lX(t),
:o that
(7.6)
X(t) =X(O)K(t)
or more explicitly, using (7.2),
(7.6')
X;(t) = E Xp).K).p,(t).
A
Since K).p,(t) is monic of degree n) - n('\)  n) (6.5), it follows that the
dominant term on the right-hand side of (7.6') is xn)K(n)p,(t) = tn(p,) (6,
Example 1), so that
(7.7) X;(t) is monic of degree n).
Green's polynomials Q;(q) are defined by
(7.8) Q;(q) = qn().)X;(q-l).

248 III HALL-LITTLEWOOD S.YMMETRIC FUNCTIONS
..,
..,:.;.;
In terms of these polynomials, the orthogonality relations (7.3') an';
(7.4') take the forms d_
(7.9)
E yp(q) -1 Q:(q )Q;-(q) = 8>"jJ.a>..(q),
I pl:zn
E a).(q)-lQ;(q)Q; (q) = 8puYp(q),
IAI=n
where a).(q) = ql>"I+2n().)b>..(q-1), yp(q) = q-I PIZp(q-l) (Chapter II, (1.6)).
(7.10)
Also (7.6') now becomes
Q:(q) = E Xp).K>"jJ.(q)
A
where K).jJ.(q) = qn(jJ.)K).p.(q-l) = qn().) + terms of higher degree by (6.5).
(7.11)
Examples
1. xn)(t) = 1 for all partitions p of n. This is a particular case of (7.7). Equiva..
lently, Qn)(q) = 1.
2. X()(t) = tn().)CPl(A)_1(t-1), or equivalently Q()(q) = 'Pl().)-l(Q). This follows from
the identity (3) of 3, Example 1:
1(>..)
n (1 - Xi Y ) / (1 - Xi) = E t n().) n (1 - t 1 - j Y ) . p). (x ; t).
i>1 ,\ ja::1
Divide both sides by 1 - Y and let Y -+ 1: since
1 ( Il 1 - xiY _ 1) __  Xi 
lim i..J 1 _ = i..J Pn(x)
y-+ 1 1 - Y i 1 - Xi i Xi n  1
we obtain
I( >..) - 1
Pn(X) = E tn().) n (1 - t-i)p). (x; t)
I>..I-n 1
which gives the result.
3. X1")(t) = ni_1(ti - 1)/nj>1(tPj - 1), or equivalently Q111)(q) = (()n(q)/
rIj> 1(1 - q Pj). This follows from (7.5) with A = (In): since Q(l")(X; t) = 'Pn(t)en(x)
we have
E Zp(t) -1 Xlll)(t)pp(X) = 'Pn(t)en(x)
p
= CPn(t) E 8pz;lpp(X)
p
so that X111)(t) = 8pz;lzp(t)CPn(t), which is equivalent to the result stated.

7. GREEN'S POLYNOMIALS
249
.J,,)(t) = (1. - t)nET 'PT(t), summed over all standard tableaux T of shape A.
..'t(/) was defIned In (5.9).)
or by (7.1) we have
. X(111)(t) = (Q).,pi) = (l-t)-n(Q).,qf)
'::::'.-;";::-.;:
\,!;j,.:}.: "-
filtT.;'-;
li:t4.S) and (5.11).
f'py taking Xi = ti-1 in (7.5) we obtain by 2, Example 1
-.::
1o.;..,.:;:: tn().)  -lX).(t)
-(. = L- Z
f!f;:. p P P
:TJ. .
tin terms of Green's polynomials,
EJ-:' ;";
::.:> - 
(.:.: .
t.tr.:all partitions A of n.
f6Y'---X ).(1) = (p p' h). > (for the scalar product of Chapter I), hence is the value of the
J1rtdud character ind(1) at elements of type p.
¥:;\\.:'-:;. . ).
i'Let w be a primitive r th root of unity (r  2), and consider the effect of
ftpiacing the parameter t by w. Let Kr = Q( w) be the rth cyclotomic field,
;11):= Z[w] its ring of integers. By (2.7) the P).(x; w) form an !Or-basis of A 0z!Or.
....r
QI1the other hand, the definition (2.12) of b).(t) shows that b).(w) and therefore
);{so Q.\(x; w) is zero if mi(A)  r for some i, that is to say if the partition A has r
'Qr:)nore equal parts. Let 9'(r) denote the set of partitions A such that mi( A) < r for
U/i, and let A(r) denote the Dr-submodule of A 0zDr generated by the P).(x; w)
i A E9'(r). Then A(r) is an O,-subalgebra of A 0z!O r.
'i1£.t A(r) = A(r) (), K" which is the K,-vector space generated freely by the
:(x; w) such that A E9'(r). Since zp( w)-1 = Z; 1 n j> 1(1 - w Pi) is zero if any Pi is
visible by r, it follows from (7.5) that A(r) is contained in the Kr-algebra B(r)
'gnerated by the power sums Pn such that n is not divisible by r. Now if A(r)' B()
"afe the components of degree n of A(r)' B(,) respectively, it follows that
n > 0 dim Afr) . un is the generating function for partitions A with no part repeated
:ore than r - 1 times, so that
= (1 - t) -n E 'PT(t)
T
E z;lQ;(q)=1
Ipl-n
E dim A{'r). un = TI (1 + ui + ... +u<r-l)i)
n;;.O i;;.1
.f:::..' .
In
= TI (1 - u,i) /(1 - ui),
i;;.1
rid En> 0 det B() . un is the generating function for partitions p none of whose
parts is divisible by r, so that
i,::  :
E dim B() . un = TI (1 - u;) -1 .
n;;.O il
i:# 0(,)
7)

250 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
From (1) ad (2) it follows that dim A{r} = im B(} and hence that A(r) === B .':'
hence A(r) IS a K,-algebra and therefore A(r) IS an CJ r-algebra, as asserted. (?):'1
The scalar product defined in '4 is non-degenerate on A(r)' and the orthogOl-
ity relations (7.3'), (7.4') are now orthogonality relations for the matrix XAt\
where ,\ runs through partitions of n which have no part repeated r times or  cu ':-:
and p through partitions of n which have no part divisible by r. ore,-:,
8. Let us write
(1)
Q:(q) = E f/Iti(p)qi
(I ILl = I pi ==n}:
where the coefficients I/J ti may b regarded as class-functions on Sn. Hotta and
Springer [H9] have shown that t/1t' is a character of Sn' for each partition Ii of n'
they define an action of S n on the rational cohomology H *(XIL) of the variety X.
(Chapter II, 3, Example) and show that p.
"'ti(W) = s(w ) trace ( w-l, H2i(X»)
where as usual s(w) is the sign of w E Sn.
If
(2) "'ti = E kiZ xA
A
is the decomposition of '" t i as a sum of irreducible characters, it follows from (1)
and (2) and (7.11) that
KAIL (q) = E kii qi
i
with coefficients kii integers  o.
When p = (In), Qtilf)(q) is the Poincare polynomial of the variety XIL, When q is
a prime-power, we have QIJ,,)(q) = IXIL(q)l, the number of Fq-rational points of Xp."
The representations I/J' i and their generalizations are studied in a number of
recent papers. We mention the paper of W. Borho and R. MacPherson [B6], whef(
is explained their relationship with the Deligne-Goreski- MacPherson sheaves O'h
closures of unipotent classes (6, Example 6).
Notes and references
The polynomials Q:(q) were introduced by Green [G11], who proved the
orthogonality relations (7.9), (7.10). For tables of these polynomials see
Green (loc. cit.) for n  5, and Morris [M12] for n = 6, 7.
8. Schur's Q-functions
We have seen in A2 that when t = 0 the Hall-Littlewood function P).(x;t)
reduces to the Schur function SA' and when t = 1 to the monomial
symmetric function mA; and in A3 that when t = q-t, q a prime power, the
PA(x; t) have a combinatorial significance via the Hall algebra. There is at

rt:': ;"
:::
8. SCHUR'S Q-FUNCTIONS
251
1eaSt one other alue of t that is of prticular intrest, namely t = -1: the
'1;-:- metric functIons Q>..(x; -1) were Introduced In Schur's 1911 paper [S5]
the projective representations of the symmetric and alternating groups,
Whre they play a role analogous to that of the Schur functions in
onnection with the linear representations of the symmetric groups
if(Chapter I,  7).
lrit-ro simplify notation, we shall write q" P)., Q). for q,(x; -1),
ii(x; -1), Q,,(x; -1), respectively. As in 2, let
. Q(t) = E q,t'
r;;.O
iithe symbol t now being available again), then from (2.10) we have
-- 1 + tx.
l8.1) Q(t) = If 1 _ tx; = E(t)H(t)
-'from which it follows that
Q(t)Q( -t) = 1
;that is to say
;t82)
E (-1)' q,qs = 0
r+s-n
:for all n  1. When n is odd, (8.2) tells us nothing. When n is even, say
n' = 2m, it becomes
m-1
) ,,( 1)' - t 1 ( 1 )m -1 2
-:(8.2' q2m = i...J - q,q2m -, +"2 - q m,
r-1
-,
'r::.: 
rhich shows that q2m E Q[qhq2,...,q2m-t] and hence (by induction on
m) that
!".
(8.3)
q2m E Q[ ql' q3' qs, . · ., q2m -1].
;' Another consequence of (8.2') is the following. Recall (Chapter I, 1,
xample 9) that a partition is strict if all its parts are distinct.
8.4) Let A be a partition of n. Either A is strict, or q A = q).l Q).2 . .. IS a
Z-linear combination of the qJL such that J.L is strict and J.L> A.
Proof. We proceed by induction, assuming the result true for all J.L > A.
The induction starts at A = (n), which is strict.) If A is not strict then for
ome pair i <j we have Ai = Aj = m, say. The relation (8.2') then shows
that q>.. is a Z-linear combination of the qp. where J.L = RjA, 1  r  m. By
Chapter I, (1.14) each such J.L is > A, and so (8.3) is true for J.L. This
completes the induction step. I

252 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
Let r denote the subring of A generated by the qr:
r = Z[ q l' q 2 , q 3' · .. ] ·
r is a graded ring: r = $n:> 0 rn, where rn = r n An is spanned by the q
such that I AI = n. A
Also let rQ = r  Q = Q[ql' q2, Q3' ...]. From (8.3) it follows that rQ is
generated over Q by the qr with r odd. In fact these elements are
algebraically independent. For from (8.1) we have
!2'(t) '(t) ]{'(t)
Q(t) = E(t) + H(t) =P(t)+P(-t)
= 2 E P2r+lt2r,
rO
and therefore
rq r = 2( PI q r - 1 + P 3 Q r - 3 + ... ) ·
The series on the right terminates with Pr- 1 ql if r is even, and with p, if r
is odd. Hence these equations enable us to express the odd power sums in
terms of the q's, and vice versa. Since (Chapter I, 2) PI' P3' ... are
algebraically independent over Q, it follows that
(8.5) We have
rQ = Q[Pr: r odd] = Q[qr: r odd]
and the qr (r odd) are algebraically independent over Q.
A partition A is odd if all its parts are odd. For each n  1, the number
Nn of odd partitions of n is equal to the number of strict partitions of n,
because the generating function for odd partitions is
1 1 - t2r
E t1A1= 0 1 2,-1 = 0 1 ,= O(1+t'),
A odd r 1 - t r 1 - t r 1
which is the generating function for strict partitions.
(8.6) (i) The q,\, A odd, form a Q-basis of rQ.
(ii) The q,\, Astrict, form a Z-basis of r.
Proof. (i) follows from (8.5), and shows that the dimension of rQ is Nn.
From (8.4) it follows that the q,\, where A is a strict partition of n, span rn
and hence also rQ. Since there are Nn of them, they form a Q-basis of rQ.
Hence they are linearly independent over Q, and therefore also over Z,
which proves (ii). I

8. SCHUR'S Q-FUNCTIONS
253
Consider now PA = P,\(x; -1) and Q,\ = Q,\(x; -1). The symmetric func-
/tions P,\ are well-defined and non-zero for all partitions A, and since the
: transition matrix M(P, m) is unitriangular, it follows that the PA form a
:..Z..basis of A. On the other hand, bA( -1) = fli  1 'Pmt('\)( -1) will vanish if
.any miCA) > 1, that is to say if the partition A is not strict. If A is strict, we
>have b,\( -1) = 21('\), and hence
;\(8.7)
{ 21('\)p
Q - ,\
A-
o
if A is strict,
othelWise.
From (2.15), with t = -1, we have
.(8.8) Let A be a strict partition of length  n. Then QA is equal to the
coefficient of t'\ = t;l t f2 . .. in
n
Q(t1,...,tn) = n Q(ti) n F(ti-1tj)
1- 1 I <J
;where
l-y
F(y) = 1 = 1 + 2 E (-1)' y',
+ y r 1
As in 2, (8.8) is equivalent to the raising operator formula
(8.8')
i-R..
n I}
Q'\=..l+Rq,\.
1 <J ij
From (8.8') it follows that Q,\ is of the form
Q,\ = q,\ + E a,\p.qp.
J.L>A
with coefficients a,\p. E Z. If A is strict, it follows from (8.4) that we have
Q,\ = q,\ + E ap' qp.
J.L>'\
where now the sum on the right is restricted to strict partitions J..L > A.
From (8.6) (ii) it now follows that
-(8.9) The Q,\, ,\ strict, form a Z-basis of r.
We come next to Schur's definition of Q,\. From (8.8) it follows that

254 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
Q(r,s) (where r > s  0) IS the coefficient of trus in Q(t, u) ::::
Q{t )Q{u)F(t-1 u), so that
(8.10)
s
Q(rts) = qrqs + 2 E (_l)i qr+iqs-i.
i-1
If r  s we define Q(rts) by this formula. Then it follows from (8.2) that
Q(St r) = - Q(r, s).
Now let A be a strict partition, which we may write in the form
A = (At, A2,..., A2n) where At > ... > A2n  O. Then the 2n X 2n matrix
M).. = (Q().." )..J»)
is skew-symmetric, and Schur's definition of Q).. (which for us is a theorem)
reads:
(8.11)
Q).. = Pf( M)..)
where Pf{M)..) is the Pfaff tan of the skew..symmetric matrix M)... (We recall
that if A = (a;j) is a skew symmetric matrix of even size 2n X 2n, its
determinant is a perfect square: det{A) = Pf(A)2, where
Pf{A) = E B{W). aw(1)w(2)'" aw(2n-l)w(2n)
w
summed over W E S2n such that w{2r - 1) < w(2r) for 1  r  n, and
w(2r - 1) < w(2r + 1) for 1  r  n - 1.)
Proof of (8.11). From above, Pf(M)..) is a sum of terms of the form
:t Q( ,ul t ,u2) · · · Q( ""211 - I' ,u211)
where (J..Ll'"'' 1-L2n) = (Ai"'" Ai ) is a derangement of (AI'''.' A2n). By
I 211
(8.8) the term just written is equal to the coefficient of t).. = t fl t;2 . .. in
:i:Q{tl)...Q(t2n)F(f;lti )...F(t;-l ti ),
1 2 2,,-1 211
and consequently Pf(M)..) is the coefficient of t).. in
(1)
Q(t1)... QCt2n)Pf( (F( ti\))).
Now F{ti-t tj) = (ti - tj)/{ti + tj)' and it is well..known that
(t.-t.) (t.-t.)
Pf · J =0 I ).
ti + tj i <j ti + tj
(2)

8. SCHUR'S Q-FUNCTIONS
255
From (1) and (2) it follows that Pf(M,\) is the coefficient of t A in
Q(t1)... Q(t2n) n F(t;-l tj) = Q(tt,..., t2n)
i<j
hence by (8.8) is equal to Q,\. I
(For a proof of (2) (which is a classical result) see Example 5 below; and
for an extension of (8.11) to skew Q-functions Q'\I",,' see Example 9.)
Next, the scalar product of 4 is not well-defined on all of A when
t == -1, because ZA(t) has a pole at t = -1 if any part of A is even. But if A
is odd it follows from the definition of ZA(t) that z,\( -1) = 2-/(A)Z,\, and
hence the restriction to r of the scalar product is well defined and positive
definite. We have
(8.12) (i) (QA' Q",,) = 21('\)SA"" if A, J..L are strict.
(in (PA' P"" > = 2 -1(A)Z A 8,\"" if A, J..L are odd.
These correspond to the series expansions
1 +x.y.
n I J = E 2-1('\)Q,\(x)Q,\(y)
i,j 1 - XiYj A strict
(8.13)
= E 2/('\)Z;lPA(X)PA(Y)'
A odd
obtained by setting t = -1 in (4.4) and (4.1).
Let A, IL be partitions such that IL is strict, A::> J..L and A - J..L is a
horizontal strip. Then mi(A)  2 for each i  1, and hence the formula
(5.8) shows that
( ) { 20('\- ",,) if A is strict,
'P,\I"" - 1 = 0 h.
ot erwlse,
where a(A - J..L) is the number of integers i  1 such that A - IL has a
square in the ith column but not in the (i + l)st column. Hence from (5.7)
we have
(8.14)
(8.15)
p q =  2°('\-P.)p
P. r  A
A
for a strict partition J..L, where the sum is over strict A:) J..L such that A - J..L
is a horizontal strip.
This result may be more conveniently rephrased in terms of shifted
diagrams. As in Chapter I, 1, Example 9, let S(A) be the shifted diagram
of a strict partition A, obtained from the usual diagram by shifting the ith
row (i - 1) squares to the right, for each i > 1. The effect of replacing

256 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
A - J..L by SeA - J..L) = SeA) - S( J..L) is to convert the horizontal strip into a
disjoint union of border strips, and a( A - J..L) is just the number of
connected components of S(A"- j.L), i.e. the number of border strips of
which it is composed.
We are therefore led to define a shifted tableau T of shape S( A - J.L) =:::
SeA) - S( J..L), where A and j.L are strict partitions, to be a sequence of strict
partitions (A(O), A(1), . .:' A(r) such that J..L = A(O) C A(l) C ... C A(r) = A, and
such that each S(A(t) - A(t-l») is a disjoint union of border strips; Or
equivalently, as a labelling of the squares of S( A - J..L) with the integer
1,2, . . ., r which is weakly increasing along rows and down columns and is
such that no 2 X 2 block of squares bears the same label, so that for each i
the set of squares labelled i is a disjoint union of border strips. If We
denote by beT) the number of border strips of which T is composed, then
it follows from (8.14) and (5.11) that, for A and J.L strict partitions, we have
(8.16) QAII-L = E 2b(T)xT
T
summed over shifted tableaux of shape SeA - j.L), where as usual xT is the
monomial defined by the tableau T.
Remarks. 1. The diagonal squares (i, i) in SeA) - S( J..L) must all lie in
distinct border strips, so that beT)  leA) -l( J..L) for each tableau T of
shape S( A) - S( J..L).
2. The formula (8.16) shows that QA/IJ.' like s).lp. but unlike QA/p.(x; t) for
arbitrary t, depends only on the skew diagram A - j.LI
For later purposes it will be convenient to rephrase (8.16). Let pi denote
the ordered alphabet {I I < 1 < 2 I < 2 < ...}. The symbols 1 I , 2 I, . .. are
said to be marked, and we shall denote by lal the unmarked version of any
a E P'. Let A and J.L again be strict partitions such that A:J J.L. A marked
shifted tableau T of shape S( A - j.L) is a labelling of the squares of
S(A - J..L) with symbols from pi such that (compare Chapter I, 5, Example
23):
(M1) The labels increase (in the weak sense) along each row and down
each column.
(M2) Each column contains at most one k, for each k  1.
(M3) Each row contains at most one k I, for each k  1.
The conditions (M2) and (M3) say that for each k  1, the set of squares
labelled k (resp. k ') is a horizontal (resp. vertical) strip.
The weight (or content) of T is the sequence (aI' a2' ...) where Uk' for
each k  1, is the number of squares in T labelled either k or k I .
Let ITI be the (unmarked) shifted tableau obtained from T by deleting
all the marks, i.e., by replacing each a E P' by its unmarked version lal.
The conditions (M1)-(M3) ensure that no 2 X 2 block of squares can bear

'
F
Qbe same label in ITI, so that ITI is a shifted tableau as defined above.
1{;Moreover, in each border strip f3 in IT I, each square of which is labelled k,
Iit is easy to see that the conditions (M1)-(M3) uniquely detennine the
\ssignment of marked and unmarked symbols (k' or k) to the squares of
;:p, with the exception of the square (i, j) E f3 nearest the diagonal (i.e.
f:$uch that (i + 1, j)  {3 and (i, j - 1)  {3), which can be marked or un-
;tnarked. Call this square the free square of (3.
RA, From this discussion it follows that the number of free squares in ITI is
rfJ6(ITI), and hence that the number of marked shifted tableaux T arising
from a given (unmarked) shifted tableau ITI is 2b(ITI). Consequently (8.16)
fay be restated as
{"{.-
8. SCHUR'S Q-FUNCTIONS
257
.;ii8.16')
QA/JJ. = L xlTI
T
Jsummed over marked shifted tableaux of shape S( A - 11,).
'iV.': Equivalently:
.< .
':(8 .16" )
,;j::."
QA/JJ. = L K - JJ., vmJl
v
t.where K _ JJ., v is the number of marked shifted tableaux T of shape A - J-L and
;.:weight v.
..
..Remarks. 1. From (5.2) we have
QA/JJ. = L f;v( -l)Qv.
JI
Since f;v(t) E Z[t], this shows that QA/P. is a Z-linear combination of the
rQJI (where we may assume v strict, otheIWise' Qv = 0). Hence it follows
Jrom (8.8) that QA/JL E r for all A, JL.
2. The formula (8.16) or (8.16') may be taken as a combinatorial definition
dof the Q-functions, and the same remarks apply as in the case of the Schur
?Junctions (Chapter I, 5). In particular, it is not at all obvious from this
definition that QA/JL is in fact a symmetric function of the Xi.
To conclude, we shall mention without proof two positivity results, and
.heir combinatorial interpretations.
(8.17) (i) If JL, v are strict partitions and
Pp.Pv = L f;vPA
A
then f;v  0 for all A.

258 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
(ii) If A is a strict partition and
PA = E g,\,us,u
1J.
then g A,u  0 for all J.L.
Since the PA (for all partitions A) form a Z-basis of A, the coefficients
f;JI( = f;JI( -1), in the notation of 3) are integers; and since P,u, PJI E r by
virtue of (8.7) and (8.9), we have f;" = 0 if A is not strict. Q
In (8.17Xii), the matrix (gA,u) is strictly upper unitriangular, and is the
inverse of the matrix K( -1) (6). Hence (8.17Xii) says that the matrix
K( -1)-1 has non-negative (integer) entries; or again, in the notation of
Chapter I, 7, Example 9, that PA E A+ for all strict partitions A.
There are at least three ways of proving both parts of (8.17). One is to
interpret the coefficients f;JI and gA,u in terms of the projective represen-
tations of the symmetric groups, in the same sort of way that the Little-
wood-Richardson coefficients C;JI can be shown to be non-negative by
reference to the linear representations of the symmetric groups. Another
way is to interpret the QA as cohomology classes dual to Schubert cycles in
the Grassmannian of maximal isotropic subspaces in C2n relative to a
skew-symmetric bilinear form of rank n. The third method, which we shall
describe below, provides a shifted analogue of the Littlewood-Richardson
rule for f;JI' and likewise for gA,u' A fourth possibility, which as far as I
know has not been investigated, would be to interpret the PA (,\ strict) as
traces of polynomial functors on an appropriate category, in analogy with
Schur's theory (Chapter I, Appendix A).
Let T be a marked (shifted or unshifted) tableau, that is to say a
labelling of the shape of T with symbols from the alphabet P', satisfying
the conditions (M1)-(M3) above. As in Chapter I, 9, T gives rise to a
word w = w(T) by reading the symbols in T from right to left in successive
rows, starting with the top row. Let w be the word obtained from w by
reading w from right to left, and then replacing each k by (k + 1)' and
each k' by k, for all k  1. For example, if w = 11'2'12'2, then w =
3'22'212' .
Suppose T contains altogether n symbols, so that w is of length n; then
ww is of length 2n, say ww = a1a2'.. a2n. For each k  1 and 0 p  2n,
let mk(at... ap) denote the number of k's among at,..., ape The word
w = w(T) is said to have the lattice property if, for each k and p as above,
whenever mk+ t(at... ap) = mk(at... ap), we have lap+ 11::1= k + 1 (i.e. ap+l
is neither k + 1 nor (k + 1)').
With this explained we can now state

8. SCHUR'S Q-FUNCTIONS
259
--
. _....
liftS.iS) (i) Let A, J-L, v be strict partitions. Then the coefficient f;v of P).. in PIJ-PV
:r equal to the number of marked shifted tableaux T of shape S{A - J.L) ad
;;.weight v such that
t':(a) w{T) has the lattice property;
+-<b) for each k  1, the last (i.e. rightmost) occu"ence of k' in w{T) precedes
':the last occurrence of k.
<.(ij) Let A, J-L be partitions, Astrict. Then the coefficient gAp. of S p. in P).. is
:{:qual to the number of (unshifted) marked tableaux T of shape J-L and weight
r' satisfying (a) and (b) above.
':;':-_ These theorems are due to J. Stembridge [528].
:'Examples
('1. The formula (2.1) for P..\(x1,..., x,.; t) no longer makes sense at t = -1, even if
:{l is strict (unless l(A.) == n or n - 1), because v..\(t) vanishes at t = -1 if 1(>..) < n - 1.
;;.:But the alternative formula (2.2) does make sense at t = - 1, and for a strict
::i:partition A. of length I  n it gives
( x.+x.)
p..\ (x 1 , . . . , X,.) = E w X A n '_ J ,
WE Sn/Sn-I Xi Xj
:,':here S n -I acts on x 1+ l' . .. , X n' and the product on the right is over pairs (i, j)
<-such that 1  i  I and 1  i <j  n. This leads to the following expression for
::Q.\(Xl,...' X,.): let 11,. denote the linear operator on the ring L,. = Z[X1:C 1,.. ., Xn:C 1]
:"of Laurent polynomials defined by
'TT'n/= E w(fxBjas)
WESn
where 8 == (n - 1, n - 2,...,1,0) and aB == n; < j(x; - Xj). Then we have, for a strict
partition A. of length I  n, .
Q..\(x1,..., Xn) = 2/"',.(x..\I/),
where I, == n(l +xjlXj), the product being taken over parrs (i,j) such that
1  i  I and j > i.
2. Suppose A is a strict partition of length n or n - 1. Then in n variables
xl,...,xn we have P..\ =s..\-sss, where 8==(n-l,n-2,...,1,0). (Use (2.1) with
t = -1, and obselVe that n; < j(xj + Xj) = n; < j(xl- xJ) j(x; - Xj) = Sc5.)
3. (a) Let CIJ be the involution on A defined in Chapter I, 2. From (8.1) we have
wqn == qn for all n > 1, hence w fixes each element of the ring f, and in particular
wQ..\ = Q..\ and wP..\ = P..\ for each strict partition A.
(b) We have PB == Ss for any 'staircase' partition 8 = (n - 1, n - 2,...,1,0). (For Pc5
is of the form
P8 = E gc5p.S,."
J.L<.8

260 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
with g88 = 1; from (a) above it follows that g8JJ. = g8JJ." and since JL  8 implies
J.L'  8 (Chapter I (1.11)) it follows that g8JJ. = 0 unless J.L = 8.
4. Let A = (aij) be a 'marked matrix', i.e. with entries aij E P' U {a}. The matT"
IAI = (lajjD obtained by deleting the marks is a matrix of non-negative intege DI
Deduce from (8.13) and (8.16) that for any two partitions J.L and II the number rSf
marked matrices A such that IAI has row sums J.Li and column sums vi is equal 
the number of pairs (S, T) of marked shifted tableaux of the same (shifted) shap
such that S has weight J.L, T has weight v, and the main diagonal in T i
unmarked.
5. Let P(tl,..., tm), where m is even, denote the Pfaffian of the m X m matrix M
whose (ij) element is (ti - tj)/(ti + tj). From the definition of the Pfaffian it
follows that
(1)
m
P ( t 1 , . .. , t m) = E ( - 1) i P ( t 1 , t i ) P (t 2 , .. · , t;, .. . , t m ) .
i- 2
Also it is clear that
U( t I' . · · , t m) = P (t 1 , . . · , t m ) TI (t i + t j)
i<j
is a homogeneous polynomial in the ti' of degree tm(m - 1). Now p2 = det M
vanishes whenever two of the ti are equal, hence the same is true of the polynomiaJ
U, and therefore U is divisible by V = ni < j(ti - tj). Since U and V have the same
degree, we conclude that U = cmV for some constant cm depending only on m, and
hence that
( ti - tj) _ TI ti - tj
Pf - cm .
ti+tj i<j ti+tj
It remains to show that cm = 1. From the relation (1), on multiplying both sides by
n;<j(t;+tj) and then comparing the coefficients of t}"-1t2-2 ...tm-1 on either
side, we obtain
m
cm= E(-1)icm_2=Cm_2
;-2
and hence cm =Cm-2 = ... =c2 = 1.
6. (a) Show that
Q(t) = exp( E 2Prtr Ir )
r odd
and hence that
qn = E z;12/(p)pp.
p odd

,;.r-"';
::: ., .
1.::,
8. SCHUR'S Q-FUNCTIONS
261
(....
[(b) Let V be a vector space (over a field of characteristic zero) with basis
t)J;H', . . ., V n' and letS n act on V by permuting the Vi. Then S n acts on the exterior
:gebra A V: if the character of this representation is (jJn, then
.. ch( n) = qn.
... .
J(We have 'Pn(w) = n?_t(l + i)' where the i are the eigenvalues of wE Sn acting
r-bn V (i.e. of the permutation matrix corresponding to w). To each r-cycle in w
ihere correspond r eigenvalues i' namely the r th roots of unity. The correspond-
:h"hg product n(l + i) is zero if r is even, and is equal to 2 if r is odd. It follows
l:hhat 'Pn( w) = 0 if w has any cycles of even length, and that CPn{ w) = 2/ if W consists
;:()f I cycles of odd length. The result now follows from (a) above.)
';'(c) From (8.1) we have
qn = E erhs = E 2s(alb)
r+s-n a+b-n-l
y Pieri's formula (Chapter I, (5.16».
.:..7. (a) Let
SA = SA(X; -1) = det(qA;-i+j)
;as in (4.5). In the notation of Chapter I, 5, Example 23, SA = SA (X/ - x). Hence
(loc. cit.) we have
SA = ExT
T
..
:summed over all (unshifted) marked tableaux of shape A.
(6) From the definition, SA is of the form
(1)
SA = qA + E CAJJ.qJJ.
IL>A
:,With integer coefficients cAJJ.' (The transition matriX M(S, q) is the same as
M(s,h), hence is the transposed inverse of the Kostka matrix K (Chapter I, 6).)
.Hence SA E r for all partitions A, and if A is strict it follows from (8.4) that
(2)
SA = q). + E C II- q II-
IL>A
where the sum is now over strict partitions J.L > A. Hence from (8.6) (ij) it follows
-that the SA' A strict, form another Z-basis of the ring r.
(c) Show that
S =  z-12/(p}vAp
A i..J p /\.p P
p odd
= SA * qn = ch( XA 'Pn)
where (Example 4 (b») 'Pn is the character of the exterior algebra representation of
Sn. -

262 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
8. For each f E f, let f-: r  r denote the adjoint of multiplication by f, so that
(f-g, h) = (g, fh )
for all g, hEr. Let
Q.1(t) = E qn.1 tn,
nO
then (5, Example 8) we have
(1)
Q.l(t)Q(u) = F( -tu)Q(u)Q.l(t)
as operators on r, where t and u are commuting indeterminates, Q(u) is the
operator of multiplication by Q(u), and F(y) = (1 - y) /(1 + y) as in (8.8).
(a) Define operators Bm: r  r for each m E Z by
B m = q m - q m + 1 q 1.1 + q m + 2 q2.1 - . . .
(where qm = 0 if m < 0, and qo = 0). Collectively we have
B(t) = E Bmtm = Q(t)Q.l( -t-1).
meZ
More generally, if t1,..., tn are commuting indeterminates, let
(2)
n n
B(t1,..., tn) = n Q(t;) n Q.1( -t;-1).
i=1 i==1
From (1) it follows that
(3)
B(t)B(u) =F(t-1u)B(t,u)
and hence by iteration that
(4)
B(t1)...B(tn) =B(t1,...,tn)n F(ti1tj).
i<j
Since Q.l(tX1) = 1, it follows from (2) and (4) that
B(tt)...B(tn)(l) = Q(t1,...,tn)
in the notation of (8.8), and hence that
(5)
Q,\ = B'\l · · · B)./(l)
if ,\ is a strict partition of length 1.
(b) Let
cp (t , u) = cp ( u, t) = F (t -1 u) + F (tu - 1 )
=2 E (-l)nt-nu",
neZ
a formal Laurent series in t and u.

" 8. SCHUR'S Q-FUNCTIONS
{.
iItShow that
t,
(6) Q(t )Q(u) cp(t, u) = cp(t, u).
:;;t_:::
te coefficient of t'uS on the left-hand side is
;>;'"
;':;: a,s = E (_1)c qaqb
::i:.:;"
263
itummed over a, b EN and C E Z such that a + c = rand b - c = s, so that
f;! + b = r + s; hence
(-1)' a,s = E (_1)a qaqb = o,'+s
", a+b-,+s
;..('
t}since Q(t)Q( - t) = 1.)
b.' From (6) it follows by taking adjoints that
!,- .
Q.l.(t )Q.l.(u) cp(t, u) = cp(t, u)
ind hence that
r(7) B(t,u)cp(t,u)=cp(t,u).
:;,L Finally, from (3) and (7) we have
"(8)
B(t, u) + B(u, t) = B(t, u) cp(t, u) = cp(t, u).
;:': Explicitly, (8) says that for all r, S E Z:
:.,(1) B,Bs = - BsB, if r + S + 0, and in particular B; = 0 if r:# 0;
\(C2) B,B_r + B_,B, = 2( -1)' if r:# 0, and BJ = 1.
,(e) Let a = (at,..., an) E zn be any sequence of n integers. We define Qer to be
"the coefficient of ta = tit... t:" in the Laurent series Q(t1,..., In), or equivalently
Qa = Bat' . . Bern (1).
,Use the commutation rules (C1), (C2) to show that
(0 if I all,. .., I ani are all distinct, then Qer is skew-symmetric in a (i.e. Qwer =
',B(W)Qa for all W E Sn)' and Qer = 0 if any a; is negative.
(ii) If a; = ai+ 1 for some i, then Qer = O.
iiO If ai = - ai+ 1 = r:# 0, then Qer = 0 if r> 0, and Qa = 2( -1)rQj3 if r < 0,
where f3 is the sequence obtained from a by deleting ai and ai + 1.
Deduce that, for each a E zn, Qer is either zero or is equal to :f:2rQA for some
integer r  0 and strict partition A [H7].
9. Let A = (AI'.." Am) and J.L = (ILl'...' J.Ln) be strict partitions such that Al >
. .. > Am > 0 and ILl > ... > J.Ln  O. Since we allow J.Ln = 0 we may assume that
m + n is even. Then we have
QA/p. = Pf(MA, p.)
where MA, p. is the skew-symmetric matrix of m + n rows and columns
( MA o'J.t)
MA,p. = -N
A,p.

264 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
in which MA = (Q(AII A}») as in (8.11), and NA, p. is the m X n matrix (q Ai- p,).
(With the notation of Example 8, we have
QA/P. = 2 -1(p.)Q: (QA)
and by (8.8) Q: (QA) is the coefficient of tAu p. in
n Q.L(Uk){ n Q(ti)) n F(ti-1tj) n F(Uk1ul)
k i i <j k < I
which by use of Example 8, formula (1) is equal to
(1)
n Q(t;) n F(t;-ltj) n F(u"k1u/) n F( -t;uk).
i i <j k < / i, k
The product of all the F's in this expression can be written in the form
n F(V,-lvs)
r<s
where (v1, V2," ., vm +n) = (- u; 1,..., - uil, - ull, t1,..., tm). Hence the same ar-
gument as in the proof of (8.11) shows that the expression (1) is the Pfaffian of the
matrix
(a,asF(u,-l us))
of m + n rows and columns, where a, = 1 if r  n, and ar = Q(v,) if r > n. On
picking out the coefficient of tAu p., we obtain the result.)
10. Define a ring homomorphism cp: A  r by
cp(en) = qn
(n  1).
(a) Show that
(0 cp(hn) = qn'
(ij) CP(Pn) = 2 Pn if n is odd, and CP(Pn) = 0 if n is even,
(iii) cp commutes with the involution 00,
(iv) CP(S(a\b-l») = !(qaqb + Q(a,b»).
(b) Let A, J.L be strict partitions of lengths m, n respectively, such that A:J J.L. Let
D(A) = (At,..., Ami A} - 1,..., Am - 1) be the 'double' of A (Chapter I, 1, Example
9), and define D( J.L) similarly as the double of J.L. Then we have
CP(SD(A)/D(p.») = 2n-mQf/p..
(From Chapter I, 5, Example 22 we have
n (CA
SD(A)/ D( p.) = (-1) det _ E
A,p.
H )
where CA = (s(A;lAj-1»)' HA, p. = (hA;- p.j) and EA, p. = (eAj- p.). By use of (a) it follows
that
CP(SD(A)/D(p.») = 2n-m det(MA,p. + B)

;-:::: ::
8. SCHUR'S Q-FUNCTIONS
265
i.:!-
'Phere MA, p, is the matrix defined in Example 9, and B = (bjj) is given by
;:;:." b _ { qA;q;"j if 1  i,j  m,
i'" ij - 0 othetwise.
f>::' '
i;'Frorn Example 9 it now follows that
, , CP(SD{).)! D{p,» = 2n-mQ!p, det( 1 + BM;:)
t'and since rank (B) = 1,
. - I
det(l + BMA) = 1 + trace(BM).) = 1
  : ..
L',6ecause M;'  is skew-symmetric.)
,(c) By applying cp to both sides of the identity of Chapter I, 5, Example 9(b), we
,;'.btain
E 2-1().)Q).(X)2 = n (1 +xixj)/(1-xixj)
A ;,j
.:..(Le. (8.13) with xi = Yi for each i).
,::t1. (a) With the notation of Example 8, show that for r odd
;- .,
':. .(1)
1 a
J.
P = - r-
r 2 apr
(Use (8.12) (ii).)
From Example 6(a) we have
,,'
qn = E 2/{p)z;1pp
p
"(2)
: .summed over odd partitions p of n. Deduce from (1) -and (2) that
" (3)
prJ...qn =qn-r
:Jor all n  1; equivalently,
(3')
p/-Q(t) = trQ(t).
Deduce that
"(4)
P r J...Q ( t 1 , . . · , t n) = P r ( t 1 , · . . , t n ) Q ( t 1 , · · · , t n ) ·
(Obsetve that prJ... is a derivation, by (1) above.) More generally, for any odd
partition p,
. (5)
p P J...Q ( t 1 , . . . , t n) = P p ( t 1 , · . . , t n ) Q ( t 1 , · · · , t n ) .
Hence PpJ... Q;.. (A a strict partition) is equal to the coefficient of tA in the right..hand
side of (5).

266 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
(b) Let A be a strict partition of length 1 and let r  1 be odd. From (4) above We
have
(6)
I
pr.l.QA = E Q.\-rei
i-I
where A - rBi = (At,..., Ai - r,.." AI). From Example 8(c) it follows that QA-re. == 0
unless either (0 Ai;> r and the numbers AI'...' Ai - r, . . . , Al are all distinct, o (in
r = Ai + Aj for some j > i.
In case 0),. ether Aj> Ai - r > Aj+ 1 for some j < m, or else Am > Ai-r' and
QA-7EI = (-l)J-'QI£ where J.L = (AI'...' Ai-I' Ai+ 1'...' Aj' Ai - r, ...). The shifted
diagram S( p.,) of p., is obtained from SeA) by removing a border strip /3 of length r,
starting in the ith row and ending in the jth row, and hence of height (Chapter I,
1) ht( /3) = j - i. Define B( /3) = (_l)ht(tn.
In case (ii) we have
(7)
Q = (_1)j-i-l+Aj2Q
A-rei J.L
where now p., is the strict partition obtained from A by deleting Ai and Aj. To
express this in terms of shifted diagrams, we define a double strip 0 to be a skew
diagram formed by the union of two border ships which both end on the main
diagonal. A double strip 0 can be cut into two non-empty connected pieces, one
piece a consisting of the diagonals in 0 of length 2, and the other piece {3
consisting of the border strip formed by the diagonals of length 1. Define B( 0) =
(-l)d, where d = tlal + ht( (3). From (6) and (7) we now have
(8)
pr.l. QA = E B( 'Y )QJ.L
summed over strict partitions p., c A such that I AI-I p.,1 = rand y = S( A - J.L) is
either a border strip (case (0) or a double strip (case (i0).
(c) From (8) we obtain the following combinatorial rule for computing XpA( -1) =
(QA'Pp), where A is strict and p is odd:
X;(-l) = E B(S)
s
summed over all sequences of strict partitions S = (A(O), A(1),..., A(m» such that
m = I( p), 0 = A(O) C A(l) C ... C A(m) = A, and such that each S(A(i) - A(i-l» = 1i is
a border strip or double strip of length Pi' and B(S) = Oi:.l B( y). (Compare
Chapter I, 7, Example 5.)
12. From Example ll(c) (or from 7, Example 4) it follows that if A is a strict
partition of n, then X(1..)( -1) is equal to the number g A of shifted standard
tableaux of shape S( A), obtained by labelling the squares of S( A) with the numbers
1,2, . . , , n, with strict increase along each row and down each column.
Just as in the case of ordinary standard tableaux (Chapter I, 5, Example 2)
there is a formula for g A in terms of hook-lengths. For each square x in S( A), the

8. SCHUR'S Q-FUNCTIONS
267
hook-length hex) is defined to be the hook-length at x in the double diagram
D(A) = (AI - 1, A2 - 1, ...1 AI' A2' ...). Then we have
(1) g). = n!j n h(x).
x e S( A)
This formula is equivalent to one due to Schur [55]:
A n! n Ai-Aj
g=-
A! i <j Ai + Aj
(2)
where A! = AI! A2! . .. . The equivalence of (1) and (2) follows from the fact that the
numbers h(x) for x in the i th row of S( A) are Ai + Ai + l' Ai + Ai + 2' ... and
Ai' Ai - 1,...,2,1 with the exception of Ai - Ai+ l' Ai - A;+2' ... .
. Since the g A clearly satisfy the recurrence relation
gA = E g).(I)
i> 1
where A(i) is the partition obtained from A by replacing Ai by Ai - 1, the proof of
(2) reduces to verifying the identity
A.+A. A.-A.-1
 n I J I J
'-' Ai = '-' Ai
i i j.. i Ai + Aj - 1. Ai - Aj
which may be proved by considering the expansion into partial fractions of the
function
n (t + Ai)(t - Ai - 1)
(2t - 1) .
i (t + Ai - l)(t - Ai)
13. Let A denote the skew-symmetric matrix
«Xi -xj)/(xi +Xj»I<i,j<
lnd for each strict partition A = (AI"." AI) of length I  n let BA denote the n xl
matrix ( x t1 ). Let
AA(XI>.'" xn) = (B A)
hich is a skew-symmetric matrix of n + 1 rows and columns.
Now define
Pf A ( X 1 , . . · , X n )
.0 be the Pfaffian of A).(Xl'"'' xn) if n + 1 is even, and to be the Pfaffian of
t,\(x1,..., xn, 0) if n + 1 is odd. (In the latter case, the effect of introducing the
xtra variable xn + 1 = 0 is to border A with a column of 1's and a row of -1'8.
)how that
p). (x l' · · . , X n) = Pf A (X 1 , . . . , X n) /Pfo (x 1 , . . . , X n) .

268 III HALL-LITTLEWOOD SYMMETRIC FUNCTIONS
(Multiply both sides of the formula (1) of Example 1 by
Pf 0 (x 1 , . . . , X n ) . = Il (Xi - X j ) / ( Xi + x j ) .
i<j
This formula for P>.. may be regarded as an analogue of the definition (Chapter I
(3.1)) of the Schur function s>..(x1,..., xn) as a quotient of alternants. '
Notes and references
As explained in the preface, we have stopped short of describing the role
played by Schur's Q-functions in the construction of the projective repre-
sentations of the symmetric groups. Apart from Schur's original paper [85]
of 1911, there are several recent accounts available: the monograph [H7]
of Hoffman and Humphreys, and the papers of Jozefiak [JI5] and Stem-
bridge [S28].
It has become clear in recent years that Schur's Q-functions arise
naturally in other contexts: as (up to scalar factors) the characters of
certain irreducible representations of the Lie superalgebra Q(n) [SII]; as
the cohomology classes dual to Schubert cycles in isotropic Grassmannians
[JI6], [P2]; as the polynomial solutions of the BKP hierarchy of partial
differential equations [Y1]; and as generating functions for the zonal
spherical funytions (Chapter VII,  1) of the twisted Gelfand pair
(S2n, Hn, '), where Hn is the hyperoctahedral group, embedded in the
symmetric group S2n as the centralizer of a fixed-point free involution, as
in Chapter VII, 2, and ( is the composition of the sign character of S"
with the projection Hn -+ Sn [529].
A combinatorial theory of shifted tableaux, parallel to that for ordinary
(unshifted) tableaux, has been developed independently by Sagan [Sl] and
Worley [W4]; it includes shifted versions of the Robinson-
Schensted-Knuth correspondence, Knuth's relations, jeu de taquin, etc. In
particular, (8.17) (ii) (originally conjectured by R. Stanley) is due to Sagan
and Worley (independently), and Stembridge's proof of (8.18) relies on
their theory. Finally, we may remark that setting t = -1 in Chapter II,
(4.11) will provide another expression for f;v = f;v ( -1).
Example 3(b) is due to R. Stanley.
Example 8. The operators Bn are the analogues, for Schur's Q-func-
tions, of Bernstein's operators (Chapter I, 5, Example 29). See [H7] and
[Jlt].
Examples 9 and 10. These results are due to Jozefiak and Pragacz [117].
OUf proof of the Pfaffian formula for QA/JL (Example 9) is somewhat
different from theirs.
Example 13 is due to Nimmo [N2].

J'-:-',
IV
"'1
"
THE CHARACTERS OF GLn OVER A
FINITE FIELD
i}.-.
:: .
.. -.
,:'
:.,
I.
t,t. The groups Land M
Let k be a finite field with q elements, k an algebraic closure of k. Let
F: x t-+xq
:'he the Frobenius automorphism of k over k. For each n  1, the set kn of
Cfixed points of Fn in k is the unique extensio_n of k of degree n in k.
;:. Let M denote the multiplicative group of k, and Mn the multiplicative
;igroup of kn, so that Mn = Mpn.
-i" If m divides n, the norm map Nn,m: Mn -+ Mm defined by
:',.
d-l
N (x) =x(qn-l)j(qm-l) = n pmix
n,m ,
i=O
'-
r'where d = n 1m, is a surjective homomorphism. The groups Mn and the
. homomorphisms Nn, m form an inverse system. Let
K= limMn
+-
'he their inverse limit, which is a profinite group. The character group of K
::. is therefore a discrete group
': .
A A
K=L = limM
-+ n
A A
where Mn is the character group of Mn. Whenever m divides n, Mm IS
, A
,.-embedded in Mn by the transpose of the norm homomorphism Nn,m.
Both groups Land M are (non-canonically) isomorphic to the group of
roots of unity in C of order prime to p = char. k.
The Frobenius map P: g t-+ gq acts on L. For each n  1 let Ln = Lpn
be the group of elements gEL fixed by Fn, so that L = U n  1 Ln' and
A
Lm cLn if and only if m divides n. The canonical mapping of Mn into L is
A A
an isomorphism of Mn onto Ln. By identifying Mn with Ln by means of
. this isomorphism, we define a pairing of Ln with Mn:
(g,x)n = g(x)
for  E Ln and x E Mn. (The suffix n is necessary because these pairings

270 IV THE CHARACTERS OF GL,. OVER A FINITE FIELD
(for different n) are not coherent: if m divides n and  ELm' x EM We
have (, x)n = (, x)m)n/m.) m
Finally, let <I> denote the set of F -orbits in M. Each such orbit is of the
d-l d
form {x, xq,..., xq } where xq = x, and the polynomial
d-l
f= n (t _xqi)
i-O
belongs to k[t] and is irreducible over k. Conversely, each irreducible
monic polynomial f E k[t], with the single exception of the polynomial t
determines in this way an F-orbit in M. We shall therefore use the sam
letter f to denote the polynomial and the orbit consisting of its roots in Ie
,
and the letter <I> to denote the set of these irreducible polynomials.
Re'!}ark. For each n  1 let J.Ln = (Z/nZXIXk) be the group of nth roots of unity
in k, and let
A _
Z(l)(k) = lim J.Ln

with respect to the homomorphisms J.Ln  J.Lm: x  xn/m whenever m divides n
([D2], 2). We have
A - A
Z(l)(k) =K=L,
A A
the character group of L, so that L is a Z-module, where
A
Z = lim (Z/nZ) = fl Zp
 p
(direct product over all primes p).
A
Let M be the character group of M. Then we have
A A A
M = Homz(L,Z),
A A A
L :: Homz(M,Z)
A A _
and therefore M = Z( -lXk).
Moreover,
M= (Q/Z)  i = (Q/Z)(l)(k),
L = (Q/Z)  M = (Q/Z)( -l)(k).
2. Conjugacy classes
Let Gn denote the group GLn(k) of invertible n X n matrices over the
finite field k. Each g E Gn acts on the vector space kn and hence defines a
k[t]-module structure on kn, such that t. v = gv for v E kn. We shall
denote this k[t]-module by . Clearly, two elements g, h E Gn are conju-
gate in Gn if and only if  and  are isomorphic k[t]-modules. We may
therefore write  in place of , where c is the conjugacy class of g in

f 2. CONJUGACY CLASSES 271
mt:
.
fv . The conjugacy classes of Gn are thus in one-one correspondence with
'.,..n
iihe isomorphism classes of k[t]-modules V such that (i) dimk V= n, (ii)
tbu== 0 implies v = o.
tr . Since k[ t] is a principal ideal domain, V =  is a direct sum of cyclic
odules of the form k[t J/(f)m, where m  1, f E <f) and (f) is the ideal in
;}k[t] generated by the polynomial f. Hence V assigns to each f E <f) a
partition fJ.(f) = ( J.Lt(f), J.L2(f), · · .) such that
:;
f(2.1) V::: EB k[t ]j(f)Ih;<J),
(', f, i
:tin other words, V determines a partition-valued function fJ. on <f). Since
";.dimk k[t ]/(f) l(f} = d(f) J.Li(f), where d(f) is the degree of f, it follows
{that fL must satisfy
:'(2.2)
flfLll = E d(f) 1fJ.(f)1 = n.
few
. .
rtonversely, each function fL: <f) .9, where ,9J is the set of all partitions,
,.which satisfies (2.2) determines a k[t]-module V of dimension n by (2.1),
and hence a conjugacy class cIL in Gn. We shall write VJL in place of  .
, For each fEci> let V{f} denote the f-primary, component of V = VJL' i..
the submodule consisting of all v E V annihilated by some power of f. In
Jhe notation of (2.1),
(2.3)
V{f) = €a k[ t ]/(f)I1;(f).
i
":The Jl(f} are haracteristic submodules of V, and V is their direct sum.
'Hence if Latt(M) denotes the lattice of submodules of a module M, and
ut(M) the automorphism group, we have
,:.;1
(2.4)
Latt(V) == n Latt(V(!»),
few
(2.5)
Aut(V) =: n Aut(V(!»).
fe 4»
Let k[t f) denote the localization of k[t] at the prime ideal (f), i.e. the
ring of fractions u/v where u, v E k[t] and v  (f). Then k[tf) is a
discrete valuation ring with residue field kf = k[t]/(f), of degree d(f)
over k, and vef) is a finite k[t ](ffmodule of type fJ.(f), by (2.3). Hence
from Chapter II, (1.6) we have
(2.6)
lAut(V{f»)1 = aJL(f){qf)
where qf= Card(kf)=qd(f), and for any partition A
aJ..(q) = qlJ..\+2n(J..)b,\(q-l).

272 IV THE CHARACTERS OF GLn OVER A FINITE FIELD
For any g E Gn, the automorphisms of the k[t]-module  are precisel
the elements h E Gn which commute with g, i.e. Aut() is the centraliz:
of g in Gn. Hence, from (2.5) and (2.6), the centralizer of each g E CIL i:
Gn has order
(2.7)
aIL = n aJJ.(f){qf).
fEcI>
Examples
1. For each f E <1>, say f = td - Ef-l aiti-I, let J(f) denote the 'companion matrix'
for the polynomial f:
J(f) =
and for each integer m  1 let
Jm(f) =
J(f)
o
o
o
o
1 0 0
0 1 0
. . . ,
0 0 1
a2 a3 ad
o
al
1d
J(f)
o
Id
o
o
o
o
J(f)
with m diagonal blocks J(f). Then the Jordan canonical form for elements of the
conjugacy class c JJ. is the diagonal sum of the matrices J JI.;(f)(f) for all i  1 and
fEci>.
2. Sometimes it is more convenient to regard J1.: cI> -+.9 as a function on M, i.e. we
define J1.(x) for each x EM to be the partition fJ.(f), where f is the F-orbit of x
(or the minimal polynomial of x over k). We have then
11....11 = E d(f) 1fJ.(f)1 = E IfL(x)1
IE <I> XEM
and fL 0 F = fL. With this notation, the characteristic polynomial of any element
gEe fA. is
n flp.(f)1 = n (t - x)Ip.(x)1
Ie <I> xeM
so that in particular
trace(g) = E IfL(X)1 x,
xeM
det(g) = n x1p.(x)l.
xEM

L 3. INDUCTION FROM PARABOLIC SUBGROUPS 273
a::.:.r....
.;-..
;: The support SUPp(f1) of p,: <I> --+.9 is the set of f E <I> such that fJ.(f)"* O. Let
Jf:e c".. Then
. 0) g is unipotent  Supp(fJ.) is the polynomial t - 1.
(ii) g is primary  Supp(fJ.) = {f} for some f E ct>.
(iiD g is regular  fJ.(f) has length  1, for each f E <1>.
,': (iv) g is semisimple  fJ.(f)' has length  1, for each fEcI>.
,.;;;;0,
(v) g is regular semisimple  fJ.(f) = 0 or (1) for each f E ct>.
r
4. Let kn be the unique extension of k of degree n, and let a: kn --+ kn be an
lsomorphism of k-vector spaces. For each x E Mn, multiplication by x defines an
ii:nvertible k-linear map kn --+ kn, hence an element T(X) E Gn: namely T(X)V =
&(x(a-lv)) for VEkn. The mapping 'T:Mn--+Gn so defined is an injective
fhomomorphism, so that Tn = T(Mn) is a subgroup of Gn isomorphic to Mn. If we
hange the isomorphism a, we replace Tn by a conjugate subgroup. Each T(X) E Tn
]18 semisimple, with eigenvalues FiX (0  i  n - 1). Hence if fx is the minimal
polynomial of x, the conjugacy class of T(X) in Gn is c,.., where fJ.(fx) = (In/d) and
c(f) = 0 for f"* fx' and d = d(fx).
;;"': Now let v = (Vt,..., v,) be any partition of n. Then Tv = TVI X ... X Tv, is a
tsubgroup of Gv X ... X GJI, hence of Gn, and is well-defined up to conjugacy in
..;. 1 ,
(Jn. Let Nv be the normalizer of Tv, and  = Nv/Tv. Then u-:, is isomorphic to the
tentralizer of an element of cycle-type v in the symmetric group Sn. In particular,
[Wn is cyclic of order n, and is obtained by transporting Gal(kn/k) via the
iisomorphism a.
;:., The groups Tv (up to conjugacy in Gn) are the 'maximal tori' in Gn. Every
!'semisimple element of Gn lies in at least one such group.
;. Let  \ Tv be the set of orbits of  in Tv. Then the number of conjugacy
-plasses in Gn is equal to
.i .
E I \Tvl.
IJlI=n
We may identify Tv with MV1 X ... X Mv" and hence an element t E Tv with a
:sequence (Xt,..., x,), where Xi E MV1 (1  i  r). Given t, we define a function
'p = pet): <I> --+.9 as follows: for each f E <1>, the parts of p(f) are the numbers
:..Vi/d(f) for all i such that Xi is a root of f. We have IIp(t)1I = n, and pet) depends
!only on the »:-orbit of t in Tv. Conversely, given fJ.: <I> --+.9 such that 1IfJ.1I = n, let v
,be the partition of n whose parts are the non-zero numbers d(f) I fJ.(f)l, f E ct>.
:,Then fJ. determines a unique -orbit in Tv.
3. Induction from parabolic subgroups
et n = n 1 + ... + n r' where the n i are positive integers, and let V(i) be
the subspace of V = kn spanned by the first n1 + ... + ni basis vectors. Let
F denote the flag
o = V(O) C V(l) C ... C v(r) = v.

274 IV THE CHARACTERS OF GLn OVER A FINITE FIELD
The parabolic subgroup P of elements g E Gn which fix F consists of all
matrices of the form
g11 g12 glr
0 g22 g2r
g=
0 0 grr
where gii E Gn. (1  i r).
,
Let ui be a class-function on Gn; (1 <; i <; r). Then the function u
defined by
r
u(g) = n Ui(gii)
t-1
is a class function on P. Let U1 0 U2 0 ... 0 Ur denote the class-function on
Gn obtained by inducing u from P to Gn:
u 1 0 ... 0 U r = ind  n ( U) .
If each Ui is a character of Gni, then u1 0 ... 0 ur is a character of Gn.
Let Gn = U i tiP be a left coset decomposition for P in Gn. For any
x E Gn we have
(UIO ... 0 ur)(x) = E U(ti-1 Xli)
with the understanding that U vanishes outside P. Now t-1xt belongs to P
if and only if it fixes the flag F, i.e. if and only if X fixes the flag tF, or
equivalently tF is a flag of submodules of the k[t]-module . If we put
tV(i) = W(i) and
h11 h12 h1r
t - 1 xl = 0 h22 h2r
0 0 hrr
then the module W(i) jW(i - 1) is isomorphic to Vh.. (1  i  r). Hence
"
(3.1) Let Ui be a class function on Gn. (1 <; i  r). Then the value of the class
,
function u1 0 ... 0 ur at a class CfL in Gn is givn by
(u1 0 ... 0 Ur)(CfL) = E g:(;)... fL<r)U1(C,..(l)) . .. ur( C,..<r»)
summed over all sequences (fJ.(1),..., fJ.(r»), where CfL<i) is a conjugacy class in
Gni (1 <; i <; r), and g:(1)... ,..<r) is the number of sequences
o = W(O) C W(l) C ... C w(r) = V,..

I 3. INDUCTION FROM PARABOLIC SUBGROUPS
If submodules of v... such that W(i) /W(i - 1) = V...<i) (1  i  r ).
"'':
>
:Jn particular, let 1T,... be the characteristic function of the class c,..., which
Itkes the value 1 at elements of c..., and 0 elsewhere. Then from (3.1) we
" '
i\n.ave
!';
:(3.2)
:W;
:: '.
A1so from (2.7) it follows that
:-:5.- :-";
(3.3 )
:-'
U'. .
c
I: ."
\Vhere the G's on the right have the same meaning as in Chapter II.
:5::;"
l':::'-.
.:--
:f Now let An denote the space of complex-valued class functions on Gn:
[tthis is a finite-dimensional complex vector space having the characteristic
[nctions 1T.... such that II fLll = n as a basis. It carries a hermitian scalar
[product:
275
11',...(1) 0 ...0 1T,...<r) = E g(1)...,...<r)1T....
J1.
g(1) .......<r) = n G}).......(r)(f)(k[t ](f»
IE <f>
[':. .
1
(u,v)= IG IE u(g)v(g).
n ge Gn
rLet
 '. .
":'1-,
A = €I) An
n;;;.O
i;'c.
f(with the understanding that Go is a one-element group, so that Ao = C).
f.The induction product u 0 v defined above defines a multiplication on A,
nd we extend the scalar product to the whole of A by requiring that the
;omponents An be mutually orthogonal.
{ From (3.2), (3.3), and Chapter II, 2 it follows that
1{3.4 )
A=HC
;",
where H is the tensor product (over Z) of the Hall algebras H(k[t](f» for
;:all f E <1>. In particular, A is a commutative and associative graded ring,
ith identity element the characteristic function 1T 0 of Go.
" Next let Rn cAn be the Z-module generated by the characters of Gn.
;.The irreducible characters of Gn form an orthonormal basis of Rn, and
An =Rn  C. Let
R=R'
W n'
n;;;.O
since the induction product of characters is a character, R is a subring of
A, and we have A ==R  C.

276 IV THE CHARACTERS OF GL,. OVER A FINITE FIELD
4. The characteristic map
Let Xi,f (i  1, f E CP) be independent variables over C. If u is any
symmetric function, we denote by u(f) the symmetric function u(Xf):::::
u(X1,f' X2,f'''.). For example, en(f) is the nth elementary symmetric
function of the variables Xi, f (i  1, f fixed), and Pn(f) = Li Xi7 f.
Let
B = C[en(f): n  1, fE <1>]
be the polynomial algebra over C generated by the en(f). We grade B by
assigning degree d(f) to each Xi,!, so that en(f) is homogeneous of
degree n. d(f).
For each partition A and each f E <I> let
P).(f) = qjn(A)PA( Xf; qjl),
Q). (f) = qY1+n(A)QA (Xf; qj 1 )
= aA(qt)PA(f),
where P>., QA are the symmetric functions defined in Chapter III, and
qf = qd(f). For each ....: <I> gIJ such that 11....11 < 00 let
p... = n p...(f)(f), Q... = n Q...(f)(f) = a...ft...
fE fE
(almost all terms in these products are equal to 1, because 11....11 is finite).
Clearly p... and Q... _are homgeneous elements of B, of degree 111111. By
Chapter III, (2.7), (P..) and (Q...) are C-bases of B. We define a hermitian
scalar product on B by requiring that these should be dual bases, i.e.
(P..., Q11 > = 8...v
for all ...., v.
Now let ch: A  B be the C-linear mapping defined by
ch( 17"...) = p...
where as before 1T is the characteristic function of the conjugacy class cfA..
This map ch we call the characteristic map, and ch(u) is the characteristic
of u eA.
(4.1) ch: A  B is an isometric isomorphism of graded C-algebras.
Proof. Since the 1T... form a basis of A and the p... a basis of B, ch is a
linear isomorphism, which clearly respects the gradings. It follows from

4. THE CHARACTERISTIC MAP
277
(3.2) and Chapter III, (3.6) that ch is a ring homomorphism, and from (2.7)
that it is an isometry. I
From Chapter III, (4.4) we have
E qIAlp;\(x; q-l )Q.\ (y; q-l) = n (1 - XiYi) /(1 - qXiYi)'
A i,j
Ifwe replace Xi by Xi,f 1, Yj by 1 f;!)Xj,f' and q by qt' this identity takes
the form
E  (I) f;!) QA (I) = n (1 - Xi,t f;!) Xj,t) /(1 - qfXi,t  Xj,f)'
A i,j
Hence, by taking the product over all 1 E <I> on either side, we obtain
(4.2) E p... f;!) Q  = n n (1 - Xi, t f;!) Xj, t ) / (1 - qf Xi, f f;!)  , f) ·
... fE i,j
Now take logarithms:
109( E p... f;!) Q) = E E (log(l - Xi,t f;!) Xj,t) -log(l - qtXi,t f;!) Xj,t))
fA. fE i,j
1
= E - E E (q; - l)Xif f;!) Xj7t
n>I n fE i,j
so that
( .., ..,) 1
(4.3) log E p... f;!) Q... = E - E (q; -1) Pn(/) f;!) Pn(/).
fL n  1 n fE 
It is convenient at this stage to modify our notation for the power sums
Pn(/). Let X EM and let f E <I> be the minimal polynomial for x over k
(or, equivalently, the F-orbit of x (1)). Define
Pn(X) = { n/if)
if n is a multiple of d = d(f),
otheIWise,
so that Pn(x) E Bn, and Pn(x) = 0 unless x E Mn. In this notation (4.3)
takes the form
(4.4)
( ) qn - 1
log E p... f;!) Q... = E
fL nl n
E p(x) f;!)Pn(x),
xEMn

278 IV THE CHARACTERS OF GL,. OVER A FINITE FIELD
Now define, for each  E L,
(4.5)
Pn ( ,) =
xeM,.
( - 1)" - 1 "E ( , x ) " fin (X) if  E L" ,
o
if Ln'
Since Pn(x) = Pn(Y) if x, yare in the same F-orbit in M, it follows that
Pn()=Pn(TJ) if, TJ are in the same F-orbit in L.
Let e denote the set of F-orbits in L. For cp E e we denote by d( ep) the
number of elements of ep, and for each r  1 we define
Pre cp) = Prd( ,)
where  is any element of ep, and d = deep). We regard the p,{ep) as the
power-sums of a set of variables Yi,rp' each of degree deep). We can then
define other symmetric functions of the ¥i, rp by means of the formulas of
Chapter I, in particular Schur functions SA ( ep) by the formula (7.5) of
Chapter I.
For each function A: e fJlJ such that
IIAI(= E d{cp)IA(cp)\<oo
rpe8
let
s = n s(cp)(ep)
rpe8
which is a homogeneous element of B, of degree I( A II.
Now let S be the Z-submodule of B generated by the S . Since the
Schur functions form a Z-basis of the ring of symmetric functions, it is
clear that S is closed under multiplication and that the S  form a Z-basis
of S. Hence S is a graded sub ring of B. The equations (4.5) can be solved
for the Pn(x) in terms of the fin{ ), namely
Pn(X) = (_l),,-l(qn_l)-l E ("x)nPn()
eL,.
for x E Mn, by orthogonality of characters of the finite group Mn. Hence
we have S  C = B.
The point of setting up this machinery is that the S  such that II A II = n
will turn out to be the characteristics of the irreducible characters of
Gn = GL,,(k).
To complete this section, we shall show that the S  form an orthonor-
mal basis of S.

4. THE CHARACTERISTIC MAP
279
If u is any element of B, we may write u == E u...p... with coefficients
u". E C, and we define u to be E u...p.... Now from Chapter I, (4.3) we have
log( E SA(X)SA(Y)) =  log(l-XiYj)-J
A ',j
1
= E -Pn(X)Pn(Y)
n1 n
for two sets of variables Xi' Yj' and therefore
log(ES,.S,.)= E  E Pll(rp)Pll(rp)
A. n>1 n lpe8
1 
= E - E Pn ( ,)  p( , ) ·
,,>1 n teL,.
But from (4.5) we have
E Pn()Pn()="E E (X)n(,Y)nPn(X)Pn(Y)
EeL,. teL,. x,yeM"
= (qn - 1) E Pn(x)  Pn(x)
xeM"
by orthogonality of characters of the finite group Mn. It now follows from
(4.4) that
(4.6)
E s).  s). == E p...  Q....
A. ....
If S). == EfL u)....p... == E... V).fLQ... it follows from (4.6) that L). u)....v).1' ==
8...1" and therefore also
E UI(...v).... = 81t)..
....
In other words, (S 1(' S).) = 81().  so that
(4.7) The S). form an orthonormal basis of s.
Example
Let u be any class function on Gno Then the value of u at elements of the class c...
is
U(CfL) == (ch(u), QfL).
For u(c...) == a...(u, 11...) == (ch(u), ch(afL 11...» == (ch(u), Q...) by (2.7) and (4.1).

280 IV THE CHARACTERS OF GL" OVER A FINITE FIELD
5. Construction of the characters
Fix a character 8 of M, the multiplicative group of k.
(5.1) Let G be a finite group and let p: G -+ GLn(k) be a modular representa_
tion of G. For each x e G let ui(x) (1 <; i  n) be the eigenvalues of p(x). Let
fEZ[ t l' . . . , t n]S" be a symmetric polynomial. Then the function
X: x  f(8(ut(x)),..., 8(un(x)))
is a character of G, i.e. an integral linear combination of characters of
complex representations of G.
We defer the proof to the Appendix. We shall apply (5.1) in the case
that G = Gn, p is the identity map and f is the rth elementary symmetric
function e, (0  r  n). We shall also assume that 0: M --+ C* is injective,
i.e. an isomorphism of M onto the group of roots of unity in C of order
prime to p = char. k. Then (5.1) asserts that
X,: g  e,(8(xt),..., 8(xn)),
where Xi E f are the eigenvalues of g E Gn, is a (complex) character of Gn.
Now the characteristic polynomial of g is (2, Example 2)
n
n {,...(/)I = n (t -xi)
feet> i-1
where J.L: <I> -+ fJ'J is the partition-valued function determined by g.
For each fE <1>, say f= nCt - Yi)' we shall write
!=n(l+Yit)
and
o(!) = n (1 + 8(y;)t).
Then since
n n
E X,(g)t'= n (1 + O(x;)t)
r- 0 i-1
we have
(5.2)
n
E X,(g)t' = n 6 (i)'fI-<f)'.
r-O feci>
We need to calculate the characteristic of Xr' Since
Xr = E X,(C...)7T...
....

5. CONSTRUCTION OF THE CHARACTERS 281
_where X,(c,.I) is the value of X, at any element of the class c..., it follows
.'from (5.2) that
0(5.3)
n
E eh( X.)t' = E n OUYP.(f)lpp.
,-0 fA. fE
sununed over all fL: <I> --+.9 such that 1IfJ.1I = n.
Now from Chapter III, fi3, Example 1 we have
E tn(A)PA(x; t) = hm(x)
IAI-m
::nd therefore
(5.4 )
E PA(f) = hm(f)
IAI-m
for each f E <1>. It follows that the sum on the right-hand side of (5.3) is
equal to
 n (- u(f)
i..J 8 f) hU(f)(f)
Of fE
ummed over all u: <I>  N such that Ef d(f)u(f) = n, and hence is equal
to the coefficient of un in
H(t, u) = n n (1- O(j)Xi.fud(f)fl.
fE4.> i>1
Hence
(5.5) ch( X,) is equal to the coefficient of tf un in H(t, u).
; To get H(t, u) into a usable form, we shall calculate its logarithm:
log H(t, u) = E E log(l- O(j)Xi.fud(f)rl
fE i> 1
1 ..., m
= E - E 8(f) Pm(f)umd(f).
ml m fE
Now if x is any root of f, we have
d-l
8(i) = n (1 + t8(piX))
i-O

282 IV THE CHARACTERS OF GL" OVER A FINITE FIELD
where d = de!), and therefore (since Fdx =x)
md-l
8(i) m = n (1 + t8(piX».
i-O
Consequently,
um m-l
10gH(t,u) = E - E Pm(x) n (1 +t8(piX)).
m>l m xEMm i-O
Now Fi (0  i  m - 1) acting on km are the elements of the GalOis
group gm = Gal(kmlk). Hence the inner sum above may be rewritten as
E Pm(x) n (1 + t8( 1X)) = E till E Pm(X) n 8( l'X)
xEMm -yE Qm legm xeMm "'lEI
=(_1)m-l E tII1pm(8m,I)
legm
by (4.5), where 8m,I is the character x  n')' E I 8( I'x) of Mm, identified
with its image in Lm. So we have
(5.6)
( _t)m-1um
log H(t, u) = E E tI11pm(8m,I).
m>l m legm
The group 9m acts by multiplication on the set of all subsets 1 of Gmo If
the stabilizer of 1 is trivial, we shall say that 1 is primitive and that the pair
(m, I) is an index. In that case all the translates 1'1 (I' E gm) are distinct,
and it follows that the F-orbt of 8m, J consists of m elements. If I is not
primitive, the stabilizer of 1 in gm is a non-trivial subgroup 1) of gm'
generated by say FS, where s divides m. Since 1 is stable under multiplica-
tion by elements of 1), it is a union of cosets of I) in 9 m' say 1 = 1) I where
I = III) may be regarded as a subset of gm/I) = gs = Gal(kslk). More-
over, by construction J is a primitive subset of 9 s' i.e. (s, J) is an index,
and it is clear that 8m,I = 8s,1 0 Nm,s, where Nm,s: Mm -+ Ms is the norm
map (1), so that 8m,I and 8s,1 define the same element of L. Since (s,J)
is an index, the F-orbit ({Js,1 of 8s,1 in L has s elements, and therefore
Pm(8m,l) = Pm/s(CPs,1).
Hence the formula (5.6) now becomes
00 ( _1)n-l n
10gH(t,u) = E E Pn(CPs,J)(tIJ1uS) ,
($ , J) n - 1 n
the sum being over all indices (s, I), where two indices (s, I) and (s, I') are

;;."...
s. CONSTRUCTION OF THE CHARACTERS
283
.r -.
l116<be regarded as the same if J' is a translate of J, i.e. if 'Ps,J = 'Ps,J'. Now
fue inner sum above is by Chapter I, '(2.10') equal to
:):f log  e (cp )(tIJ1us)m
LJ m s.J
m>O
::'ihd therefore we have proved
:(5.7) For each pair of integers r, n such that 0  r  n, the coefficient of t' un
;:,.'
::  .!; ;
H(t, u) = n ( E em('PS,J)(tIJluS)m)
($,1) m>O
: .'
:"the characteristic of a character of G n.
C\::We shall use (5.7) to prove
(5.8) Let 'P be an F-orbit in L, let d = d('P) and let n  O. Then en('P) is the
characteristic of a character of Gnd.
Ijoof. By induction on the pair (n, d) ordered lexicographically: (n, d) <
(',d') if either n<n', or n==n' and d<d'. The result is true when
n= d == 1, because then cp == {} where  ELI is a character of G1 = Ml'
and
el( ,) = Pl(') = E (x) Pl(X) = ch()
xeG1
(since ch( 1Tx) == Pl(X) for x E G It 1Tx being the characteristic function of
(x}).
'::Let Xn,d denote the coefficient of tnund in H(t, u). By (5.7) Xn,d is the
paracteristic of a character of Gnd, and we have
Xn.d == E n eV(S,J)( 'PS,J)
v ($, J)
.ummed over all N-valued functions v on the set of indices such that
E III v(s, I) == n,
($, J)
E SV(S, J) = nd.
($, J)
3ach index (s, I) that actually occurs has v(s, J) <; n, and s <; d if
J(s, J) = n, so that (v(s, J), s) <; (n, d), with equality if and only if v(s, J)
= n (so that IJI = 1) and s = d. Hence Xn,d has 'leading term' en( 'Pd,{1})'
vhere 1 is the identity element of 9 d' and 'Pd,{l} is the F-orbit of (J I Md.
ow we can choose (J so that the F -orbit of (J I Md is 'P, and then we have
Xn,d = en( 'P) + ... ,

284 IV THE CHARACTERS OF GLn OVER A FINITE FIELD
where the terms not written are products of two or more e's, which by th
inductive hypothesis are characteristics of characters of groups G e
where (n', d') < (n, d). Hence en( <p) is the characteristic of a characte;'f
Gnd. I
6. The irreducible characters
We can now very quickly obtain all the irreducible characters of the groups
Gn = GLn(k). Since each Schur function SA is a polynomial in the e's with
integer coefficients, it follows from (5.8) that each SA ( 'P) and hence each
S). is the characteristic of a character X). of Gn (where n = " All).
By (4.1) and (4.7) we have ( X 1(, X). ) = 51().' and therefore each X). such
that II A II = n is, up to sign, an irreducible character of Gn. Moreover, the
rank of the free Z-module Rn generated by the irreducible characters of
Gn is equal to the number of conjugacy classes in Gn, i.e. the number of
functions fl.: <I> --+fJiJ such that 1Ifl.1I = n. Since the groups L, M are isomor-
phic, this is also the number of functions A: e --+!?)J such that II A II = n, and
consequently the X). form an orthonormal basis of Rn. Hence the
characters :t X). exhaust all the irreducible characters of Gn.
It remains to settle the question of sign, which we shall do by computing
the degree d). = X ).(1 n) of X)..
Let X; denote the value of X). at elements of the class c.... Then
X). = E X;1TfL
....
summed over all J.1: <t> --+!?)J such that 11....11 = II A II, and therefore
(6.1)
 ).-
S). = '-' XfL PfL
....
which shows that X; = (S)., Q...>, and hence by (4.7) that
(6.2)
- ).-
Q... = L., X... S)..

Now suppose that c... is the class of the identity element 1n E Gn, and let
11 denote the polynomial t - 1. Then fI.(/l) = (In) and ....(/) = 0 for f+ fl'
Hence
Q- = Q- (I) = qn(n+ 1)/2{f) {q-l)p (X, q-l)
... (1") 1 'Tn (1") 11'
= t/1n(q )en(/l)

6. THE IRREDUCIBLE CHARACTERS
285
['.bY Chapter III, (2.8), where
n
t/1n(q) = f1 (qi - 1).
i-I
. .
;.»
'!:
;f'flence from (6.2) we have
t/1n(q)en(!I) = L dA.SA..
II"-n
?{6.3)
Now let S: B --. C be the C-algebra homomorphism defined by
_'(6.4)
5(Pn(x)) = (_1)n-l /(qn - 1)
if x =1= 1,
if x = 1.
:(From (4.6) and (4.4) it follows that
:.:: .
: : ; 
. '
log(  S(S).)S).) = log (  S(P,.)Q,.)
(_I)n-1
= E fin(l)
n;;.l n
= log f1 (1 + Xi,!I)
i
and therefore
L ene/I) = E S(SA.)SA.'
n;;.O 
, so that
ene/I) = L 8(SA.)SA.'
II"-n
By comparing this formula with (6.3), we see that
(6.5)
dA. = t/1n(q)S(SA.)'
To calculate the number S(SA.)' obselVe that from (6.4) and the defini-
tion (4.5) of Pn() we have
S(Pn(E» = (qn _1)-1
for  E Ln, and hence for all 'P E e
S(Pn('P» = (q; _1)-1 = L q;in,
;;;.1

286 IV THE CHARACTERS OF GLn OVER A FINITE FIELD
where qqJ == l/d(tp). In other words, S(PIl«({J)) is the nth power sum of the
numbers q;' (i> 1), and hence rom Chapter I, 3, Example 2 we have
8(s..\«({J» ==S..\(q;l ,q;2,...)
= q;IAI-n(A) n (1 _ q;h(X») -I,
xe,\
where hex) is the hook-length at x E A. Since
I: hex) == IAI + n(A) + n(A')
xe,\
(Chapter I, 1, Example 2) it follows that
(6.6)
S(s..\( ((J») == q;().1>fi).(qtp)-1
where
fi..\(qrp) == n (q;(.t) - 1).
xe,\
From (6.5) and (6.6) we obtain
(6.7)
d). == «P1I(q) n q;().(rp)1)fi).(rp)(qrp)-1
qJe8
which is clearly positive. Hence X). (and not - X).) is an irreducible
character of Gn. To summarize:
(6.8) The functions X). defined by ch( X).) == S)., where ).: e 9' and
11).11 == n, are the i"educible characters of the group Gn = GL,,(k). The degree
d). of X). is given by (6.7), and the character table of G" is the transition
matrix between the bases (S) and (P.,.).
Examples
1. Let v =: (v,..., v,) be a partition of n, and let 0v be a character of the 'maximal
torus' Tv (2, Example 4). If we identify Tv with MVI X ... X Mvr, Ov may be
identified with (ft,..., r)' where fi E LVI (1 <; i <; r). Then 0'i-l PVI( f) is the
characteristic of an (in general reducible) character B(Ov) of Gn, which depends
only on the F-orbits lfJi of ;, i.e. B(Ov) depends only on the -orbit of Ov. in the
A A
character group Tv. The distinct characters B(O), where 0 e U v( \ Tv)' are
pairwise orthogonal in An; and since by 2, Example 5 there are exactly as many of
them as there are conjugacy classes in Gn, they form an orthogonal basis of the
space An of class functions on Gn. Green [011] calls the B(O) the basic characters.
B(Ov) is irreducible if and only if d(epi) = Vi (1 <; i <; r), that is to say if and only if
0v is a regular character of Tv (i.e. its stabilizer in  is triviaO.

r:
,': "
:.!:"",
lJ.: "
6. THE IRREDUCIBLE CHARACTERS
287
fi;. The valu of B( 8.,) at a unipotent element of type A is (4, Example 1) equal to
i'KfI Pv/( i)' QA(f1»' where /1 = t - 1. By (4.5) this is equal to
:'. (-1)E("I-I)(Pv(/I), QA(/1» = (_1)n-l(v)Q:(q)
:here Q;(q) is the Green polynomial (Chapter III, 7).
:.:'.:. Also B( 8.,) vanishes at regular semisimple classes which do not intersect the
:torus Tv.
:.2/ As in 2, Example 2 it is sometimes more convenient to regard A.: e -+!fIJ as a
.function on L, i.e. we define A.( g) for  e L to be the partition A.( (,0), where cp is
:the F-orbit of . With this notation let
(A.) = n I).(f)l eLl.
eL
,".Let a E k* = MI. Then
(*)
x(a .In) = ((A.), a)1 X(ln).
,This can be shown by the method used in the text to compute X (ln)' using the
homomorphism 6a: B -+ C defined by 6a{Pn(x» = (_I)n-l /(qn - 1) if x = a, and
fJa(Pn(x» = 0 if x::l= Q. Then X (a .In) = r/1n(q )6a(S).), and 6a has the effect of
,replacing the cp-variables li, rp by (, a)d . q;i, where  E cp and d = d( cp). The
formula ( *) now follows easily.
3. Let Un be the set of unipotent elements in Gn. Then for every irreducible
character X  of G 11 we have
Ex). (u) = ( _1)a().) qN()')X)' (In)
u e Un
where
a(A.) = n - EfP IA.«(,O)I and N(A.) = tn{n -1) + Erp d(cp)(n(A.(cp» - n(A.«(,O)'».
Let e = Eu e U X (u). Then we have
"
e). = IGnl E ap(q) -1(S)., Qp(/1»
Ipl-n
where /1 = t - 1, and hence from (5.4)
e = IGnl(S)., hn(/l»
so that
qn(n-I)/2r/1n(q)hn(!I) = E e).S)..
IIlll-n
Let 8: B -+ C be the C-algebra homomorphism defined by e(Pn(l» = (qn - 1) -1,
e(Pn(X» = 0 if x::l= 1. Then as in the text we find that
hn(/l) = E e(S).)S).
IIlll-n

288 IV THE CHARACTERS OF GLn OVER A FINITE FIELD
so that eA. = qn(n-l)/2.pn(q)e(S}..), and e(S}..) can be computed in the same way
8(S A.) in the text. as
4. Let .p be a non-trivial additive character of k, and X A. an irreducible character
of Gn. We shall compute
w A. = X). (1 n) - 1 E f/I (trace g) X ). (g )
geGn
(see [K14], [M2]). For this purpose we introduce the following notation: if x E k
let f/ln(x) = f/I(tracek,,/ k(X», and for  E Ln let 11
Tn ( ,) = ( - 1) n - 1 E (, , X) n t/ln ( X ) .
xeMn
If / E , let 1/1(/) denote f/ld(X), where d = d(f) and X is a root of f. If cp E 0, let
T(<,O) == Td( ), where d = d( cp) and, E <,0.
If g E cfJ.' we have trace(g) = LxeM 1,..,(x)1 x (2, Example 2) and therefore
f/I(trace g) == fI .p(f)lfL(f)1 = f/lp., say.
Ie ct>
We have
X }..(In)w). = IGnl E a;.lx;f/lp.
1IfJ.II-n
and since X; = (SA.' QfJ.) it follows that
(1)
E f/lp.PfJ. = /Gnl-1 E XA.(ln)w).S).o
fL }..
Now define a C-algebra homomorphism e: B  C by e(Xi,t) = f/I(f)qii, for all
/ E  and i  1. Then from Chapter III, 2, Example 1 it follows that S(QfL) = t/Jp.,
and hence from (1) and (4.6) we obtain
(2)
e(SA.) =IGnl-1 X).(ln)w)..
For each x E Mn, we have e(Pn(x» = f/ln(x)/(qn - 1), and therefore
(_I)n-l
e(Pn(» = n 1 E ("x)nf/ln(x)
q - xeMn
= Tn(,)/(qn -1)
for E Ln. Now the Hasse-Davenport identity (see e.g. [WI]) states that Tnd(g) =
Tn( )d for all d  1, and hence we have e(Pn(CP» = T(cp)n /(q - 1), where q'P =
qd(fJ), so that the effect of the homomorphism s is to replace the <p-variables Yi, rp
by ".( <,O)q;i. Consequently
e (S A.) = n T ( cp ) I). ()I . 8 ( SA.)
lpe8

6. THE IRREDUCIBLE CHARACTERS
289
[.:-where 8 is the homomorphism defined in (6.4). From (2), (3), and (6.5) we obtain
:!
w).=qn(n-l)/2 TI T()IA()I.
tpE 8
] ":;
:_5. In this example we shall compute the sum of the degrees of the irreducible
: representations of Gn. Since (6.5) d>., = tJln(q)8(S>.,), we need to compute
:-:II).II-n 8(S>.,). For this purpose we shall compute the sum of the series
S = E 5(S).)tll).11
A.
'where t is an indeterminate, and then pick out the coefficient of tn.
. We have
S = TI E 8(s,\ (lp ))tl'\ld(),
tp A
-:and since the homomorphism 8 replaces the variables ¥i,  by q; 1 (i  1), it
follows from Chapter I, 5, Example 4 that
-1 -1
E 8(S,\(lp))tIAld() = IJ (1- (tq-i)d(») n (1- (t2q-i-j)d(»)
A l 1<)
and hence that
:(1) log S = E ( \og(l- (tq-i)d(tp») -I + . log(l _ (t2q-i-i)d(tp») -I).
tp I l<)
Let dm denote the number of F-orbits  with cardinality d(lp) = m, for each
m  1; then we have
(2)
qm - 1 = E rdr
rim
from considering the action of F on the group Lm. We can then rewrite (1) as
( (tq- i) mr (t2q-i- j) mr )
log S = E dm E E + E E
m;.l i;.l r;.l r i<j rl r
d tmr ( )
= E E....!!!.. mr _ 1 + E tmrq-2imr
m;.l rl r q 1 il
which by use of (2) is equal to
( tN t2Nq-2iN)
E N+E N .
N>l ;;.1

290 IV THE CHARACTERS OF GLn OVER A FINITE FIELD
It follows that
111
s==- 0 2 -2; =(1+1)0 2 2'
1 - t ;> 1 1 - t q ;> 0 1 - t q- ,
==(I+t) E t2m/q;m(q-2)
m>O
(Chapter I, 2, Example 4). On picking out the coefficient of tn, and multiplying by
IJI,.(q), we obtain
E d == (q - 1)q2(q3 - 1)q4(qS - 1) ...
11).11-,.
(n factors altogether). This number is also equal to the number of symmetric
matrices in On' or equivalently to the number of non-degenerate symmetric
bilinear forms on k".
6. A calculation analogous to that of Example 5 shows that
E d.,. == (q - l)q2(q3 - 1)q4... (q2n - 1)
where the sum on the left is over fL such that II fLll == 2n and the partition ...( )' is
even for each tp. This number is equal to the number of non-degenerate skew-sym-
metric bilinear forms on k2n. In fact we have
E X'" = indg211(1)
II
summed over fL as above, where Cn = SP2n(k) is the symplectic subgroup of
O2,. == OL2n(k).
7. Let Nil denote the subgroup of upper unitriangular matrices in Gn. For each
sequence b  (bI,..., br) of positive integers with sum n, define a character t/lb of
Nn by the formula
.pb«h;j» == IJI( E h;,;+l)
where'" is a fIXed non-degenerate additive character of k, and the sum is over the
indices i other than bh bI + b2, . . ., bl + ... + br-t. Then the multiplicity of each
irreducible character X in the induced character ind(.pb) is equal to
II
E 0 K('P),a('P)+
a e8
where the sum is over all functions a: e ..... (z +)r such that EfP d( q; )a( q;) == b, and
a(tp)+ denotes the partition obtained by rearranging the sequence a(tp). (For a
proof, see [Z2].) In particular, ind(.p(n» = E X., summed over v such that for
each q; the partition ..,( tp) has length  1 (this result is due to Gelfand and
Graev).
Notes and references
The result of Example 4 is due to T. Kondo [K14], and the results of
Example 5 to A. A. Kljacko [Ktt] (also to R. Gow [GIO] in the case of

APPENDIX: PROOF OF (5.1)
291
Example 5(a) with q odd). Example 7 was contributed by A. Zelevinsky.
Appendix: proof of (5..1)
Since I is a polynomial in the elementary symmetric functions e" it is
enough to prove the proposition for 1= e, (1  r  n). Replacing p by its
exterior powers, it follows that we may assume that f == e1 == t 1 + ... + tn'
(1) Suppose first that g == IGI is prime to p == char(k). Then q mod. g is
a unit in Z/(g), hence g divides q' - 1 where r = 'P(g) (Euler's function).
Since ui(x)g == 1 for all x E G and 1  i  n, it follows that all the eigen-
values ui(x) lie in M" and hence only the restriction of (J to M, comes
into play. Since M, is a cyclic group it is enough to prove the proposition
A
for a prescribed generator (Jo of M" for then we have (J 1M, = (Jo say, and
we can apply the proposition (supposed proved for (Jo) to the symmetric
function I(tf,..., t).
Let 0 be the ring of integers of the cyclotomic field generated by the
(q' - l)th roots of unity in C, and let p be a prime ideal of 0 which
contains p. Then o/tJ = k" and p is the reduction mod p of a representa-
tion u of G with coefficients in o. Let U be the group of (q' - l)th roots
of unity in o. Reduction mod tJ defines an isomorphism of U onto M" and
we take 80= M,  U to be the inverse of this isomorphism. If the eigenval-
ues of u(x) are vi(.x) (1  i  n), each vi(x) is a complex gth root of unity;
hence lies in U, and its reduction mod p is ui(x) (for a suitable numbering
of the vi(x)). Hence X(x) = E (Jo(ui(x)) == E vi(x) is the trace of u(x).
(2) Now let G be any finite group. By a famous theorem of Brauer (see
for example Huppert, Endliche Gruppen I, p. 586), it is enough to prove the
proposition in the case that G is an elementary group, i.e. the direct
product of a cyclic group <x) and an I-group" where I is a prime which
does not divide the order of x. Then G is the direct product of a p-group
P and a group H of order prime to p. If yEP, the eigenvalues of p(y) are
p-power roots of unity in k, hence are all equal to 1, and since y
commutes with each x E H the matrices p(x) and p(y) can be simultane-
ously put in triangular form; hence the eigenvalues of xy are the same as
those of x, whence X(xy) = X(x). But X I H is a character of H, by the
first part of the proof, hence X is a character of G. I
Notes and references
Our account of the character theory of GLn(k) follows Green's paper
[Gl1] in all essential points. For another account, oriented more towards
-. the structure theory of reductive algebraic groups, see Springer [S20].
Another approach to the characters of GLn(k), based on the theory of
Hopf algebras, is developed in [Z2] and [S21].

v
THE HECKE RING OF GLn OVER A
LOCAL FIELD
1. Local fields
Let F be a locally compact topological field. We shall assume that the
topology of F is not the discrete topology, since any field whatsoever is
locally compact when given the discrete topology. A non-discrete locally
compact field is called a local field.
Every local field F carries a canonical absolute value, which can be
defined as follows. Let dx be a Haar measure f9r the additive group of F.
Then for a E F, a =1= 0, the absolute value lal is defined by d(ax) = lal dx.
Equivalently, for any measurable set E in F, the measure of aE is lal
times the measure of E. To complete the definition we set 101 = O. Then it
can be shown that la + bl  lal + Ibl for any a, b E F, and that the distance
function dCa, b) = la - bl determines the topology of F.
There are now two possibilities. Either the absolute value lal satisfies
the axiom of Archimedes, in which case F is connected and can be shown
to be isomorphic to either R or C: these are the archimedean local fields.
The other possibility is that the absolute value lal is non-archimedean, in
which case F is totally disconnected (the only connected subsets of Fare
single points): these are the non-archimedean local fields.
The classification of the non-archimedean local fields can be simply
described. If F has characteristic 0, then F is a finite algebraic extension
of the field Qp of p-adic numbers, for some prime p. If F has characteris-
tic > 0, then F is a field of formal power series in one variable over a
finite field.
From now on, let F be a non-archimedean local field. Let 0 =
{a E F: lal  I} and p = {a E F: lal < I}. Then 0 and p are compact open
subsets of F; moreover 0 is a subring of F, and p is an ideal in o. The
ring 0 is the ring of integers of F; it is a complete discrete valuation ring,
with F as field of fractions;  is the maximal ideal of 0, and k = olp is a
finite field (because it is both compact and discrete). Let q denote the
number of elements in k, and let 7T' be a generator of p. We have
111:'1 = q-l, and hence the absolute value lal of any a =1= 0 in F is a power
(positive or negative) of q. The normalized valuation v: F* -+ Z is defined
by
vex) = n x = U7Tn

2. THE HECKE RING H(G, K)
293
:witb u a unit in 0 (i.e. lul = 1).
: 2. The Heeke ring H( G, K)
Let F be a non-archimedean local field, and let G = GLn(F) be the group
of all invertible n X n matrices over F. Also let
G+= G nMn(o)
:be the subsemigroup of G consisting of all matrices x E G with entries
Xij E 0, and let
K= GLn(o) = G+n(G+)-l
so that K consists of all x E G with ntries xij E 0 and det(x) a unit in .
We may regard G as an open subset of matrix space Mn(F) = Fn ,
,.whence G inherits a topology for which it is a locally compact topological
group. Since 0 is compact and open in F, it follows that G+ is compact
::and open in G, and that K is a compact open subgroup of G.
A Haar measure on G is given by the formula
dx = ( n dxjj) det xln
I,) / I
"and is both left- and right-invariant. In particular, we have dx = dx', where
x' is the transpose of x.
Let dx henceforth denote the unique Haar measure on G for which K
has measure 1.
Next, let L(G, K) (resp. L(G+, K)) denote the space of all complex-
valued continuous functions of compact support on G (resp. G+) which
are bi-invariant with respect to K, i.e. such that
f(k1xk2) = f(x)
.for all x E G (resp. G+) and k1, k2 E K. We may and shall regard
'L(G+, K) as a subspace of L(G, K).
We define a multiplication on L(G,K) as follows: for f, gEL(G,K),
(f * g )(x) = J f{xy-l )g(y) dy.
G
(Since f and g are compactly supported, the integration is over a compact
set.) This product is associative, and we shall see in a moment that it is
commutative. Since G+ is closed under multiplication it follows immedi-
ately from the definition that L(G+, K) is a subring of L(G, K).

294 V THE HECKE RING OF GL,. OVER A LOCAL FIELD
Each function f E L( G, K) is constant on each double coset KxK in G.
These double cosets are compact, open and mutually disjoint. Since f has
compact support, it follows that f "takes non-zero values on only finitely
many double cosets KxK, and hence can be written as a finite linear
combina tion of their characteristic functions. Hence the characteristic
functions of the double cosets of K in G form a C-basis of L(G, K). The
characteristic function of K is the identity element of L(G, K).
If we vary the definition of the algebra L(G, K) (resp. L(G+, K)) by
requiring the functions to take their values in Z instead of C, the resulting
ring is called the Hecke ring of G (resp. G+), and we denote it by B(G, K)
(resp. H(G+, K)). Clearly we have
(2.1) L(G, K) =:: H(G, K) z C, L(G+, K) =:: H(G+, K) z C.
Our first aim is to show that the Hecke ring H(G, K) is closely related
to the Hall algebra H(o) of the discrete valuation ring o.
Consider a double coset KxK, where x E G. By multiplying x by a
suitable power of 'IT (the generator of .p) we can bring x into G+. The
theory of elementary divisors for matrices over a principal ideal domain
now shows that by pre- and post-multiplying x by suitable elements of K
we can reduce x to a diagonal matrix. Multiplying further by a diagonal
matrix belonging to K will produce a diagonal matrix whose entries are
powers of 7T', and finally conjugation by a permutation matrix will get the
exponents in descending order. Hence
(2.2) Each double coset KxK has a unique representative of the form
7T' It = diag( 7T' A1 , .. . , 7T' An )
where Al  A2  ...  An. We have An  0 (so that A is a partition) if and
only ifx E G+. I
Let c A denote the characteristic function of the double coset K 7r AK.
Then from (2.2) we have
(2.3) The cA (resp. the c,\ such that An  0) form a Z-basis of H(G, K) (resp.
H(G+, K)). The characteristic function Co of K is the identity element of
H(G, K) and H(G+, K). I
We can now prove that H(G, K) (and hence also H(G+, K), L(G, K)
and L(G+, K)) is commutative:
(2.4) Let f, g E H(G, K). Then f * g = g * f.
Proof. Let t: G -. G be the transposition map: t(x) = Xl. Clearly K is
stable under t, and since by (2.2) each double coset KxK contains a

2. THE HECKE RING H(G, K)
295
diagonal matrix, it follows that KxK is stable under t. Hence f 0 t = f for
all f E H( G, K), and therefore
(J*g)(x)= If(xy)g(y-l)dy
G
= 1 f(ytx')g{ (yt) -1) dyl
G
= (g * f)(xl) = (g * f)(x).
(2.5)
H(G, K) = H(G+, K)[ C(ln\].
Proof. Since 1T(1n) = 1T1n is in the centre of G, an easy calculation shows
that, for any A and any integer r, we have
C.\ * C(rn) = CA+(rn)
where A + (rn) is the sequence (AI + r,..., An + r). In particular it follows
that C(1n)' the characteristic function of K1T(1n)K = 1TK, is a unit in H(G, K)
and that its rth power is c(rn), for all r E Z. (2.5) now follows directly. I
In view of (2.5), we may concentrate our attention on H(G+, K), which
by (2.3) has a Z-basis consisting of the characteristic functions C A' where A
runs through all partitions (AI'.'" An) of length  n.
Let I-L, v be partitions of length <; n. The product C p. * Cv will be a linear
. combination of the CA' In fact
(2.6) clL * Cv = E g;v(q )c,\
,\
. summed' over all partitions A of length  n, where g;v(q) IS the 'Hall
,polynomial' defined in Chapter II, 2.
:'Proof. Let h;v denote the coefficient of c,\ in cp. * cV' Then
h.\ = (c * C ) ( 1T A) = 1 C (1T Ay - 1 ) C (y) dv
#LV P. V #L v '.l .
G
Since cv(y) vanishes for y outside K 1T v K, the integration is over this
double coset, which we shall write as a union of right cosets, say
K1TvK= U KYj
j
(Yj E 1TVK)
. Likewise let
K1TILK= UKxi,
i
(Xi E 1T ILK)

296 V THE HECKE RING OF GL" OVER A LOCAL FIELD
both unions being disjoint. Then we have
h;" = L f . Cp.(7TAy-l) dy
j KYj
=  JK Cp.( 7TAYj-1k) dk
}
= L CJJ.( 1T AYj-l )
j
since K has measure 1.
Now
CJJ.( 'IT AYj-l ) =1= 0  1T AYj-l E Kxi for some i,
C:$ 1T A E KxiYi for some i.
Hence h;" is equal to the number of pairs (i, j) such that
1TA = /ex.y.
, }
for some k E K (depending on i, j).
Let L denote the lattice 0" in the vector space F". Let gt,..., g" be the
standard basis of Fn, and let G act on Fn on the right (i.e. we think of the
elements of Fn as row-vectors rather than column-vectors). Then L'TT' A is
the sublattice of L generated by the vectors 1T At (1 <; i <; n), and therefore
M = L/L7T'A is a finite o-module of type ,\ (Chapter II, 1).
Consider Lxi' Since Xi E 1T ILK we have Lxi = L1T JJ.k for some k E K,
hence L/Lxi=LjL1TJJ., and is therefore a finite o-module of type J.L.
Next, consider N = Lxi/L7T' A = Lxi/LxiYj' Since Yj E 1T"K it follows as
before that N is of type v. Hence for each pair (i, j) such that 1TA E KxiYj
we have a submodule N of M of cotype J.L and type v. Conversely, each
submodule N of M with cotype J.L and type v determines a pair (i, j) such
that 1T A E KxiYp Hence h;" = g;,,(q). I
From (2.6) it follows that the mapping uA  C,\ is a homomorphism of
the Hall algebra H(o) onto H(G+, K) whose kernel is generated by the uA
such that l(,\) > n. Hence from Chapter III, (3.4) we obtain a structure
theorem for H(G+,K) and L(G+,K):
(2.7) Let An[q-l] (resp. An,e) denote the ring of symmetric polynomia in n
variables Xl'"'' xn with coefficients in Z[q-l] (resp. C). Then the Z-linear
mapping () of H( G + , K) into A ,,[ q -1] (resp. the C-linear mapping of
L(G+, K) into An,e) defined by
() ( C A) = q - n (A) PA (x 1 , . . . , x" ; q -1 )

2. THE HECKE RING H(G, K)
297
for all partitions A of length  n, is an injective ring homomorphism (resp. an
isomorphism of C-algebras). I
Remark. From (2.7) and Chapter III, (2.8) we have
8(C(1"») = q-n(n -1)/2X1 x2 . .. Xn'
Hence from (2.5) it follows that 0 extends to an injective ring homomor-
phism of H(G, K) into the algebra of symmetric polynomials in
x l 1, . . . , X n-J: 1 with coefficients in Z[ q - 1 ], and to an isomorphism of L( G, K)
onto the algebra of symmetric polynomials in X1-J: 1,..., x! 1 with coeffi-
cien ts in C.
In the next section we shall need to know the measure of a double coset
K1TAK. This may be computed as follows. For fEL(G+,K), let
p.(j) = f f(x) dx.
G
Then JoL: L(G+, K)  C is a C-algebra homomorphism, and clearly
(2.8) JoL(cA) = measure of K7rAK.
In view of (2.7) we may write J.t = J.t' 0 8, where JoL': An c  C is a C-algebra
. I
homomorphism, hence is determined by its effect on the generators e,
(1  r  n). Since e, = P(l')(X1,.. ., xn; q-l) (Chapter III, (2.8)) it is enough
to compute the measure of K 7r AK when A = (1') (1  r  n).
Now the measure of K7rAK is equal to the number of cosets Kxi
contained in K7r AK. For each of these cosets we have a sublattice Lx; of
L = on such that LILx; is a finite a-module of type A. Hence the measure
of K7TAK is equal to the number of sublattices L' of L such that LIL'
has type A. In particular, if A == (1'), each such L' will be such that
7T(L/L') = 0, i.e. 7rL cL'. Now L/7rL = kn is an n-dimensional vector
space over the finite field k, and L' I 7r L is a vector subspace of codimen-
sion r. The number of such subspaces is equal to the Gaussian polynomial
[; ](q) (Chapter I, 2, Example 3) and therefore
P.(C(l'») = [;](q) (1 rn).
From (2.7) it follows that p.'(e,) = q,(,-l)/2[; ](q), which ([DC. cit) is the
rth elementary symmetric function of qn-l, qn- 2,...,1. Hence JoL' is the
mapping which takes Xi to qn-i(1  i  n). It follows therefore from (2.7)
and (2.8) that the measure of K7rAK is q-n{).>PA(qn-l,qn-2,...,I;q-l),
which by Chapter III, 2, Example 1 is equal to
( q - n ( A) / VA (q - 1 ) ) q E( n - ;)A, V n ( q - 1 ).

298 V THE HECKE RING OF GL,. OVER A LOCAL FIELD
Hence we have the formula
(2.9) measure of K1rAK = qE(n-2i+l)A;vn(q-l )/VA(q-l),
= q2(A,p>Vn(q-l )/VA(q-l)
where p = !(n - 1, n - 3, ..., 1 - n).
3. Spherical functions
A spherical function on G relative to K is a complex-valued continuous
function lJ) on G which satisfies the following conditions:
(a) lJ) is bi-invariant with respect to K, i.e. lJ)(k1 xk2) = w(x) for x E G and
k1, k2 E K;
(b) w * f = AflJ) for each J E L(G, K), where AI is a complex number: in
othelWords, lJ) is an eigenfunction of all the convolution operators defined
by elements of L(G, K);
(c) w(l) = 1.
From (b) and (c) the scalar At is given by
A,=(W*f)(1) = jf(x)w(x-1)dx.
G
Regarde as a function of lJ), AI is called the Fourier transform of f and is
written f( w). Regarded as a function of f for fixed lL>, it is written we!).
For J, g E L(G, K) we have from (b)
At. g lJ) = lJ) * (f * g) = (w * f) * g = AtAg lJ)
and hence Af. g = >''1. Ag, or equivalently
(3.1)
GJ(f* g) = w(f)GJ(g).
Also it is clear that lJ) * Co = w, where Co is the identity element of
L(G, K) (the characteristic function of K). Hence w(co) = 1 and therefore
(3.1) shows that
(3.2) w: L(G, K) --+ C is a C-algebra homomorphism.
Conversely, it can be shown that all the C-algebra homomorphisms
L(G, K) --+ C arise in this way from spherical functions, so that the set
O(G, K) of spherical functions on G relative to K may be identified with
the complex affine algebraic variety whose coordinate ring is L( G, K).
From the remark following (2.7), L(G, K) is isomorphic to the C-algebra
C[ :t I :t 1 ]s C[ :t I :t: 1]. th d · t · f (C * )n
Xl '...'Xn "; now Xl ,...,xll IS e co or lnae ring 0 ,
and therefore O(G, K) may be canonically identified with the nth symmet-
ric product of the punctured affine line C.. Hence the spherical functions

f.
:_;0t.
r
3. SPHERICAL FUNCTIONS
299
h:Jnay be parametrized by n-tuples z = (z l' . . . , zn) of non-zero complex
2numbers, the order of the components z; being immaterial. However, it
l\will be more convenient to take as parameter s = (St, .., sn)' where
fl-Z' = q!(n-l)-sl (so that Sj E C (mod. 21TiZjlog q).)
'" ,
t Let w, denote the spherical function with parameter s. We have
G:{.rJ = ww, for all W E Sn.
,' s
,or
;;(3.3) The Fourier transform of the characteristic function cA is given by
C A ( W,) = q (A, p) PA (q -s 1, . . . , q -S,. ; q -1 ).
f:-Proof. From the discussion above and (2.7) it is clear that w, is the
::.composition of 8 with the specialization x;  z; = q!(n-I)-Sl (1  i  n).
>The result therefore follows from (2.7). I
Now we have
(\(w.)= 1 cA(x)ws(x-l)dx
G
= W,(1T-A) X measure of K1TAK.
From (3.3) and (2.9) it follows that
!:'
q-(A,p)
( -A)_ ( -I)p( -SI -s,.. -I)
W, 'IT - ( -1) VA q A q , · · · , q , q
Vn q
and therefore, from the definition (Chapter III, 2) of the symmetric
.Junctions P,\, we have (always under the assumption that Al  ...  An)
q-(A,p) ( q-s,_q-<SJ+l»)
(3.4) W.(1T-A) = (-I) E W q-(A..} n . -s/ -SJ ·
vnq weS i<j q-q
,.
Since w, is by definition constant on each double coset of K in G, the
formula (3.4) gives the value of the spherical function w, at each element
of G.
In particular, when s = p, we have wp( 'TI"-A) = 1 for all A, so that wp IS
the constant function 1 (the trivial spherical function).
From (3.4) it follows easily that
(3.5) w,(x-I) = w_,(x)
for all x E G.
Example
w, is bounded C$ Re(s) lies in the convex hull of the set Sn P == {wp; WE Sn}.
Let (7'= Re(s), i.e. ui == Re(s;) (1  i  n). We may assume, since ww, == Ws for all

300 V THE HECKE RING OF G.L,. OVER A LOCAL FIELD
WE SII' that Ul  ...  Un' Then the leading term in (3.4) is q-().'s+P), with
absolute value q-(A, u+ p). Hence Ws is bounded if and only if q-(A. u+p)  1 for all
A such that Al  ...  An' that is to say if and only if (A, a + p)  0 for all such A
and it is easy to see that this condition is equivalent to a E Cony (Sn p). '
4. Heeke series and zeta functions for GLn(F)
For each integer m  0 let
G;;'= {x E G+: v(det x) = m}
where v is the normalized valuation on F (1). The set G is the disjoint
union of the double cosets K'IT' AK, where A runs through all partitions of
m of length <: n. In particular, Gri = K. Let 'Tm denote the characteristic
function of G;:;, so that
'Tm= L CA"
I,\I-m
The Hecke series of (G, K) is by definition the formal power series
oc
(4.1)
'T(X) = L 'Tmxm E H(G+, K)[[X]].
m-Q
The results we have obtained enable us to calculate 'T(X) with no
difficulty. For by (2.7) we have
8(Tm) = L q-n(A>PA(X1,...,xn;q-l)
I,\I-m
=hm(x1,...,Xm)
by Chapter III, 3, Example 1. Hence
(4.2)
oc n
L 8('Tm)Xm = n (l-XiX)-l.
m-O i-I
Since
n n
n (I-XiX) = L (-1)' e,X'
i-I r-O
n
= L (-1)'P(l,)(X1,...,xn;q-l)X'
r-O
n
= L (-1)' q,(,-1)/20(C(1'»)X'
r-O

4. HECKE SERIES AND ZETA FUNCTIONS FOR GL,.(F) 301
;;O;bY (2.7) again, it follows from (4.2) that
(4.3) T(X) = ( t (_1)r qr(r-l)/2C(lr)Xr) -1,
r-O
a formula first obtained by Tamagawa [T2].
Now let w be a spherical function on G relative to K, and let s be a
:: complex variable. The zeta function (s, w) is defined by
((s, CtI) = 1 rp(x )lIxIlSCtl(x-1 ) th
G
where as usual dx is Haar measure on G, normalized so that Ix dx = 1; 'P
":is the characteristic function of G+, and IIxll = Idet(x)1 = q-v(detx). For the
',moment we shall ignore questions of convergence and treat s (or rather
;'o:q-S) as an indeterminate. We have
(4.4)
00
((s, CtI) = E q-ms 1 Tm(X)CtI(x-1) th
m-O G
00
= E W(Tm)q-ms
m-Q
= ( t ( _1)r qr(r-l)/2w(C(lr»q-rs)-1
r-O
by (4.3).
If w = W" it follows from (4.2) that
(4.5)
11
1 -1
(s, W,) = n (1 - qt(n-l)-:sI-S) .
i-I
These formal calculations will be valid, and the integral (4.4) will
converge, provided that Iqt<n-l)-s;-Sl < 1 for 1  i  n, i.e. provided that s
lies in the half-plane Re(s) > uo, where
0"0 = m ( n ; 1 - Re(s;» .
1 <, <n
In this half-plane the integral (4.4) defines a meromorphic function of s,
given by (4.5), which is the analytic continuation of (s, w) to the whole
s-plane.
In particular, for the trivial spherical function wp = 1 we have Si =
t(n + 1) - i, and therefore
(4.6)
n
(s, 1) = n (1 - qi-l-s) -1.
i-I

302 V THE HECKE RING OF GL,. OVER A LOCAL FIELD
5. Hecke series and zeta functions for GSP2n(F)
In this section we shall show how knowledge of the spherical functions for
GLn(F) enables us to compute explicitly the Heeke series for the group of
symplectic similitudes GSP2n(F), thereby completing a calculation started by
Satake in [S3], to which we refer for proofs.
Let i denote the n X n matrix with 1'8 along the reverse diagonal and zeros
elsewhere, and let
. ( 0
J= .
-l
) .
The group of symplectic similitudes G = GSP2n(F) is the group of matrices
x E GL2n(F) such that xjxl is a scalar multiple of j, say
xjxt == IL (x) j
where IL(X) E F*. Let
K == GSP2n(fJ) = G n GL2n(o),
G+= G n M2n(o).
Then G is a locally compact group, K is a maximal compact subgroup of G and is
open in G, and G+ is a subsemigroup of G. The Heeke rings H(G, K), H(G+, K)
and the C-algebras L(G, K) ::: H(G, K) z C, L(G+, K):: H(G+, K) z C are de-
fined as in 2. The characteristic functions of the double cosets KxK c G form a
Z-basis of H( G, K). Also H( G, K) is commutative, for the same reason as in the
case of GLn(F) (2.4).
Spherical functions CJ) on G relative to K are defined as in 3, and are in
one-one correspondence with the C-algebra homomorphisms w: L( G, K) -+ C. The
spherical functions are parametrized by vectors s = (so, S17...' sn) E Cn+1 ([83],
Chapter III).
Define 8j: Cn + 1 -+ cn + 1 (1 <:. i <:. n) by
8;(SO,SI'...' Sn) = (so +S;,SI,,,.,S;-I, -Sj,Si+l,,,.,Sn).
The 8; generate a group E of order 2n. Also the symmetric group Sn acts by
permuting sl,...,sn. Let WcGLn+1(C) be the group generated by E and Sn'
which is the semidirect product E X Sno Then we have [S3]
Ws = Wws
for all W E W, where CJ)s is the spherical function on G == GSP2n(F) with parameter
s.
Now let
G= {x E G+: v( IL(X») = m}

,:
tf;'
;:.
5. HECKE SERIES AND ZETA FUNCTIONS FOR GSP2n(F)
303
:Jor all n  0, and let Tm denote the characteristic function of G. The Heeke
tseries T(X) and the series T(a, X) are defined as in 4:
::
.;
GO
c
1..-,:..
i';
-::,;,
r
T(X) == E TmXm,
m-O
GO
!o::
T(a, X) == E Tm(ws)Xm.
m-O
i.NOW from [S3], Appendix 1 we have the following expression for Tm(W.):
. .
:;.,'(5.1)
Tm(Ws)" E q-(A,P)CA(Wl)qm(N-so)
.\
;.;'where a == so' SI'.. ., 9n), a' = (SI'. .., sn)' N == tn(n + 1); CA( ws') has the same
:.:':l11eaning as in 3, and the summation is over aU partitions A such that Al  m.
;'.'From (3.3) and (5.1) it follows that
Tm( WI) == E PA (q-Sl,..., q-S,,; q-l )qm(N-so)
.\
! and hence that
(5.2)
Ter, X).. E PA(q-Sl,..., q-S,,; q-1 )(qN-soX)m
m, ,\
': summed over m  Al  ...  An  o.
Now the sum on the right-hand side of (5.2) is one that we have calculated in
Chapter III, 5, Example 5. Hence we obtain the following expression for T(a, X)
as a sum of partial fractions:
(5.3) T(a, X) == E w( cI>(q-Sl,..., q-S,,; q-t )(1 - qN -so X) -1)
weE
where
n
cI>(X1,...,xn;t) == n (1-x;)-1 n (1-a;xj)(1-x;xj)-1.
i-I i <j
From (5.3) it follows that T(a, X) is a rational function f(Xo) /g(Xo) where
Xo == qN-soX, with denominator
g(Xo) = n (1- q-sJXo),
J
the product being over all subsets J of (I, 2,... , n), and S J == Ei E jS;. Also it is not
difficult to show that the degree of the numerator f(Xo) is 2n - 2.
Finally, we can attach to the spherical function Ws a zeta function as in 4: we
define as before
'(S, WI) == i lp(x )lIxIlSws(x-1) dx
G

304 V THE HECKE RING OF GLn OVER A LOCAL FIELD
where s is a complex variable, cp is the characteristic function of G+, and
!Ix\! == ! j.L(x)!. Then we have
00
({s, Cl)s) = E Tm{ (Us)q-ms
m-Q
= T(s, q-S)
and therefore from (5.3) we obtain the following formula for this zeta function:
(5.4) (s, (Us) = E W(<I>(q-Sl,...,q-SII;q-l)(l_qN-SO-S)-l).
weE
Notes and references
For background on algebraic groups over local fields, their Hecke rings
and spherical functions, we refer to [Ml] and [S3] and the references given
there.
We have derived the formula (3.4) for the spherical function by exploit-
ing our knowledge of the Hall algebra. It is possible to obtain (3.4) directly
by computing the spherical function from an integral formula. This is done
in [Ml], Chapter IV in much greater generality, and in the case of GLn
provides a more natural (though less elementary) proof of the structure
theorem (Chapter III, (3.4)) for the Hall algebra.
The formula (3.4) was also obtained by Luks [L14].

VI
SYMMETRIC FUNCTIONS WITH TWO
PARAMETERS
1. Introduction
The Schur functions SA (Chapter I, 3) are characterized by the following
two properties:
(a) The transition matrix M(s, m) that expresses the Schur functions in
terms of the monomial symmetric functions is strictly upper unitriangular
(Chapter I, (6.5)), that is to say SA is of the form
SA = mA + E K)..p.mp.
p.,<>"
for suitable coefficients KAp. (the Kostka numbers).
(b) The SA are paiIwise orthogonal relative to the scalar product of
Chapter I, 4, which may be defined by
(1.1)
(PA' Pp.) = SAp.ZA.
Again, the Hall-Littlewood functions PA(x; t) (Chapter III, 1) are
characterized by the same two properties (a) and (b), except that in (b) the
scalar product is now that of Chapter III, 4, defined by
1(>..) ,
(PA' Pp.) = SAp.ZA(t) = SAp.ZA n (1- tAi-1).
i-I
(1.2)
Another example is provided by the zonal polynomials (Chapter VII).
Up to a scalar factor they are characterized by the same properties (a) and
(b), but with a different choice of scalar product, namely
(1.3)
( ) - 21()') 
PA' Pp. - z). u).p..
More generally, Jack's symmetric functions (10 below) are character-
ized by (a) and (b), the scalar product now being defined by
"
(1.4)
( ) - I().) 
PA'Pp. - a z).u).p.
where a is a positive real number. They reduce to the zonal polynomials
when a = 2, and to the Schur functions when a = 1.

306 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
In this Chapter we shall study a class of symmetric functions PA (x; q, t)
depending rationally on two para.meters q and t, which include all the
examples above as particular cases. They will be characterized by the same
two properties (a) and (b), relative now to the scalar product defined by
I(A) 1 - qAt
(1.5) (PA' P,) = (PA' p,)q,t = 8A,. z\J] 1 - tA, ·
When q = t, they reduce to the Schur functions sA' and when q = 0 to the
Hall-Littlewood functions P).(x; t). To obtain Jack's symmetric functions,
we set q = ta and let t  1, so that (1 - q')/(1 - t') -+ a for each r  1;
and hence in the limit the scalar product (1.5) becomes that given by (1.4).
Example
Let F be a field of characteristic zero, and let ( , ) be a non-degenerate
symmetric P -bilinear form on A F with values in P, such that the homogeneous
components A'j.. of AF are pairwise orthogonal. If (ui),(v) are dual bases of A'F
(i.e. (u;, Vj) = tJij) then
Tn{xly) = E Ui(X)Vi(y)
;
is independent of the choice of dual bases (so that in particular Tn(xly) = Tn(ylx».
Let
T(xly) = E Tn(xly)
n>O
(the 'metric tensor').
Let : AF -+ AF  AF be the diagonal map (or comultipliction) defined by
f == f(x, y) (Chapter I, 5, Example 25).
A family (aA) of elements of a commutative monoid, indexed by partitions '\, will
be called multiplicative if a.\ all- = a A U II- for each pair of partitions '\, J.L. Equiva-
lently, aA = aA1a'\2". for each partition A == (AI' A2' ...), where ar = a(r) for each
integer r  1, and ao = 1. For example, the bases (e.\), (h.\) and (p.\) of AF are
multiplicative.
Show that the following conditions on the scalar product ( , ) are equivalent:
(a)  is the adjoint of multiplication, i.e.
(&f,g0h) = (f,gh)
for all f, g, h E AF.
(b) There exists a multiplicative family (v.\) in P* such that
(P.\,Pp.) = tJAII-Z,\v,\
for all partitions A, 1-£.

1. INTRODUCTION
307
_:LCc) There exists a character (i.e. F-algebra homomorphism) X: A F -+ F such that
,I.:_ X(Pn) :1= 0 for all n > 1 and
(f,g) == x(f* g)
,;. ..
f-i"-for all f, g E AF, where f * g is the internal product defined in Chapter I, 7.
;/-
;:;:.(d) The basis of A F dual to (m A) is multiplicative.
;\: (e) The metric tensor T satisfies
T(xly, z) == T(xly)T(xlz).
(Let P: =: z; IPA for all partitions A. Then
!:ap: = Ep: P;
.-. summed over pairs of partitions (a, ) such that a U {3 == '\. Hence (a) implies that
'(1)
(P: ,ppPu) = E (P:, pp)(p; ,Per)
::::summed over (a, (3) such that lal = I pi, 1131 = lul and aU f3 ==,\.
By iterating (1) we obtain
-. (2)
(P: 'Pit) = E (P:<1),Plt1)(P:(2),plt2)...
summed over all sequences (a(l), a(2), ...) of partitions such that I a(i)1 = JLi for
each i > 1, and U a(i) =,\. The sum on the right-hand side of (2) will be zero
::unless the partition ,\ is a refinement of J.L (Chapter I,  6), hence (PA'PIt) = 0
unless ,\ < J..L, and therefore also (by symmetry) unless J.L  '\. Thus (PA'Pp.) = 0
unless ,\ == J.L, in which case (2) gives
(P: ,PA) == n (P:"PA)' .
;> 1 I
Hence if we define vn = (P:, Pn) for each n  1 (and Vo = 1) we have
(PA'PIt) == AJ1.ZAVA
where vA == VA1VA2 . .. . Thus (a) implies (b).
Next, since (Chapter I, (7.12» PA * Pit == AJ1.ZAPA' it follows easily that (b) and (c)
are equivalent, the character X being defined by X(Pn) = vn for all n  1.
To show next that (b) implies (d), obselVe that it follows from (b) that the bases
(PA) and (zA" 1 VA-1pA) of AF are dual to each other, so that
T(xly) = E Z;lVA-1pA(X)PA(Y)
A
== exp( E V;;1 Pn(X)Pn(Y»).
n> 1 n

308 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
Let
exp( E .0;1 pntn) == E qnt",
n>l n n>O
then we have
T(xly) = IJ ( E qn(X)Yl)
J n>O
= E q.\(x)mA(y)
A
where q A == q Al q A2 · .. · This shows that the basis of A F dual to (m A) is (q »), which is
multiplicative. Conversely, if (q A) is multiplicative then we have
T(xly) == Il T(xIYj)
j
and so (d) implies (e).
Finally, to show that (e) implies (a), let (UA), (VA) be dual bases of AF' indexed by
partitions, and let
(3)
v.\(y,z) = E a;vvp,(Y)vv(z)
p.., v
Then
T(xly, z) = E UA(X)VA(Y' z)
A
(4)
= E a;vUA(X)Vp,(Y)Vv(Z)
A, p.., v
and on the other hand
(5)
T(xly)T(xlz) = E uj.C.(x)uv(x)vp,(Y)vv(z).
IL, v
Since the right-hand sides of (4) and (5) are equal, we conclude that
(6)
Uj.C.Uv == E a;VUA.
A
From (3) and (6) it follows that
(4V.\,up, Uv> =a;v= (VA,up,Uv>
which completes the proof.)
The scalar products considered in the text all satisfy these conditions: we have
X(Pn);;: 1,(I-t")-1,2, a,(1-q")/(1-tn)
corresponding to (l.t), (1.2), (1.3), (1.4), (1.5) respectively.
(n  1)

2. ORTHOGONALITY
309
tes and references
l earlier account of the symmetric functions P)..(x; q, t) appeared in [M6].
r some indications of the historical background, see the notes and
:erences at the end of this Chapter.
Orthogonality
tl)
t q, t be independent indeterminates, let F = Q(q, t) be the field of
tional functions in q and t, and let AF = A  F denote the F-algebra of
mmetric functions with coefficients in F. If we define
l(A) 1 _ q)..t
z)..(q,t) =z)..n
i - 1 1 - t At '
le scalar product (1.5) takes the form
2.2)
(P).., P p.) = (PA' P p.) q, I = SAp. Z A (q , t ).
)n each homogeneous component A'F of AF, this scalar product differs
Inly by a scalar factor from that defined by the parameters q-t, t-I. For
ve have
Z)..(q-l,t-I) = (q-It)'A'ZA{q,t)
lod hence
(2.3)
(f,g)q-l,t-1 = {q-lt)n<f,g)q,t
for f, g E A}.
 If a is an indeterminate we denote by (a; q)oo the infinite product
00
;(2.4)
{a; q)«J = n (I - aq')
r-O
regarded as a formal power series in a and q.
Let now x = (Xl' X2, ...) and Y = (Yt, Y2' ...) be two sequences of inde-
pendent indeterminates, and define
(2.5)
n (txiYj;q)«J
ll(x, y; q, t) = . . (x . ) ·
" J i Yj' q 00
Then we have
(2.6)
ll(x,y; q,t) = E z)..{q,t)-I PA{X)PA(Y).
A

310 VI SYMMETRIC FUNCTIONS W.ITH TWO PARAMETERS
Proof. We calculate
10gII(x,y;q,t) = E E (log(1-q'ttiYj) -log(1-q'x;Yj))
i,j ,>0
= E E E 2.(1 - tn)(qrxiy/'
i,j r>O n>l n
1 1 - tn
= E E - 1 _ n (XiY/'
i,j n> 1 n q
1 1 - tn
= E - 1 n Pn(x)Pn(Y)
n> 1 n - q
from which it follows that
( 1 1 - tn )
II(x,y;q,t}= n exp - 1 nPn(X)Pn(Y)
n> 1 n - q
wbich as in Cbapter III, (4.1) is seen to be equal to the right-band side of
(2.6). I
(2.7) For each n  0, let (u,\), (v,\) be F...bases of A'F, indexed by the partitions
of n. Then the following conditions are equivalent:
(a) (u..\, v}J.> = 8A}J. for all A, J..L;
(b) E u,\(x)v,\(y) = II(x,y; q,t).
A
Proof. The proof is almost the same as that of Chapter I, (4.6). Let
p: = z,\(q, t)-Ip,\, so that (p:, P}J.> = 8,\}J.. If
u,\ = E a,\pp; , vJ.L = E bJ.LuPu
p u
then we have
(u,\, v}J.> = E a,\pb}J.p
p
so tbat (a) is equivalent to
(a')
E a,\pb}J.p = 8,\}J..
p
On the other hand, by virtue of (2.6), tbe condition (b) is equivalent to
E u,\(x)u,\(y) = Ep;(x)pp(Y)
A p

2. ORTHOGONALITY
311
and hence to
(b/)
E a).pbACT = 8pCT'
A
Since (a/) and (b/) are equivalent, so are (a) and (b).
Now let gn(x; q, t) denote the coefficient of yn in the power-series
expansion of the infinite product
(2.8)
n (tx;y; q)ex> 
. ( .) = i.J gn(x;q,t)yn,
.>1 x;y, q ex> n>O
and for any partition A = (AI' A2, .. .) define
gA(X; q, t) = n gA (x; q, t).
. 1 I
,>
Then we have
gn(x; q, t) = E ZA(q, t) -1 PA (x)
IAI-n
by setting Y2 = Y3 = ... = 0 in (2.6), and hence
n(x,y;q,t)= n( E gn(X;q,t)yp)
} n>O
(2.10) = E g,\(x; q, t)m.\(y).
,\
(2.9)
It follows now from (2.7) that
(2.11)
< g,\ (x; q, t) , m p. (x) > = 8A p.
so that the g,\ form a basis of Ap dual to the basis (mA). Hence
(2.12) The gn (n  1) are algebraically independent over F, and AF ==
F[ g l' g 2' ...]. I
Next we have
(2.13) Let E: Ap  Ap be an F-linear operator. Then the following condi-
tions on E are equivalent:
(a) E is self-adjoint, i.e. (Ef, g) = (f, Eg) for all f, g E AF;
(b) Exn(x,y;q,t)==EyTI(x,y;q,t), where the suffIX x (resp. y) indicates
operation on the x (resp. y) variables.
-_Proof. For any two partitions A, JL let
e,\p. = (Em,\, mp,).

/\) v

312 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
Then (a) is equivalent to
(c)
eAp' == ep.A.
Next, from (2.11) we have
EmA == E eAj.Lgj.L
JL
and since by (2.10)
II(x, y; q, t) == E mA(x)gA(Y) = E mA(y)gA(X)
A A
it follows that (b) is equivalent to
E eAp.gp.(x)gA(Y) == E eAp.gp.(y)gA(X)
A,JL A,JL
and hence to (c). I
For u, v E F such that v + :t 1, let ll)u,v denote the F-algebra endomor-
phism of A F defined by
(2.14)
,-1 1 - u'
(J)u,v(Pr) = ( -1) 1 _ vr Pr
for all r  1, so that
(2.14')
l( A) 1 - u A;
ll)u,v(PA) = SAPA n A
i-I 1 - v j
for any partition A, where SA == (_1)IAI-/{A). Clearly
-1
ll)v,u == ll)u,v
(if u + :t 1), and ll)u, u is the involution ll) of Chapter I, 2.
From (2.9) and Chapter I, (2.14') it follows that
(2.15)
Cl)q,l(gn(X; q, t» = en(x),
the nth elementary symmetric function.
The endomorphism ll)u, v is self-adjoint:
(2.16 )
(CJ)u,vl,g) == (I, ll)u,vg)
for all f, g E A F. By linearity it is enough to verify this for 1 = PA and
g = PJL' where it is immediate from the definitions.


2. ORTHOGONALITY
313
Also we have
(2.17)
< wt,qf, g )q,t = < wf, g)
for f, g E AF, where the scalar product on the right is that of Chapter I.
Again we may take f = PA and g = P,.,., where the result becomes obvious.
From (2.17) it follows, for example, that (SA) and (Wt,qSA') are dual bases of
AF.
Finally, it follows from (2.6) and Chapter I, (4.1') that
(2.18) Wq,tll(x,y;q,t) = n(1 +xiYj)
i, j
where wq, t acts on the symmetric functions in the x variables.
So far we have worked in the algebra AF of symmetric functions in
infinitely many variables Xl' x2,. . ., with coefficients in F. However, in the
next section we shall need to work temporarily in An,F = F[x1,..., xn]Sn,
and to adapt the scalar product (1.5) to An, F' The original definition (1.5)
no longer makes sense in the context of AntF' because when X = (Xl'.'" Xn)
the power sum products PA (x) (for all partitions A) are no longer linearly
independent. We therefore proceed as follows.
(2.19) The gA(X; q, t) such that leA)  n form an F-basis of An,F.
Proof. Suppose first that q = t = O. Then gA becomes hA in the notation of
Chapter I, 2, and from Chapter I, 6 we have
(1) h,.,. (x) = E K,\,.,.s,\ (x).
A
Now in the ring An the Schur function SA(X) is zero if L(A) > n, and the
s,\(x) with L(A)  n from a Z-basis of An' Since the matrix (K,\,.,.) is
unitriangular, so is its principal submatrix (KA"")/('\)t /(,.,.) < n' Hence it follows
from (1) that the hA(x) such that L(A)  n form a Z-basis of An'
Now let q, t be arbitrary. Since the monomial symmetric functions
m,.,.(x) with l( J.L)  n form a Z-basis of An' we may write
(2)
gA(X; q.! t) = E b,\,.,.(q, t)mjJ.(x)
J.L
with coefficients bA,.,.(q, t) E F. From above, the matrix B(q, t) =
(b'\JA,(q, t»'('\),/(,.,.)< 11 is well-defined and nonsingular when q = t = O. Hence
its determinant, which is a rational function of q and t, cannot vanish
identically. In other words, the matrix B(q, t) is invertible over F, and
hence (2) shows that the gA(X; q, t) with L(A)  n form an F-basis of An,p,
as required. I

314 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
We now define a (symmetric) scalar product on An,F by requiring that
the bases (gA)/(A)< nand (mA)/(A)< n be dual to each other:
(2.20 )
(gA' mil) = SA,.,.
whenever '\, JL are partitions of length  n. With the obvious modifica_
tions, (2.7) remains valid for this scalar product, since by virtue of (2.8) We
have
(2.21)
n(x,y;q,t)= E gA(x;q,t)m).(y)
I(A)< n
when x=(x1,...,xn) and Y=(Y1,...,Yn).
Examples
1. For each integer m > 0 let
m-1
(a;q)m = n (l-aqr)
r-O
== (a; q )«J/(aqm; q)oo
and for any partition p, == ( ILl' P,2' ...) let
(a;q)p.= n(a;q),.,.;.
;;J 1
Then we have
 (t;q),.,.
gn(X; q, t) = i-J ( . ) mp.(x).
1,.,.1- n q, q JL
This follows from the identity
(I)'; q )«J
(y;q)«J
 (t; q)m
- i..J ym
m>O (q;q)m
which is a particular case of Chapter I, 2, Example 5 (with a, b, t there replaced
by 1, t, Y respectively).
Since (q; q )n/(q; q)". is a polynomial in q for all partitions IL of n (because it is
a product of Gaussian polynomials), it follows that (q; q )ngn(x; q, t) is a linear
combination of the monomial symmetric functions with coefficients in Z[q, t], i.e. it
lies in the subring A[q, t] of Ap.
2. Show that
cuq"gr(x;O;t-1) == (_t)-r gr(x;O,q).

lip.
3. THE OPERATORS D
315
3. The operators D;
':{i'In this section we shall work with a finite set of variables x = (Xt,..., xn),
:j.and we shall define F-linear operators
D:; An,F  An,F
.i'Jor each integer r such that 0  r  n. It will appear in fi4 that the
>symmetric functions PA.(x; q, t) are simultaneous eigenfunctions of these
/operators.
V., For each u E F and 1  i  n we define the 'shift operator' TUtX1 by
. .
L (3.1)
(Tu,xl!)(Xt,..., xn) = !(x1,..., ux;,..., xn)
'Jor any polynomial f E F[xt,.. . , xn]. Next, let X be another indetermi-
:rnate, and define
n
'(3.2) DII(X;q,t)=a8(x)-1 E s(w)xW8n(1+Xt(W8)/Tq,x,)
weS,. i-I
.<where as usual 8 = (n -1, n - 2,...,1,0), aB(x) is the Vandermonde deter-
'minant (Chapter I, *3), e(w) = :f: 1 is the sign of W E Sn, and (w); is the
"ith component of W.
For r = 0,1,..., n let D: denote the coefficient of X' in Dn{X; q, t):
(3.3)
n
Dn(X; q, t) = E D;X'.
r-O
:'We have D = 1, and
(3.4)
n
D = E A;(x; t)TqtXf
i-I
where
Ai(x; t) = aB{x)-t E s(w)t(WB)ixw8
weS,.
= a8(x) -1,Xia8(x)
so that
(3.5)
IX. -x.
A;(x;t)= n I J.
j+i Xi -Xj
ore generally, for any r = 0,1,..., n we have
(3.4 ),
D;= EA/(x;t)P Tq,xl
I leI

316 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
summed over all r-element subsets I of {1, 2,..., n}, where
A/(x;t) =as(x)-l(f1 T'"x.)as(x)
ieI '
(3.5),
lx. -x.
= t,(,-I)/2 f1 l ] .
ieI Xi -x.
jI ]
Each of the operators D: maps symmetric polynomials to symmetric
polynomials, and hence is an F-linear endomorphism of An,p. To establish
this fact we shall compute Dn(X; q, t)m).(x), where A = (AI'''.' An) is any
partition of length  n. First of all, for any vENn we have from (3.2)
n
Dn(X;q,t)xv= E 8(Wl)f](1+Xt(w18)lqJ't)Xw18+v,
w1eSII i-1
since Tq,xt(xV) = qVIXV. Now replace v by W2A, where w2 E Sn, and sum
over W2. We shall obtain
IS:IDn(X; q, t)m).(x)
n
= as(x)-t E s(wl) f] (1 +Xt(w18)tq(w2).)t)xw)8+W2).,
i-I
W17 W2
where S: is the subgroup of Sn that fixes A.
In the sum on the right, let us put W2 = wlw. Then it becomes
n
as(x)-l E s(wl) f] (1 +Xtn-iq(w).);)xW1(w).+8)
i-I
W,Wl
n
- E f] (1 + Xtn- iq(w).); )sw). (x),
we S i-I
,.
so that finally we have
n
(3.6) Dn{X; q,t)m).(x) = E f] (1 +Xtn-iq(lt)Sa(x)
a i-1
summed over all derangements a E Nn of A.
From this formula we see that each operator D is a degree-preserving
F-linear endomorphism of An,F. Moreover, since each Schur function Sa
that occurs on the right-hand side of (3.6) is either zero or else is equal to
:tSIL for some partition p., < A (unless a = A), and since the transition

3. THE OPERATORS D
317
t:"J11atrix M{s, m) is strictly upper unitriangular (Chapter I, (6.5)) we con-
r"clude from (3.6) that
tC3.7) Dn{X; q, t)mA == E aAp.(X; q, t)mp.
#L<A
:":with coefficients aAp' E Z[X, q, t], and in particular
(3.8)
n
aA)..(X; q, t) = n (1 + Xtn-iqA;).
i-I
(Thus the matrix of each operator D:, relative to the basis of An,F formed
\by the monomial symmetric functions, is strictly upper triangular, and its
: eigenvalues are the coefficients of X' in the polynomials (3.8). In particu-
lar, we have
(3.9)
DmA == E cAjJ.(q,t)mp.
#L<A
with coefficients cAjJ. E Z[q, t], and the eigenvalues of D are
(3.10)
n
C AA (q , t) == E q Al tn - i ,
i-I
which are visibly all distinct.
The second basic fact about the operators D; is
(3.11) Each operator D: is self-adjoint for the scalar product (2.20), that is to
say
(Df,g>n == (f,Dg>n
for allf,g E An,F and Orn.
It follows from (2.3) that (3.11) is equivalent to the identity
(3.12) Dn{X; q, t)xI1(x, y; q, t) == Dn{X; q, t)yII(x, y; q, t)
where II{x, y; q, t) is the product (2.5) (with i, j running from 1 to n) and
the suffix x Crespo y) indicates operation on the x (resp. y) variables.
To prove (3.12) we obselVe that from the definition (2.5) of Il =
n(x,y; q, t) we have
n 1-x.y.
n-1T Il = n I J
q,xi j-11-txiYj
which is independent of q. It follows now from (3.2) that
1l-1D(X; q, t)xn

318 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
is independent of q. Hence to prove (3.12), or equivalently (3.11), we may
assume that q = t.
Now for any polynomial f(XI'''.' xn) and W E Sn we have
xw8t(w8)iT f== T (xW8f)
t,x, t_x;
and therefore
n n
xw8n (1 +Xt(w8)/T )f== n (1 +XT. )(xw81")
t _ x, t, Xi J,
i-I i-1
so that
n
Dn(X; t, t)f== ail n (1 + _x )(asf).
. 1 I
1-
In particular, therefore,
n
Dn(X; t, t)SA == ail n (1 + XTt,x )a,\+ 8
. 1 I
,-
n
== n (1 +XtA;+n-i)s,\
i-1
for any partition ,\ of length  n.
Now the Schur functions s,\ such that 1(,\)  n form an orthonormal
basis of An_ F relative to the scalar product (2.20) when q == t, by virtue of
the Cauchy formula (Chapter I, (4.3». Hence we have (Dn(X; t, t)s,\, sp,)n
== 0 if ,\ =F J.L, and therefore
(Dn(X; t, t)s,\, sp,>n == (SA' Dn(X; t, t)sp,)n
for all '\, J.L of length  n. This shows that each operator D: is self-adjoint
when q == t, and completes the proof of (3.11). I
Examples
1. (a) The coefficient of mp, in Dn(X; q, t)m). is explicitly
n
a).,.,.(X; q, t) == E e( w )K1rjA. Il (1 + Xqaitn -i)
;-1
summed over all triples (w, a, 11') where we Sn' a E N" is a derangement of '\, and
11' is a partition, such that a + 8 == w( 11' + 8); and K'lrjA. is a Kostka number
(Chapter I, A6).
(b) Suppose that Al :a J.Lt. Deduce from (a) that
a).jA.(X;q,t) == (1 +Xq).lt"-l)a'\*jA._(X;q,t)
where A* == (A2, A3, ...), J.L* == (1L2' J.L3' ...).

3. THE OPERATORS D
319
2. (a) Show that, if T;> 0 and x = (Xl'...' Xn),
n
(t - 1) E Aj(x; t)x[ = tng,(x; 0, t-1) - 80,.
i-I
(Express the product
n tX -Xi
n X"-X.
i-I ,
as a sum of partial fractions.)
(b) Let II = ll(x, Y; q, t), where X = (Xl'...' Xn) and Y = (Y1'...' Yn), and let "o =
Wq,t" = ll(1 + XiY) (2.18). Then
II -1 Tq, x I" = E g, (y ; 0, t -1 ) t' X [ ,
rO
nolTq,x,no = E (-1)' g,{y;O,q)xi.
, 0
(c) Let E = £q,t = t-n{1 + (t - l)D). Deduce from (a) and (b) that
ll-lin = E g,(x;0,t-1)g,(y;0,t-1)t',
,o
llOl£llo= E (-1)'g,(x;O,t-1)g,(y;0,q),
,o
where in each case E acts on symmetric functions in the X variables.
"(d) Deduce from (c) and 2, Example 2 that
( -1 - ) -1 -
lJJq,t II Eq,tn = llo Et-1,q-l.llo
where 6Jq, t acts on the x-variables, and hence that
6Jq,tEq,t =Et-l,q-l6Jq,t.
3. Let a be a positive real number, X an indeterminate and let
n
Dn(X;a)=as(x)-l E &(w)xWn(X+(w8)i+axiDi)
weS is: 1
n
in the notation of (3.2), where Di = a/ax;. We have
n
Dn(X; a) = E xn-rD
,-0
say, where the D: are linear differential operators on functions of Xl'...' xn. In
particular, when a = 1, we have
n
Dn(X; 1)/= ail n (X + xiDi)(a8f)
i-I

320 VI SYMMETRIC FIJNCTIONS WITH TWO PARAMETERS
for a function f= f(Xh...' xn).
(a) Show that if A is a partition of lengh  n, then
11
Dn{X; a)m,\(x) = E n (X + n - i + a{3i}sfj(X)
f3 i-I
summed over all derangements f3 E Nn of A.
(b) Show that each operator D is self-adjoint with respect to the scalar product
defined by the metric tensor (1, Example) T(xly) = ni,j(1 - x;Yj)-a.
(The proofs of (a) and (b) are analogous to those of (3.6) and (3.11).)
(c) Let Y = (t - 1)X - 1. Then
Y + t(w8);T 1 - t(w8); 1 - T 1 - q
q,xi =x + + t(w8)i q,x; . .
t-l I-t 1-q I-t
If (q, t) -+ (1,1) in such a way that (I - q)/(1 - t) -+ a (for example, if q = ta and
t -+ 1), then this operator tends to X + (w8); + axiDi. Hence Dn(X; a) is the limit
of (t - 1)-nynDn{y-l; q, t).
(d) If f is a homogeneous polynomial of degree r, show that
D2f= f,
Df= (ar+ tn(n -1»f,
D;f= (- a2Un - a + cn)f,
where
n
Un = t E x;D'f,
i-I
 = E (Xi -Xj)-lxlDi'
;+j
and
Cn = t a2r{r - 1) + t arn(n - 1) + i4 n{n - l)(n - 2)(3n - 1).
(e) The Laplace- Beltrami operator D, is defined by
Dnaf= (aUn +  - (n - l)r)f
for f homogeneous of degree r, as in (d) above. Thus
D;f= (-a Dna + c)f,
where c = cn - a{n - l)r.
Let E: be the operator defined by
E:f=: ail/aUn(a/af) - {ail/aUna/a)f.
Show that
E:=+a-l.

4. THE SYMMETRIC FUNCTIONS PA(x; q, t)
321
Notes and references
Example 3. The differential operators D (depending on a) were intro-
duced by Sekiguchi [S9] and later by Debiard [D1]. See also [M5].
4. The symmetric functions P)..(x; q, t)
The operators D:: An,F  An,F defined in 3 are (for fixed r and varying
n) not compatible with the restriction homomorphisms Pm,n: Am,F  An,F
of Chapter I, 2. However, at least when r = 1 (which will be sufficient for
our purposes) it is easy to modify them. Let En be the operator on AntF
defined by
(4.1)
n
E = t-n D1 - '" t-i.
n n L.J
i-I
If A = (AI' . . . , An) is a partition of length  n, we have
mA = E sa
a
in An' where the sum is over all distinct derangements a E Nn of A, as one
sees by multiplying mA = E xa by a8 = L e{w )xw8 an rearranging (Chapter
I,  6, Example 11). Hence it follows from (3.6) that in An,F we have
EnmA = E (.r. (qal -l)t-i)Sa
a ,-1
(4.2)
summed over derangements a of A as above.
This formula shows that
Pn,n-l 0 En =En-l 0 Pn,n-l
where Pn,n-l: An,F  An-1,F is the homomorphism defined by setting
xn=O (obseIVe that sa(xt,...,xn-t,O)=O if an =1=0). Hence we have a
well-defined degree-preseIVing operator
(4.3)
E = Eq I = lim En : A F  A F
, 
such that for each partition ,\
(4.4)
EmA = E eAp.mp.
p,<A
with coefficients eAp' E Z[q, t-1], and in particular
(4.5)
eAA = E (qAt - l)t-i.
i 1

322 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
Since D is self-adjoint for the scalar product (/, g)n (3.11), so is En' and
hence
(4.6) E is self-adjoint for the scalar product (1.5) on Ap.
We come now to the main result of this section.
(4.7) For each partition A there is a unique symmetric function P).::::
PA(x; q, t) E Ap such that
(a)
PA = E uAp.mp.
p,<A
where uAp. E F and UAA = 1,
(b)
(PA, Pp.) = 0 if A =1= J.L.
Proof. We shall construct the PA as eigenfunctions of the operator E. If PA
satisfies (a) above and
(c)
then by (4.4) we have
EPA = e,\APA
e)"AuAv= E uAp.e,.,.v
v< p- < A
for all pairs of partitions v, A such that v < A, or equivalently
(1)
(eAA - evv)uAv = E uA,.,.ep.v.
v< 11-< A
Since by (4.5) we have eAA =1= evv if v =1= A (i.e. the eigenvalues of the
operator E are all distinct), this equation determines uAv uniquely in terms
of the uAp. such that v < J.L  A. ence symmetric functions PA exist
satisfying the conditions (a) and (b). But then we have by (4.6)
eA).. (PA, Pp.) = (EPA, Pp.)
= (P).., EPp.) = e,.,.p.(PA, Pp.)
and since eA).. =1= ep.p. if A =1= 1-£, it follows that the PA satisfy condition (b).
Finally, to show that the PA are uniquely determined by (a) and (b), let A
be any partition and assume that Pp. is determined for all 1-£ < A. Then by
(a), PA must be of the form
PA = mA + E vAp.Pp.;
11-< A

r..:,_.
I;'::",
11/
4. THE SYMMETRIC FUNCTIONS P).(x; q, t)
323
m{taking the scalar product of each side with PIL, we obtain
f,:,;.
:-:.:.- VAIL = - (mAt PIL) I(Pp., PIL)
:.:.r:.''-. '
land hence PA is uniquely determined. (Note that (PIL, PIL) =1= 0, because
:!J:::when q and t are taken to be real numbers between 0 and 1 the scalar
j'product (1.5) is positive definite.) I
.}f}Remark. In fact, conditions (a) and (b) of (4.7) overdetermine the symmet-
J-}ic functions PA. For if we arrange the partitions of each positive integer n
t:f:jn lexicographical order (so that (1") comes first and (n) comes last), we
t'can use the Gram-Schmidt orthogonalization process to derive a unique
;:basis (PA) of A'j.. satisfying (a') and (b), where
;:r(a') PA = mA + a linear combination of the mp' for J.L preceding A in
: lexicographical order.
;1EIf we replace the lexicographical ordering by some other total ordering
0:'compatible with the natural partial ordering A  J.L, and apply
;Gram-Schmidt as before, we shall obtain the same basis (P,\): this is the
?';content of (4.7).
}} From (4.7) (a) we have in particular
',--(4.8)
PO') = er
.:;'(since m(I') = er). Also
s.{4.9)
(q; q)r
p(r)':= (. ) gr.
t, q r
':-For by (2.11), gr is orthogonal to mIL for all partitions J..L =1= (r), hence to all
,:Pp, except for J.L = (r). It follows that gr must be a scalar multiple of p(r)'
.'and the scalar factor is given by 2, Example 1.
Next we have
..( 4.10) PA (x 1 , . . . , X n; q, t) = 0 if n < I ( A) .
Proof. If J..L  A, then p.,'  A' (Chapter I, (1.11)), so that
l( J.L) = iiI  Xl = l(A) > n,
and hence m,...{x1,..., xn) = 0 for each J.L  A.
Let
(4.11)
-and
(4.12)
b,\ = bA (q , t) = (PA, PA) -1
QA ':= bAPA,

324 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
so that we have
(PA, QIL) = BAIL
for all '\, J.L, i.e. the bases (P)..), (Q)..) of A F are duals of each other for the
scalar product (1.5). Hence by (2.7) we have
( 4.13) E PA (x ; q, t ) Q A (Y; q, t) = II (x, y ; q, t) .
x
We shall next consider some particular cases.
(4.14) (i) When q = t we have
PA (x ; t , t) = Q).. (x ; t , t) = SA (x)
(Chapter I, (4.8) and (6.5)).
(ii) When q = 0 we have
PA (x; 0, t) = P).. (x ; t ) ,
QA(X; 0, t) = QA(X; t),
the Hall-Littlewood symmetric functions studied in Chapter III. This
follows from Chapter III, (2.6) and (4.9).
(iii) Let q = t a and let t -+ 1. The resulting symmetric functions are Jack's
symmetric functions, which we shall consider in 10 below.
(iv) From (2.3) and (4.7) it follows that
PA ( x; q - 1 , t - 1 ) = P).. (x; q, t) ,
Q ( . -1 (-1)_( -l)IAIQ (. )
A x, q, - qt A x, q , t .
(v) When t = 1, we have
PA{X; q, 1) = mA(x).
For it follows from (3.2) that Dn(X; q, 1) = TIf-l(l + XTq,x), so that
Dn{X; q, l)mA = [1{1 + XqAi)mA. Hence the mA are the eigenfunctions of
D, and hence of En' when t = 1, which proves the assertion.
(vi) When q = 1, we have
PA(x, 1, t) = eA,(x)
for all t. We defer the proof of this statement to 5.
(4.15) If x = (Xt,..., xn) and 1(,\)  n, we have
n
D n (X; q , t ) PA (x ; q, t) = n (1 + Xq Ai t n - i) PA (X ; q , t) .
i-I

4. THE SYMMETRIC FUNCTIONS PA(x; q, t)
325
Proof. As J.L runs through the partitions of length  n, the mIL form a
. basis of An,F; hence so do the PIL, by (4.7) (a), and therefore also the QIL.
Hence Dn{X; q, t)PA is of the form
Dn(X; q, t)PA = L VAIL (X; q, t)Qp..
JL
By (4.7) and (3.7), the matrix (VAIL) is strictly upper triangular, and by (3.11)
it is symmetric, hence diagonal. It follows that D,,(X; q, t)PA is a scalar
multiple of PA, and by (3.7), (3.8) the scalar multiple is
n
n (1 +XqA;tn-i). I
i-I
(4.16) The operators of D(O  r  n) on An,F commute with each other.
For by (4.15) they are simultaneously diagonalized by the basis (PA).
(4.17) Let A be a partition of length n. Then
PA ( Xl' · · · , X n ; q, t) = Xl" · X n PIL (X 1 , . · · , X" ; q , t )
where J.L = (AI - 1,..., An - 1).
Proof. We have
Tq,x,(Xl ... x"PIL) = qX1 ... xnTq,xi(PIL}
and therefore
D  (x 1 · · · x" Pp.) = qx 1 · · · X" D  ( P,... )
= q ( ,r. qIJ.;t1l-i) Xl ... XnPIL
,-I
by (4.15). Hence both PA and Xl ... nPIL are eigenfunctions of the opera-
tor D for the eigenvalue EqA;tn-,. Since the eigenvalues of D are all
distinct, it follows that PA and Xl... X n P can differ by at most a scalar
factor; but they both have leading term XA, and so they are equal. I
Examples
1. For any partition A, the coefficient of xt1 in P).(x; q, t) is equal to P)..(x*; q, t),
where A* == (A2, A3, ...) and x. == (X2' x3' ...). (With the notation of (4.7), the
coefficient of X;1 in P). is equal to
E u).vmv.(x.)
v

326 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
summed over partitions v < A such that VI = AI' where v * = (V2' V3' ...). Hence .
has to be shown that uAv _ uA.v. for all such partitions. The coefficients u It
deterined recursively by the equtions (1) in the proof of (4.7). From 3. ExID.:
l(b) It follows tha eA*p.* == teAP. If J.L < A, and eA*A* - ev*v* = t(eA). - evv). Hence
u>..v = U).*v*' as requIred.)
2. From 3, Example 3(a) it follows that
D,.(X; a)m>..(x) =- E c>..p.(X; a)mp.(x)
IJ-<A.
with coefficients CAp. E Z[X, a] and
n
CAA = n (X + aA; + n - i).
i-I
Since a > 0, the sequence (a Ai + n - i)l < i < II is strictly decreasing, and therefore
C A), :# cp.p. if A:# J.L.
(a) Let C denote the matrix (c Ap.). Show that there is a unique strictly upper
unitriangular matrix U such that UCU-1 is a diagonal matrix. For each partition A
of length  n define
(1)
PAa(x) = E uAp.mp.(x)
IJ-<A.
where x == (Xl'...' XII). Show that
(2)
Dn(X; a)P>..a = c>..>..(X; a)P>..a
and deduce from f3, Example 3(b) that
(3)
(p>..a,pp.a)Q = 0
if A:# J.L.
(b) The symmetric polynomials p>..a(x) so defined are characterized by the proper-
ties (1) and (3). Hence the coefficients U Ap. can be computed recursively by the
Gram-Schmidt process. Deduce that the u >..p. are rational functions of a, indepen-
dent of X and of n. The symmetric functions pAa so defined are Jack's symmetric
functions (flO below).
(c) The differential operators D defined in 3, Example 3 commute with each
other. (By (2) above, they are simultaneously diagonalized by the P>.. a.)
3. Let Dna be the Laplace- Beltrami operator (3, Example 3(e» acting on sym-
metric polynomials in xl'...' X II .
(a) If p = Pn, II -1: An -. AII-1 is the restriction homomorphism, show that for
f E Xn we have
p(Unf) = Un-1( pf),
p(v;,f) = (-1 + r)( pf)
and hence that p 0 Dna = D:_ lOP. It follows that
oa = lim D a
+- II

lli;f.
".;':::-.
fs a well-defined operator on A F, where F = Q( a).
f}(b) Show that
5. DUALITY
327
DO: P>.. a = e>.. ( a ) PA 0:
:'where e>..(a) = n(A')a - n(A).
f';', If A> J.L we have n(A) < n(,.,,) and n(A') > n( J.L'), hence e>..(a) + elJ-(a).
'2(C) Show that, if A is a partition of length n,
:-:::"_':i
DO:m>.. = eA(a)mA + E (Ai - Aj) E m(A-rct+rcJ)+
1<i<j<n r>1
-.;:;r'
I;";
-'.where (A - rB; + rBj) + is the partition obtained by rearranging the sequence
;'(Al'...' Ai - r,..., Aj + r,..., An), and in the inner sum r runs from 1 to [teA; - Aj)].
r.;(d) Let U >..IJ- denote the coefficient of mIJ. in PA a, as in Example 2 above. Deduce
;t.'from (b) and (c) that if J.L < A we have
U>"IJ. = (eA(a) -elJ-(a))-l E( J.Li - J.Lj + 2r)uA,(IJ-+rci-rCj)+
,E;summed over (i, j, r) satisfying 1  i <j  I( J.L) and r  1, such that (J.L + rBi - rBj)+
};,: is a partition < A.
';': The coefficients U >..IJ- may be computed recursively from this formula. In particu-
;:)ar, it shows that u>"IJ. is a rational function of a whose numerator and denomina-
i/1or are polynomials in a with positive integral coefficients.
...' 5. Duality
!;The effect of the involution w (Chapter I, 2) on the Schur functions is
<; given by
WSA = SA'
.(Chapter I, (3.8)). This result is generalized in the following proposition, in
which Wq,t is the automorphism of AF defined in (2.14):-
(5.1) We have
Wq,tPA(X; q, t) = Q>..,(x; t, q),
Wq,tQA(X; q,t) =PA,(x;t,q).
- Since Wt,q = w:, these two assertions are equivalent.
Clearly (5.1) is equivalent to
(wq,t PA,(q, t), Pj.£ (t, q) )t,q = 5AIJ.
-- and hence by (2.17) to
(5.1')
( wPA,(q, t), PIJ. (t, q) = 5AIJ.

328 VI SYMMETRIC FUNCTIONS WITH. TWO PARAMETERS
in which the scalar product is that of Chapter I. Next, let
A{q, t) = M{P{q, t), s),
the transition matrix from the P's to the Schur functions. Then (5.1') is
equivalent to
(5.1" )
JA(q, t )JA(t, q)' = 1
where J = (8A'#L) is the conjugation matrix and A(t, q)' is the transpose of
A(t, q).
To prove (5.1") we introduce the matrix of scalar products
U(q, t) = (s,\, Sp.)q,t)
which has the following properties:
(5.2) (a) U(q, t) = U(t, q)-I,
(b) U{q, t) = JU(q, t)J,
(c) D{q, t) =A(q, t)U(q, t)A(q, t)' is a diagonal matrix.
Proof. Let X = (XpA) be the matrix of characters of Sn (Chapter I, 7).
Then the (p, JL) entry of XU{q, t) is
E XpA(s,\, Sp.) q,t = (pp, Sp.) q,t
A
1 - q Pi
= X: n 1 _ t Pi
from which it follows that XU(q, t)X-1 is a diagonal matrix whose inverse
is XU(t, q)X-t. This proves (a), and since Xl = eX, where e is a diagonal
matrix of :f: l's, it also proves (b). Finally, (c) expresses the orthogonality of
the PA(q, t). I
Now the matrix A{q, t) is strictly upper unitriangular, and hence
B =JA(q,t)JA(t,q)'
is strictly lower unitriangular. We now compute
C = D(t, q )B-1 = D(t, q )A(t, q ),-IJA(q, t) -1 J
=A(t, q )U(t, q )JA(q, t) -1 J by (5.2)(c)
=A(t,q)JU(q,t)-1 A(q,t)-1 J by (5.2)(a),(b)
=A(t, q )JA(q, t)' D(q, t) -1 J by (5.2)(c)
=B'JD(q,t)-IJ.

5. DUALITY
329
So the matrix C is lower triangular (because B-1 is) and upper triangular
(because B' is), hence diagonal. It follows that B = 1, which proves (5.1")
and hence also (5.1). I
In particular, we may take q = 1 in (5.1 '). Then PIL{t, 1) = mIL by (4.14Xv),
and hence .(5.1') shows that wPA,{l, t) = hA, or equivalently PA,(l, t) = eA, as
stated in (4.14Xvi).
From (5.1) and (4.12) it also follows that
(5.3)
bA(q, t)bA,(t, q) = 1,
and by applying ClJq,t to both sides of (4.13) that
(5.4)
n{l+xiYi)= EpA{x;q,t)PA,(x;t,q)
i.i A
= E QA(X;q,t)QA,(X;t,q).
A
Finally we have
(5.5)
Q(n){X; q, t) = gn{x; q, t)
by (2.13), (4.8), and (5.1).
Examples
1. If A is any partition let
fA(q, t) = (1 - t) E (qA; - 1)ti-1.
i 1
(a) Show that
fA (q , t) = fA' (t , q )
where A' is the conjugate partition.
(b) The eigenvalues of the operator (t - t)E (4.3) are fA{q, t-1).
2. We shall use Example 1 above and 3, Example 2(d) to deduce another proof of
the duality theorem (5.1). The operator E of 3, Example 2 is equal to 1 + {t - l)E,
and hence
E t -1 , q - 1 W q , t PA (q , t) = W q tiE q , I PA ( q , t)
= {1 + fA ( q , t - 1 ) ) W q , I PA ( q, t)
= (1 + fA' (t -1 , q )) Wq, t PA (q , t).

330 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
It follows that 6),,1 PA (q, t) is an eigenfunction of £,-1, f 1 with eigenvalue
1 +!A,(t-1,q), and hence must be a scalar multiple of P).,(t- ,q-1)=p).,(t,q). To
complete the proof of (5.1) it remains to show that
< W q , t P). (q , t) , P)., (t , q ) ) t , q = 1
or equivalently (2.17) that
(1)
( wPA (q , t ) , PA, (t , q » = 1
with respect to the scalar product of Chapter I. Finally, prove (1) by expressing
PA(q, t) and p).,(q, t) as linear combinations of Schur functions, and using the fact
that wSp. == sp.'.
3. Let a be a positive real number, let F = Q( a) and let lU" be the F-algebra
automorphism of AF defined by
_ ( 1),-1 ,
waP, - - a p,
for each r  1.
(a) Show that
1 -1
W O"w- =: - a 0"
" "
where 0" is the operator defined in 14, Example 3(a). (Verify that
UnPA =n(A')PA +S,
PA == «n -1)IAI- n(A'))p). + S',
where Un'  are the operators defined in 3, Example 3(d) and S (resp. S') is a
linear combination of power-sum products Pp. such that I ILl = I AI and I( IL) = I( A) - 1
(resp. I( #-t) = I( A) + 1). Hence show that
( w" Oa + a Oa-1w" )PA = 0
for all partitions A.)
(b) Let Q: = W; 1 PA'!-l (fi4, Example 2(b)). Deduce from (a) that Q: is an
eigenfunction of 0" with eigenvalue e).(a) (4, Example 3).
(c) Show that (Q:) is the basis dual to (P).G) for the scalar product (1.4). (Let
aAp' == (PAG, Q;). By expressing P)." and pp.-l as linear combinations of Schur
functions, show that the matrix (a.\p.) is strictly upper unitriangular, so that Q).). = 1
and a).p, == 0 unless A  #-t. Finally, if A > #-t use the self-adjointness of 0° and the
fact that eA(a) +- ep.(a) (A4, Example 3) to conclude that a).p. = 0.)
Notes and references
The proof of the duality theorem (5.1) in the text was suggested by A.
Garsia. See [G3].

6. PIERI FORMULAS
331
6. Pieri formulas
For any partition JL and positive integer r we have (Chapter I, (5.16))
S#-,s(,) = E s).
,\
-summed over all partitions A:) JL such that A - IL is a horizontal r-strip,
: and dually
8#-,8(1') = E s).
,\
summed over partitions A:) p., such that A - p., is a vertical r-strip. In this
section we shall generalize these facts to the symmetric functions P).(x; q, t).
For each partition JL of length  n, define a homomorphism (or
specialization)
u,..,: F[Xt,..., xn] -+ F
by u,..,(xi) = q#-'ltn-i(1  i  n). We extend u#J. to those elements of the field
F(x1,..., xn) for which the specialized denominator does not vanish. In
particular,
uo({) = {(tn -1 , tn - 2,... , t, 1)
for any polynomial {E F[Xt,..., x,,].
In this notation (4.15) takes the form
(6.1)
1)P). =uA(e,)P).
for 0 < r  n and A of length < n. Hence by (3.4), we have
(6.2) u"(er)P,, = EA1( n Tq,x;)P"
I leI
summed over all r-element subsets I of {1, 2,..., n}, where
AI = ail ( n 1',,): )as
. I I
Ie
n
l<i<j<n
X.t8'_X.t8J
I J
x.-X.
, J
where 8i = 1 if i E I, and 8; = 0 otherwise. Hence for a partition IL of
length  n we have
q#-'ltn-i+8,_ q#-'Jt,,-j+8j
u (AI) = n . .
#-' . i<j q#-'It-I - q#J.jtn-J

332 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
(6.3) up.(A J) -+ 0 if and only if J.L + 8 = ( J.Ll + 81,.. ., J.Ln + 8n) is a partition.
Proof. Suppose that Up.(A 1) = o. Then there exist i < j such that J.Li == J.L.
and i - j = 8i - 8j. Since 18; - 8jl  1, it follows that j = i + 1, 8; = 0, and
8j = 1, so that J.Li = J.Li+ 1 and J.Li + 8i < J.Li+ 1 + 8i+ l' which means that
J.L + 8 is not a partition.
Conversely, if J.L + 8 is not a partition there exists i  n - 1 such that
J.Li = J.Li+ l' 8i = 0, and 8;+ 1 = 1, whence Up,(A]) = O. I
Now let J.L, v be partitions of length  n such that v:) J.L and () = lJ - J.L
is a vertical strip, and define
(6.4)
B = tn(v)-n(p.) n
vip.
l<i<j<n
1 - q p.j - p. j t j - i + 8; - 9 j
1 - qp.;-p'Jtj-i
so that Bv/p, = U,u(A I) where I = {i: 8; = I}. Then by applying up. to (6.2) we
obtain
(6.5)
u,\(e,)up.(P,\) = E Bv/p.uv(P,\)
v
where by virtue of (6.3) the sum is over partitions v:) J.L of length  n such
that v - J.L is a vertical r-strip.
Observe next that uo(P,\) is not identically zero, because when q = t we
have uo(P,\) = uo(s,\) = s,\(tn-1, tn - 2, .. ., 1) which is not zero (Chapter I, 3,
Example 1). Let us temporarily write
P,\ = P,\/uo(P,\).
We shall now prove simultaneously
(6.6) (Symmetry) For all partitions A, J.L of length  n we have
u,,( PI') = ul'( 1\).
(6.7) (Pieri formula) For all partitions u of length  n and all positive
integers r we have
Pue, = E Bv/uPv
v
summed over partitions v:) u of length  n such that v - u is a vertical
r-strip.
Proof. We shall proceed by induction. First of all, (6.6) is true for all A
when J.L = 0, because u,\(Po) = u,\(1) = 1 and uo(P,\) = 1. So let J.L be a
non-zero partition of length  n, and assume as inductive hypothesis that

6. PIERI FORMULAS
333
:

J6.6) is true for all A and for all 'IT' such that either '1TI<IJ.LI, or 11T1=IJLI
({and 'IT' < J.L.
t Let r  1 be such that J.Lr > J.Lr+ 1 and let u be the partition defined by
:1.di = J.Li - 1 for 1  i  r, and Ui = J.Li for i> r. We shall show that (6.7) is
?-true for this u.
r:. Consider the product PeTer' The leading monomial in PO' is xu, aI!d the
i'1eading monomial in er is Xl". X,. Hence the leading monomial in PO' e, is
'xP-, and therefore we have
.(1)
erpO' = E bvPv
v< lJ.
for suitable coefficients bVI Next, from (6.5) we have, for any partition 'IT',
-(2)
u".(er)uO'(P".) = EBv/O'uv(p1T)
v
summed over partitions JI::> u such that JI - u is a vertical r..strip.
Suppose that I 'IT' I = I J.LI and 'IT'  !-t. By the inductive hypothesis we have
UU( P1T ) = u1T( PO' )
because lul < I JLI, and
Uv( P1T ) = u1r( pv)
for all v  J.L (this is assured by the inductive hypothesis if either 'IT' < J.L or
v < !-t, and the only other possibility is 1T' = V = J.L). Hence (2) now takes the
form
(3)
u".( erPO' ) = E Bv/O' U1T (pv ).
v
On the other hand, by applying u". to both sides of (1) we obtain
(4)
u1T( erPu) = E bVu1T( pv).
v< lJ.
At this point we require
(6.8)
det{ ul'( p..) )1',..",. + O.
Assuming this for the moment, it follows from comparison of (3) and (4)
that bv = Bv/CT if JI - u is a vertical r-strip, and that bv = 0 otheIWise, so
that (1) becomes
(5)
erPu = E Bv/uPv
v
which is (6.7).

334 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
Now let A be any partition of length  n, and apply uA to both sides of
(5):
(6) uA(e,)uA( pu) = E BV/UuA( pv).
v
By the inductive hypothesis we have uA(PU) = uu() since 1001 < I J.LI, and
uA(PV) = uv() if JI =F J.L. Hence (6) takes the form
(7)
u,\(e,)uu (p,\) = B/u u,\( p) + E B,,/u u,,( P,\).
v< J.L
On the other hand we have from (6.5)
(8)
uA(e,)uU( PA) = BJ.L/uuJ.L() + E Bv/uuv( P,\).
v< lJ.
Since BJ.L/CT =F 0, comparison of (7) and (8) shows that u'\(PJ.L) = uJ.L(P,\), and
completes the induction step.
Thus it remains only to prove (6.8). In the determinant we may replace
Pv by Pv and then by mv' since the transition matrix M(P, m) is unitrian-
gular. So it is enough to show that
det(u1T(mv))1T,v<J.L =1= O.
Regarded as a polynomial in t, the dominant term in u'7J'(mv) is u1T(XV) =
q(v,1T)t(v,8). Hence it is enough to show that
det(q(v,1T»)v,1T<J.L =1= O.
which is a particular case of Example 1 below.
We can restate (6.7) in the form
(6.7')
P,.,.e, = E «/J/J.LP'\
,\
summed over partitions A::> J.L (of length  n) such that A - J.L is a vertical
r-strip, and
(6.9)
t/1/J.L = B,\/J.Luo(PJ.L) /uo(P,\).
When A =, J.L + (1'), consideation of the coefficient of x,\ on either side of
(6.7') shows, that. t/1/,.,. = , and hence
(6.10) uo(P,\) = BA/J.LUO(PJ.L)
when A = J.L + (1').
Since BA/J.L is given explicitly by (6.4), we can use (6.10) to compute
uo(P,\) and then (6.9) to compute the coefficients «/J;/J.L in the Pieri formula

6. PIERI FORMULAS
335
(6.7'), thus making it fully explicit. To state the result, we define
4+=4:(q,t)= n
l<i<j<n
( -1.)
XiXj ,q 00
( -1.)'
txixj ,q 00
Then we have, for any partition A of length  n,
(6.11)
U 0 (PA) = PA (1, t , . . . , t n - 1 ; q, t )
= tn(A)UA (4 +) /uo( +).
Proof. Since (6.11) is true when A = 0 (both sides being equal to 1) it is
enough by (6.10) to verify that
(1) UA( +)/Up.( +) = tn(p.)-n(A)BA/p.
when JL = (AI - 1, '\2 - 1, ...) is the partition obtained from A by remov-
ing the first column. Both sides of (1) are products of terms of the form
(1 - qGtb):t 1, and to verify (1) and similar identities we shall encounter
later it is helpful to switch to additive notation.
Thus, if P is any (finite or infinite) product of the form
P = n (1 - qatb)nab
a, b;. 0
with exponents nab E Z we define
(6.12 )
L(P) = E nabqatb E Z[[q,t]].
a, b __ 0
The mapping L is injective, so that if P and Q are two such products and
L(P) = L(Q), we may conclude that P = Q.
For example, if P = (qatb; q)oo we have
qatb
L(P) = qatb(l + q + q2 + ... ) = 1
-q
Hence
1- t
L(UA(4+)/U(4+»)= E (UA(XiXil)-U(XiXj-l)) _ '
l<i<j<n 1 q
and since
UA( XiX} 1) - Up. ( XiX} 1) = (qAt-A) - q #J.;-p.j)tj-i
= {6/(q -l)tj-i
if ir<j,
otheIWise,

336 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
where r = I( A), it follows that
r n
(2) L( uA(d +)/u,u( +») =0 E E q,ultj-i(t - 1).
i-I j-r+ 1
On the other hand, from (6.4) we have
(3) L(tn(,u)-n(A)BA/,u) = E q,u;-,u1t j-i (t8;-8) - 1)
1 < ; <j < n
where 8; = 1 if 1  i  r, and 8; = 0 if i > r. The right-hand sides of (2) and
(3) are visibly equal, and therefore (1) is established. I
Next, from (6.9) and (6.11) we have
 . . 8 8  . . 8 8 I-t
L(.p/,u) = '-' q,uI-,uJtJ-'(t 1- J -1) - i...J q,u;-,uJtJ-'(q 1- j -1) 1
-q
= E qll-.-II-Jtj-;(t81-8} -1) - (q8;-8) -1) 1- t ),
1-q
all sums being der pairs (i, j) such that 1  i <j  n. The pairs (i, j) such
that 8; = 8j give a zero contribution, and so do the pairs (i, j) such that
8; = 1 and 8j = O. When 8; = 0 and OJ = 1, the expression in braces above is
equal to
(t-1 -1) - (q-1 -1)(I-t)/(l-q) =t-1 +q-1t-l-q-1,
and hence we have
(1 - q ,ul- ,u J t j - i-I) (1 - q AI - A J t j - ; + 1 )
"'/p. = n (1 - q p.;- p.jtj - i)(1 - q A, - Ajtj - j)
where the product is taken over all pairs (i, j) such that i < j and Ai = J.Li'
A. = II.. + 1.
J r-J
(6.13)
When q = t, the formula (6.11) reduces to the formula
(1)
SA(I,t,...,tn-1)=tn(A) n
1 <i<j<n
1 - t A;-Aj-; +j
1 - tj-i
of Chapter I, 3, Example 1. Now (loc. cit.) this can also be written in the
form
(2)
1 - tn+c(s)
(1 t tn-1) - tn(A) n
SA " · · · , - h(s)
sEA I-t
where c(s) is the content and h(s) the hook-length of the square sEA.. In

6. PIERI FORMULAS
337
::the first version (1), the dependence on n appears only in the range
C.1  i <j  n of the product, whereas in the second version (2) it appears as
Lan exponent in each factor.
, There is a corresponding reformulation of (6.11) that reduces to (2)
'when q = t. For each square s = (i, j) in the diagram of a partition A, let
'c.( 6 .14 )
a(s) = aA(s) = Ai - j,
l(s) == LACS) =  - i,
a'(s)=j-l,
l'(s)=i-l,
so that L'es), l(s), a(s), and a'es) are respectively the numbers of squares
in the diagram of A to the north, south, east, and west of the square s. The
'numbers a(s) and a'es) may be called respectively the arm-length and the
arm-colength of s, and l(s), L'es) the leg-length and the leg-colength. The
hook length h(s) is thus a(s) + l(s) + 1, and the content c(s) is a'es) -l'es).
With this notation established, we have
(6.11 ')
1 - qa'(s)tn -I'(s)
PAO, t,..., tn-I; q, t) = tn(A) }J. 1- qa(s)tl(s)+1 ·
Proof. By use of the operator L (6.12) we reduce to showing that
1- t
E (qa'(s)tn-I'(s) - qa(s)tl(S)+l) = E (qAi-AJ -l)tj-i
seA 1 - q l<i<j<n
(6.15)
for a partition A of length  n. First we have
r
E qD'(S)tn -I'(s) = E (1 + q + ... +qAi--l )tn-i+ 1
sEA i-I
where r = l( A), so that
(1)
n
(1- q) E qa'(s)tn-I'(S) = E (1- qAi)tn+1-i.
seA i-1
Next consider the sum ESEA qa(s)tJ(s)+ 1. The contribution to this sum
from the squares in the ith row of A is
r
E (q Ai - A J + . .. + q A; - A) + 1 + 1 ) t j - i + 1
j-i
r
= (1- q)-l E (qAi-A) - qAi-Aj+l)tj-i+l.
j-i

338 VI SYMMETRIC FUNCTIONS WIT-H TWO PARAMETERS
Summing for i = 1,2,.. . , r we obtain
(2)
n
(1 - q) E qD(S}t'(S}+ 1 = E (t - qA1tn+ I-i) - (1 - t) E qAI-Ajtj-i
SEA i-I l<i<j<n
and (6.15) follows from (1) and (2) without difficulty. I
Now let u be an indeterminate, and define a homomorphism
8u,t: /:1.F --+ F
by
1- u'
8u,,(p,) = 1 _ t'
for each integer r  1. In particular, if u is replaced by tn we have
(6.16)
1 - tn'
E," ,,< P r) = 1 _ t r = P r (1, t , . . . , t n - 1 )
and hence for any ! E A F
8111,,(f) = f(1,t,. ..,tn-1),
previously denoted by uo(f). We now have
t I'(s} - qD'(S}U
Eu"PJ. = sQ 1 - qQ(S)t1(s)+ 1 ·
(6.17)
Proof. By (6.11'), this is true when u = tn for any n  leA), because
LSEA l'(s) = n(A). Both sides of (6.17) are polynomials in u with coeffi-
cients in F, and agree for infmitely many values of u, hence are identically
equal. I
Next, recall «4.11), (4.12)) that
Q A (q, t) = bA (q , t ) PA (q , t )
where
bA (q , t) = < PA, PA) -1 .
We shall now compute bA(q, t). For this purpose we require
(6.18) LetfE AF be homogeneous of degree r. Then
-,
Eu,tWt,q(!) = (-q) 8U,q-l(f).

w
6. PIERI FORMULAS
339
'Proof. Since both Bu,t and CJ)"q are algebra homomorphisms it is enough
'to rfy this when f= Pf' in which case it follows directly from the
[;,. defmltlons (6.16), (2.14).
ii' We now calculate
BU"PA(q,t) == BU"W"qQA,(t, q)
== (_q)-IAIBU,q-lQA,(t, q)
== ( - q ) -I AI bA, (t , q ) B u , q -1 PA, (t - 1 , q - 1 )
'.
.' by (5.1), (6.18), and (4.14) (iv). From this it follows that
.. (_q)'AIBu,tPA(q,t)
bA, (t , q ) = p (t - 1 - 1 )
Bu,q-t A' , q
in which, by (6.17),
q-I'(S) _ t-a'(s)u
&U.Q-IPA.(t-l, q-l) = n 1 _ rD(s)q-l(s)-l
sE,\'
tl'(S) - qO'(S)U
= (-q )IAI},Jl _ qD(S)+ It'(s)
. (since ESE A a(s) == Ese A a'(s) and Ese A l(s) == ESE A l'(s)). Hence we obtain
1 - qQ(s)+ 1tl(s)
b,,(t, q) == n ( ) I() 1
" sE'\ 1 - qO S t S +
or, on replacing (A, q, t) by (A', t, q ),
1 - qQ(S)tl(s)+ 1
(6.19) bA(q, t) = }J 1 _ qD(s)+1t'(s) ·
Remark. When q == t we obtain bACt, t) = 1, in agreement with (SA' SA) = 1.
When q = 0, the denominator in the product (6.19) is 1, and the only SEA
that contribute to the numerator are those for which a(s) = 0, i.e. the end
squares of each now. Thus we obtain
m;(,\)
bA(O, t) = n n (1 - tj)
i>1 j-I
in agreement with Chapter III, (2.12).
Let us now return to the Pieri formula (6.7'), in which the coefficients
were computed explicitly in (6.13). We shall now give alJ. alternative

340 VI SYMMETRIC FUNCTIONS W"ITH TWO PARAMETERS
expression for IJI;/p.' For this purpose we define, for each partition A and
each square s,
(6.20 )
1 - qQ).(S)tJ).(S)+ 1
b).(s) = b).(s; q,t) = 1- qa).(s)+ltIA(s)
1
if s E '\,
otherwise.
If s = (i,j) E A, let s' = (j, i) E A'. Then
(6.21) a).(s) = l).,(s'), l).(s) = a).,(s'),
so that
(6.22 )
b). (s; q, t) = b)., (s' ; t, q) -1 .
Furthermore, if A and JL are partitions such that A::> JL, let C)./JJ. (resp.
R)./p.) denote the union of the columns (resp. rows) that intersect A - J..L.
Then we have
(6.23)
, - n b).(s)
.pAl p. - b ( )'
seC)./,.,.-RA/p. p, S
Proof. From (6.13) we have
L ( IJI/ p,) = (t - q ) E q P,l- P. J - 1 (t j - i - t j - i-I )
summed over all pairs (i, j) such that i <j, Ai = J.Li and Aj = JLj + 1. To
each such pair we associate the square s = (i, Aj) E C)./p. - R)./p.. The
contribution to L( t/J;/p.) from the pairs (i, j) such that (i, Aj) = s is easily
seen to be equal to
(t - q )qQ).(s)(t').(S) - t1p.(s»)
and hence L( IJI/p,) is the sum of these expressions, for all s E
JL n (C)./p, - R)./p)' But this is precisely the image under L of the product
(6.23). I
By applying duality to (6.7') we shall obtain Qp,gr as a linear combina-
tion of the Q).. Altogether there are four 'Pieri formulas':
(6.24) Let J.L be a partition and r a positive integer. Then
(i) Pp,gr = L). 'P)./p,P).,
(ii) Qp.g, = L). IJIA/p,QA'
(iii) Qp, e, = L). 'Pl/p.Q).,
(iv) Pp'er = LA t/1{/p,P)..

:
'0"
i':'
ff#:' (i) and (ii) (resp. (Hi) and (iv» the sum is over partitions A such that A - p-
'a horizontal (resp. vertical) r-strip, and the coefficients are given by
2';
,,:
f1: b). (s)
i(j 'PAIl' = n b (s) ,
i, S E CAlI' I'
,,'-.
.,.... 
(::fil )
t-
6. PIERI FORMULAS
341
J'"
t/J)./p, = n
SERA/ - c)./
blL(s)
b). (s) ,
tf' , _ n bll(s)
i(iii) 'PAIl' - ,; ( )'
,f;:. S E RA/ ). s
IfiV) ;Ip. = n :AS .
:;', see )./ - R)./ P. S
,.
l'Proof. Equation (iv) is a restatement of (6.7') and (6.23). By applying
i:duality (5.1) to (iv) we obtain (ii), with coefficients
tt A/p.(q, t) = ;'Ip..(t, q)
':"
Which by virtue of (6.21) and (6.22) shows that t/J)./p. is given by (ii) above.
:Finally the coefficients in (i) and (iii) are
", . b b -t.I, , b -1 b .1,'
"'.. 'P)./p. = ). IL 'l'A/IL' 'P)./p. =). p. 'l'A/p.
Pi.:
[and (6.19) shows that they are given by the formulas above.
-:_ .
i.'1, "
i!'Remarks. 1. When q == 0, P). (resp. Q).) is the Hall-Littlewood symmetric
iJunction denoted by the same symbol in Chapter III, and (6.24) (i), (ii), and
',(iv) reduce respectively to the formulas (5.7), (5.7') and (3.2) of Chapter
1]11.
;::. We have
'PA/p. (q, t) == 'PA'/p.' (t , q),
t/J;/p.(q, t) = t/J).'/p.,(t, q).
;:Examples
1. Let Vt,..., vn be distinct vectors in a real vector space V equipped with a
positive definite scalar product (u, v). Then
E (Vi - VU(i)' Vi - VU(i» > 0
for any permutation u + 1 in Sn, and hence
E (Vi' Vi) > E (Vi' VU(i».
Deduce that det(q(V"Vj» is not identically zero as a function of the real variable q.

342 VI SYMMETRIC FUNCTIONS WIT.H TWO PARAMETERS
2. (a) Let A, J.L be partitions such that A::> J.L and A - J.L is a horizontal strip. Then
t-q .
L(cp I ) a=  (qA.;-Aj - qA/-lLj - qlLl-AJ+1 + qlLi-ILJ+I)ti-i
'AIL l-q i..J
summed over 1 <; i <;j <; l(A), and hence
t(q A.j- Ajti-i) t(q JLI- #-'J+ lti-i)
fPA/", = l<ir!/W !(qA'-"'ltJ-i)!(q",,-AI+ltJ-i)
where f(u) == (tu; q )oo/(qu; q )00.
(b) With A, J.L as in (a) we have
t-q . .
L(.pA/",) = 1 _ q E (q""- "'I - qA,- "'I - q""- A1+1 + q A,- AI+1 )lJ-'
summed over 1  i j  l( J.L), and hence
t(q JLi-ILJtj- i) f(q A.l- Aj+ lti-i)
.pA/", ... . n !(qA'-"'ltJ-I)!(q",/-AI+ltJ-i) ·
1..; J";J"; I( JL)
(c) Let A, J.L be partitions such that A::> J.L and A - J.L is a vertical strip. Then
L ( CP/IL) == (t - q ) E 'q JL,- #-'j (ti - i - ti - i-I )
summed over all pairs (i,j) such that 1 <; i <j < 00 and Ai> J.Li' Aj == lLi. Hence
(1- qA.;-AJti-i-l)(l_ qJLI-ILJti-i+l)
fP/", = n (1- qAi-A1tJ-i)(1_ q""-"'ltJ-i)
the product being taken over the same set of pairs (i, j).
3. Let A == (A1, A2' ...) be a partition, thought of as an infinite sequence, and let
GA(q,t)==(1-t)2 E (1_qA,-AJ)ti-i-1.
1..;i<j<oo
Then
E qa(')tl(.) = GA(q, t)
sEA (1-q)(1-t)
and hence GA(q, t) == GA,(t, q).
4. Let A be a partition of length  n and let
VA (q, t) == n (qA.I-A.Jti-i; q) k
1..; i <j..; n
V (q, t) == n (qA,-AJ+ltj-i-1; q)k
l..;i<j..;n

.
;,"
.,
7. THE SKEW FUNCTIONS PA/p., QA/p.
343
:1_-."
', k
[when t == q . Then
fl/
r
n ( ut 1- i; q ) ).1
Bu, ,(PA) a: tn(A)VA (q, t) n
i-I (t; q) Ai + k(n - i)
i1and
I: '
n ( ut 1- i ; q ) AI
Bu,,(QA) = tn(A)v{q, t) n
i-I (q; q) A/+k(n -i) .
f'5. Show that
! .
B,It,t(PA(q, t» = tn(A) n
l<i<j<n
(tj-i+1. q)'\o-A.
. , , J
(j-i )
t ; q Aj - Aj
,;and
:i"z
(qtj-i; q)Aj-Aj
Bql It - 1 , I ( Q A (q , t » = t n( A) n 1 '
1 < i <j <n (qti-i- ; q) Aj- Aj
: :
/if A is a partition of length  n.
j;Notes and references
,"
{Proposition (6.6), together with the proof of (6.6) and (6.7) given in the
text, is due to T. Koomwinder [K15].
"
;"7. The skew functions PA/' QA/
i.
"ror any three partitions '\, JL, JJ let
t;(7.1)
I:., = I;.,(q, t) = (QA' Pp,P.,) E F.
t"Equivalently, f;'., is the coefficient of PA in the product Pp,P.,:
)(7.1') Pp. P., = E I;., PA.
,\
'-In particular:
:;(7.2) (i) I;.,(t, t) is the Littlewood-Richardson coefficient c;., = (SA' Sp,S.,)
;:(Chapter I, 9).
i(ii) 1;,,(0, t) is the Hall polynomial I;.,(t) (Chapter III, 3).
(iii) I;., (q,1) is the coefficient of x'\ in mp,mv.
(iv) If" (1,1)= 1 if A = J.L + JJ, and i.IJ zero otherwi.IJe.
(v) Ip.., (q, t) = I;,,(q-l, t-1).
These assertions are all consequences of (4.14). As to (iv), when q = 1
,we have P;., = eA.', hence PfJPv = e",. ,ev, = e",.'u v' = e(p, +v)"

344 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
By applying Wq,t to each side of (7.1') and using duality (5.1) we obtain
Qp.,(t, q)Qv,(t, q).= Et;v(q, t)QA,(t, q)
,\
or equivalently
Qp.(q,t)Qv(q,t) = Et;:v,(t,q)QA(q,t).
,\
Comparison of this relation with (7.1') yields
(7.3) t;v(q, t) = f;,'v,(t, q )bA(q, t)/bp.{q, t)bv(q, t)
by (4.12).
Clearly, t;v vanishes unless I AI = I ILl + I v I, for reasons of degree. More-
over
(7.4) f;v = 0 unless A :J JL and A::> v.
Proof. For each partition J.L, let Ip. denote the subspace of AF spanned by
the PA such that A:J JL. The Pieri formula (6.24Xi) shows that grIp. CIJL for
all r> 1. Since by (2.12) the gr generate AF as an F-algebra, it follows
that II{- is an ideal in AF. Hence Pp.Pv E IJL n Iv, which proves
(7.4). I
Now let A, IL be partitions and define QAIJL E AF by
(7.5) QA/p. = Et;vQv
)J
so that
(7.6) (QJ..lp., Pv) = (QA' Pp.Pv)
{and hence, by linearity, (QAIJL' f) = (QA' Pp.t) for all f E AF. From (7.4)
it follows that
(7.7) (i) Q Alp. = 0 unless A ::> IL;
(ii) If A::) IL, QA/p. is homogeneous of degree I AI- I ILl.
Likewise we define PJ../p. by interchanging the P's and Q's in (7.6):
(7.6')
( PAl p. , Qv) = (PJ..' Q p. Qv ) ·
Since QJ.. = bAPA it follows that
(7.8)
QA/p. = bJ.. b; 1 PA/p..

7. THE SKEW FUNCTIONS P)./IJ.' Q)./I-'
345
From (7.5) we have
E Q,,/(X)PA(Y) = Ef;vQv(x)PA(y)
A A,v
= E Qv(X)P(y)Pv(Y)
v
= P(y )II(x, y)
i\by (4.13), where ll(x, Y) = n(x, Y; q, t). Consequently
E Q,,/(x)P,,(y)Q(z) = E P(y)Q(z)II(x,y)
A, 
= II(x, y)II(y, z)
- .
which by (4.13) again is equal to
E Q" (x, z) P" (y ) ·
A
:Jt follows from this that
{o{7.9)
Q,,(x, z) = E Q,\/(x)Q(z)

:..where the sum on the right is over partitions JL C A, by (7.7). Likewise we
; have
.'(7.9')
P,,(x, z) = E PA/(X)P(z).,

All of this parallels exactly the developments of Chapter I, 5 (for the
_iSchur functions) and Chapter III, 5 (for the Hall-Littlewood functions).
:Just as in those cases, the formulas (7.9) and (7.9') enable us to express the
::symmetric functions p,,/ and Q,,/ explicitly as sums of monomials. Since
(mv) and (gv) are dual bases of AF (2.11), it follows that
Q,,/ = E (Q,\/, gv)mv
)J
:(7.10)
= E (Q", Pgv)mv'
)J
Now we can use (6.24) (i) to express P",gv as a linear combination of the
PA. Let T be a (column-strict) tableau of shape A - J.L and weight v,

346 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
thought of as a sequence of partitions (A(O),..., A(T») such that J.L = A(O) C
A(1) C ... C A(T) =,\ and such tht each ,\(i) - A(i-l) is a horizontal strip.
Let
(7.11)
r
CPT ( q , t) = n CPA (I)/A (i -1) ( q , t ) ,
i-I
then iteration of (6.24) (i) shows that
(7.12)
PlJ.g" = L ( L fPT )PA
A T
summed over all partitions ,\::) IL such that IA - ILl = v, the inner sum
being over all tableaux T of shape A - J.L and weight v.
From (7.10) and (7.12) it follows that
(7.13)
QA/Il- = L 'PT(q, t)xT
T
summed over all tableaux T of shape A - IL, where (as in Chapter I, (5.13))
XT is the monomial defined by the tableau T.
Likewise, if we define
(7.11 ')
r
I/JT (q, t) = n f/lA(I)/A(i-l)(q, t)
i-I
for a tableau T as above, we have
(7.13')
PA/Il- = L f/lT(q, t)xT
T
summed over tableaux T of shape A - IL, as before.
Remarks. 1. In tbe case where there is only one variable x, (7.13) gives
(7.14)
QA/p,(X; q, t) = CP'\/Il-(q, t)XIA-p,1
if ,\ - JL is a horizontal strip, and QA/p,(X; q, t) = 0 othelWise. Likewise,
from (7.13'), we have
(7.14 ')
PA/Il-(X;q,t) = f/lA/Il-(q,t)XIA-1l-1
if ,\ - IL is a horizontal strip, and PA./p,(x; q, t) = 0 othelWise.
2. If x == (Xl'. .., xn), it follows from (7.13) that

7. THE SKEW FUNCTIONS PAIli-' Q).II£
347
(7.15) QA/II-(Xt"", x,.; q, t) = 0 unless 0  A - Ji;  n for each i  1.
For if X; - Ji; > n for some i, there is no tableau of shape ,\ - IL formed
- with the symbols 1,2,..., n.
., 3. Duality (5.1) extends to the skew functions PAIli- and QA/II-: namely
(7.16) Cl)q,IPA/II-(q,t) = QA'III-,(t,q),
(7.16')
CI) q, I Q AIII- ( q , t) == PA'I 11-' ( t , q ) ·
For we have
Cl)q.I QA/p.(q ,t) = E f;., (q, t )P.,,(t, q)
v
by (7.5) and (5.1), and by (7.3) this is equal to
 A' b,.,.,(t, q)
i.J fp,.".(t, q) b ( ) Q".(t, q)
v A' t, q
bll-,(t, q)
= b ( ) Q>"/p,.(t,q) =P>"/p,.(t,q).
A' t, q
This proves (7.16'), and (7.16) follows by replacing (q, t, A, IL) by
(t, q, ,\', J.L'). I
4. The explicit calculation of the structure constants f;.,(q, t) for arbitrary
A, J.L, v remains an open problem. (The cases where J.L or v is a single row
or a single column are covered by (6.24).) For example, it is not known to
the author whether the vanishing of the Littlewood-Richardson coeffi-
cient c;., implies the vanishing of f;.,(q, t).
Examples
1. In this and the following examples we shall sketch an alternative approach to
the results of fi6. We shall not, therefore, make use of any result in 6 (with the
exception of the elementary fact (6.18) in Example 2).
Let IJ. be a partition of length n, and let r be a positive integer. Then QII-g, is a
linear combination of the QA' say
(1)
QII-(X; q, t)g,(x; q, t) == E f/!A/IJ-(q, t)Q>.(x; q, t)
A
with certain (as yet undetermined) coefficients t/JA/IJ- e F. By duality (5.1) we obtain
(2) PII-,(x;t,q)e,(x) == E f/!>./p,(q,t)PA,(x;t,q).
A
(a) Suppose that 'PA/I!:I= O. By considering the leading monomials in (1) and (2),
show that ,\  1.£ + (r) and that ,\'  1.£' + (1'), and hence that 1(,\) = n or n + 1.

348 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
(b) Suppose first that I(A) = n, and let x =: (Xl'.." Xn). Use 4, Example 1 to show
that
(3)
.p)./J.I- =: .p).e/J.l-e b).e b,.,,/b). bJ.l-e.
where A* = (AI - 1, A2 - 1, ...) and J.L* == (J.L1 - 1, J.L2 - 1, ...).
(c) Suppose next that l( A) = n + 1, and consider the coefficient of xl + I on either
side of (2). Use 4, Example 2 to show that
(4)
.p)./,." = .p). e /J.I-* ·
(d) Deduce from (b) and (c) that tP)./J.I- = 0 unless A:) J.L and A - J.L is a horizontal
strip.
2. In this example we shall indicate an alternative proof of the specialization
formula (6.17). Let
<1>). (u ; q, t) = B u, t P). (x ; q, t)
which is a polynomial in u with coefficients in F, of degree  IAI.
The developments of 7 use from 6 only the result of Example l(d), as the
reader may easily verify. In particular, we have from.(7.9') and (7.14')
(1) P).(x;q,t) = E tPA/J.I-(q,t)XI).-IL1PJ.l-(X*;q,t)
p,
where x. = (X2' x3' ...).
(a) By setting X = (1, t,..., tn-I) in (1), where n is any positive integer, deduce that
the relation
(2)
<I>).(u; q, t) = E tPA/J.I-(q, t)tl""'<I>p,(t-1u; q, t)
p.
is true for u = t, t 2, ... and hence identically.
(b) Use duality (5.1) together with (6.18) to show that
(3) <I>).(u;q,t) = (_q)-I).lbA,(t,q)<I>A,(U;t-1,q-l).
(c) The polynomial 4>.\(u; q, t) vanishes for u = 1, t,..., tl(A)-1 by (4.10). Deduce
from (3) that <I>).(u; q, t) is divisible in F[u] by nt t(qj-1u - 1).
(d) In each partition J.L on the right-hand side of (2) we have Al  J.LI  A2  ...,
since A - J.L is a horizontal strip (Example l(d». Hence by (c) above each summand
on the right of (2) is divisible by the product nt.: l(qj-1U - t), and therefore
<l»A(U; q, t) is also divisible by this product. By repeating this argument, conclude
that <l>A is divisible in F[ u] by
l( A) AI
n n (qj-1 u - t;-1) = n (qa'(s)u - tl'(s».
i-I j-I sEA
(e) Since the degree in u of <1>). is at most I AI, it follows from (d) that
(4) <I>).(u; q, t) = v).(q, t) n (qa'(s)u - tl'(s»
sE A

7. THE SKEW FUNCTIONS PA/IJ.' QA/IJ.
349
for some v).(q, t) e F. If leA) == n it follows from 4, Example 1 that
(5)
<I> (tn. q t) == tn(n -1)j2m (tn. q t)
). " "¥,\* " ,
where A* == (At - 1,..., An - 1). Deduce from (4) and (5) that
n
vA(q,t) ==v,\.(q,t) Il (q).;-1tn-i+1 _1)-1
i-I
and hence that
(6)
VA (q, t) = Il (qa(s)t1(s)+ 1 - 1) -1 .
sE ).
3. From (3), (4), and (6) of Example 2 we can now obtain the formula (6.19) for
b).(q, t). Then, knowing b)., we can derive the formula (6.24XiO for "',\/ by
induction from the relations (3) and (4) of Example 1.
4. For each partition A, let
bil == Il b).(s),
sEA
I(s) even
ba == Il b,\(s)
sEA
a(s) odd
(the superscripts el and oa stand for 'even legs' and 'odd arms' respectively). Then
we have
(tx i X)"; q )00
(i) E b:Jpv == Il
v i<j (XiXj;q)oo'
(ii) E b;lp). = Il (aj; q )00 Il (ajxj; q )00 ,
A i (xi;q)oo i<j (XiXj;q)oo
(1.1.1.)  oa = Il (qtx'f; q2)00 Il (txiXj; q)oo
i.J bp. p (2 2) (. ) ,
JL i Xi;q 00 i<j XiXj,q 00
" Il (qtxl; q2)00 Il (txiX)"; q)rrJ
(iv) i.J bfa p,\ =
A i (1-xi)(q2x;; q2)00 i<j (XiXj; q)oo ·
In these sums, A runs through all partitions, J..L through all even partitions (i.e. with
all parts even), and v through all partitions such that v' is even (so that all the
columns of v are of even length).
We shall prove (0 first, and then deduce the other three identities from (0. It is
enough to prove (0 for a finite set of variables Xl'.'" xn. By induction on n, it is
enough to show that if <I>(x) denotes the left-hand side of (0, then
Iln (txiY; q)oo
<I>(Xtt...,xn,y)=4>(Xt,...,xn) ( .).
i-I Xiy,q 00
(1)
Now the left-hand side of (1) is by (7.9') and (7.14') equal to
(2)
E b:1.p).jP (Xl'. . . , Xn) y/A - I
A,P.

350 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
summed over pairs of partitions '\, p, such that '\' is even, '\::> p, and ,\ - J.t is a
horizontal strip. Again, the right-hand sid of (1) is by (2.8) and (6.24) equal to
E b:1 P., ( x I , · · · , X n ) g r ( X I , . . . , X n ) Y r
lI,r
(3)
-  beL- P ( ) IJL-"I
- i-I .,¥JJL/" JL Xh...' Xn Y
11-, II
summed over pairs of partitions p" v such that v' is even, J.L::> v and J.t - v is a
horizontal strip.
For a given partition p, of length '" n both the partitions ,\ and v are uniquely
determined: we have ,\ == v; == p!; if p!; is even, and ,\ == p!; + 1, v: == J.t - 1 if J.t is
odd. Hence 1,\ - p,1 == I p, - vi, and a column contains a square belonging to A - J.L
if and only if it contains a square belonging to p, - v: the condition in either case is
that it should be a column of odd length in p,.
To prove that (2) and (3) are equal therefore reduces to showing that
( 4) b:\i1)./JL == b:\cJL/'"
For this purpose it is convenient to introduce the following notation: if ,\ is any
partition and 5 is any set of squares, then b). (5) denotes the product of the b). (s)
for s E S. With this notation, if C (resp. C) denotes the union of the columns of
odd (resp. even) length in p" we have 'Pp./., == bJL(C)/b.,(C) and .p)./JL == bJL(C)/b).(C)
by (6.24). Let Rl (resp. R2) denote the union of the odd (resp. even) numbered
rows, and let C; JIll: C n Ri, Ci == C n Ri. Then bil == b).(R2) and b:1 == b.,(R2), since
all the columns of ,\ and v have even length. We may therefore rewrite (4) as
b).(R2)bJL(C)b.,(C) == b).(C)bJL(C)b.,(R2)
or, since C==C1 UC2,C=C1 UC2,R2=C2UC2 (the unions in each case being
disjoint) as
(5) b). (C2)b". (EI )bJL{ C2)b.,{ CI) == b). (CI )bJL (Ct )b".( C2 )b.,{ C2).
For each square s E C n A we have l).{s) == l".(s) == I.,(s); for each square s E
R1 n,\ we have a).{s) = a".(s), and for each s e R2 n p, we have a".(s) = a.,(s).
Hence we have
(6)
b).(Ct) = bjJ.(Ct),
bJL(C2) = b.,(C2).
Furthermore, for each square s E C 2 n,\ let t E C 1 be the squae immediately
above s. Then we have Q).{s) == aJL{t) and I).(s) == IJL(t); also aJL{s) = a.,{t) and
Ip.{s) == I.,(t), and therefore
(7)
b>.(C2) == b".(Ct),
bJL(C2) == b.,(Cl).
These relations (6) and (7) together imply (5), and complete the proof of (0.
Next consider (ii). By (2.8) and (0, the right-hand side of (ii) is equal to
(8)
E b:1p.,gr == E b:\c"./.,P".
l1,r p., II

l'
351
7. THE SKEW FUNCTIONS PAlp" QAlp,
summed over partitions IL::> II such that II' is even and IL - II is a horizontal strip.
From the proof of (0 above we have
b:\r;p,/v == bv(R2)bp,( C) /bJl( C)
== bv(C2)bp,(Ct)bp,(C2)/bv(Cl)
== bp. (C1 )bp, (C2) == b;1
by (6) and (7). Hence (8) is equal to Lp. b;lpp, summed over all partitions IL, which
proves (i0.
Finally, (iii) and (iv) are obtained from (0 and (iO respectively by duality, i.e. by
operating on both sides with CJ)q, t and applying (5.1).
When q == 0, so that PA becomes the Hall-Littlewood symmetric function, the
identities (O-(iv) above reduce to Examples 3, 4, 2, 1 of Chapter III,  5,
respectively.
5. For partitions A, j.L, 'IT such that A::> IL and I AI == JILl + ,17' I, let
A A/p" 'IT == E CPT
T
summed over all (column-strict) tableaux T of shape A - J.L and weight 17', so that
by (7.13) we have
(1) QA/p, == E A,\/p" fr m'IT
1r
and hence
(2)
AA/p" fr == <Q,\/P,' gfr)
from which it follows that
(3)
g1T == E AJlfrPJI.
II
From (2) and (3) we have
A A/p" ft' == E A JI'IT < Q,\/P,' Pv)
II
== E AV'IT(Q,\, Pp,PJI)
II
== E AV1Tt;v.
v
The matix (AV1T) is strictly lower triangular and invertible; if its inverse is (BJlfr) we
have
t;JI == E A,\/p" fr BfrJl
.".
and hence the structure constants t;'JI are in principle computable in terms of the
CPT' which are given explicitly by (7.11) and (6.24).

352 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
6. Let A, J.L be partitions. Then
(a) E pp/.\(x; q, t)Qp/p,(Y; q, t) = n (x, y;'q, t) E pp,/O'{x; q, t)QA/U(Y; q, t).
p u
The proof is the same as that of Chapter I, 5, Example 26.
(b) By applying ClJq,t to the symmetric functions of the x variables in (a) we obtain
E Qp'/A,(X;t,q)Qp/p,(y;q,t) = I1o(x,y) E Qp,,/O',(x;t,q)QA/O'(y;q,t),
p u
where I1o{x, y) = ni,j{l + xiYj).
(c) Likewise
E Pp/A (x; q, t) pp'/p,'(Y; t, q) = I1o(x, y) E pp,/O' (x; q, t) PA'/O' ,(y; t, q).
p u
Notes and references
Examples 1-3. The results of these Examples, in the context of Jack's
symmetric functions, are due to R. Stanley [S25].
Example 4(i). This identity, again in the context of Jack's symmetric
functions, is due to K Kadell.
8. Integral forms
For each partition A we define
(8.1)
C A ( q , t) = n (1 - q a( s) t I( s) + 1 ) ,
seA
(8.1')
c(q, t) = n (1- qa(S)+ltl(s»,
seA
so that
(8.2)
and by (6.19)
c ( q, t) = C). I (t , q )
bA (q , t) = C A ( q , t ) / c ( q , t) .
Now let
(8.3) JA = J). (x ; q, t) = c). (q , t ) PA (x; q, t) = c (q , t ) Q). (x; q, t) .
The symmetric functions JA are in some sense 'integral forms' of the P). (or
QA)' It seems likely that when they are expressed as linear combinations of
the monomial symmetric functions, the coefficients are polynomials, not
just rational functions, in q and t. We shall make a more precise conjec-
ture later in this section.

8. INTEGRAL FORMS
353
(8.4) Remarks.
(i) When q = t we have
C A (t , t) = C (t , t) = n (1 - t h( s» ,
sEA
the hook polynomial of A, denoted by H,\(t) in Chapter I, 3, Example 2.
Hence J,\ = H,\(t)SA when q = t.
(ii) When q = 0 we have C(O, t) = 1 and CA(O, t) == bA(t), so that fA(x; 0, t) is
the Hall-Littlewood function QA(X; t).
(Hi) When q = 1,
cA(l, t) = n (1- tl(S)+1) = (t; t)A'
SEA
in the notation of 2, Example 1, and hence by (4.14) (vi)
J,\ (x ; 1, t) = (t; t ) A' e A' ( X ) .
(iv) When t == 0, J,\(x; q, 0) = PA(x; q, 0).
(v) When t = 1 we have cA(q, 1) = 0 if A '* 0, so that JA = O.
(vi) Let q = t a and let t -. 1. The symmetric functions
fA ( X ; t a , t)
J1a)(x) = lim
t-+ 1 (1 - t )IAI
are the integral forms of Jack's symmetric functions (see 10 below).
We have
C,\(q-l ,t-1) = n (1- q-a(s)t-l(s)-I)
SEA
== (_I)IAlq-n(A')t-n(A)-/Alc,\(q,t),
since Ese A a(s) = n(A') and ESE A l(s) == n(A), and hence
(8.5) J,\(x; q-1,t-1) == (_l)IA'q-n(A')t-n(A)-IAIJ;\(x; q,t)
by virtue of (4.14) (iv).
Next, duality (5.1) now takes the form
(8.6) W q, t fA (q , t) = fA' (t , q )
and we have
(8.7)
( J,\, JA) q, t = C A (q , t ) C (q , t) = (JA" fA' ) t , q ·

354 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
The specialization formula (6.17) gives
Bu,tlA (x; q, t) = n (t"(s) - qtJ'(S)u)
sEA
(8.8)
== n (ti-1_qj-1U).
(i,j)e A
As in Chapter III, (4.5) let SA (X; t) denote the Schur functions associated
with the product n(1 - tx;)/(1 - Xi)' so that (Chapter I, 7)
(8.9) SJ,.(x; t) = E Z;lXpApp(X; t)
p
where
(8.10)
l( p)
pp(x; t) == pp(x) n (1- tp.).
i-I
The SJ,.(X; t) form an F-basis of Ap, and hence we may express the
Jp,(x; q, t) in terms of them, say
(8.11) Ip.(x; q, t) == E KAp,(q, t)SA(X; t).
A
When q = 0 we have Jp.(x; q, t) == Qp,(x; t) (8.4 (ii», hence
(8.12) KJ,.p.(O,t) ==KAp,(t)
where the KJ,.p,(t) are the polynomials defined in Chapter III, 6. In
particular,
(8.13)
KJ,.p,(O,O) = 8Ap,'
KAP,(O, 1) == KAp,'
where the KJ,.p, are the Kostka numbers defined in Chapter I, 6, so that
KJ,.p, is the number of (column-strict) tableaux of shape A and weight p.,.
From (8.9) and (8.10) it follows that SA(X; t-1) = (-t)-IAISA,(x; t) and
hence by (8.5) that
(8.14)
KJ,.p,(q, t) = qn(p,')tn(p,)KJ,.'p,(q-l, t-1).
Again, it follows from (8.9) that
Wq,tSA(X; t) = E Z;lXpASppp(X; q)
p
== SA'(X; q)
and hence by (8.6) that
(8.15)
KAJA.(q,t) ==K).'JA.,(t,q).

8. INTEGRAL FORMS
355
For each integer n  0, let Kn{q, t) denote the matrix (K>.p,{q, t» where
A and JL run through the partitions of n. Unlike the matrices Kn(t) =
Kn(O, t) of Chapter III, the matrices KlI(q, t) (for n > 1) are not upper
triangular; indeed, it follows from (8.15) that K).p.(q,O) = K>"p.,(q), so that
Kn{q,O) is lower triangular.
Another special case in
(8.16)
K>.p.{l, 1) = X(n) = n!/h{A)
(where A, I-L are partitions of n).
Proof. From (8.9) and (8.11) we have
Ip.(x; q, t) = E K>.p.(q, t) E Z;lXpApp(X; t).
A p
On setting q = t we obtain from (8.4Xi)
Hp,(t)sp.(x) = E Z;lXpAK>.p.(t,t)pp(x;t).
A,p
But also (Chapter I, i7)
Sp.(x) = E Z;lX,fpp{X)
p
and therefore
E KJ,.p.{t, t) Xp>' = x,fHp.(t) /0 (1 - t Pi)
A
so that we obtain
(8.17) KAp.{t,t)= EZ;lXpAX:Hp.(t)/O(l-tPi)
p
by orthogonality of the characters of the symmetric group Sn' Now let
t -+ 1, and we obtain
KAp.(l, 1) = X(n)
since the only term that survives on the right-hand side of (8.17) is that
corresponding to the partition p = (In). I
The matrices KlI(q, t) have been computed for n  8. The results
suggest the conjecture
(8.18?) K>.p.(q, t) is a polynomial in q and t with positive integral coefficients.

356 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
Some further partial evidence in favour of this conjecture is contained in
the Examples at the end of this section. In particular, we know from the
theorem of Lascoux and Schiitzenberger (Chapter III, (6.5» that KAp.(O, t)
is a polynomial in t with positive integral coefficients. By (8.15) the same is
true of K)..p.(q,O).
Since S)..(x; t) is a linear combination of the monomial symmetric func-
tions m).(x) with coefficients in Z[t] (see the table of transition matrices in
Chapter III, 6), the conjecture (8.18?) would imply that the JA are linear
combinations of the m,.,. with coefficients in Z[q, t].
On the assumption that (8.18?) is true, the number of monomials qatb
in K)..p.(q, t) would by (8.16) be equal to the number of standard tableaux
of shape A. One may therefore ask whether there is a combinatorial rule
that attaches to each pair (T, J.L), where T is a standard tableau containing
n symbols and p., is a partition of n, a monomial qo(T,JL)tb{T,p,\ so that
KAP,(q, t) is the sum of these monomials as T runs through the standard
tableaux of shape A.
Finally, we shall introduce generalizations of the polynomials XpA(t) of
Chapter III, 7. For each pair of partitions A, p we define XpA(q, t) E F by
(8.19)
J).. (x ; q, t) = L Z; 1 Xp).. (q , t ) p p (x; t)
p
(so that Xp)..(q, t) = 0 unless I AI = I pD. By (8.4) (ii) and Chapter III, (7.5) we
have
Xp)..(O, t) = Xp)..(t).
From (8.9) and (8.11) it follows that
J)..(x; q, t) = L Kp,A(q, t) L Z;lX:pp(X; t)
p, p
and hence that
(8.20 )
X;(q,t) = L X,;KP.A(q,t).
I.L
By orthogonality of the characters of the symmetric group, this relation is
equivalent to
(8.20')
K""A (q, t) = L Z;lX: XpA(q, t).
p

8. INTEGRAL FORMS
357
The conjecture (8.18?), together with (8.20), would imply that XpA(q, t) E
Z[q, t], since the X: are integers.
From (8.14) and (8.15) we deduce that
(8.21) XpA(q-l ,t-1) = Bpq-n(A')t-n(A)XpA(q,t),
(8.22) X:(t, q) = BpXpA'(q, t).
The XpA(q, t) satisfy orthogonality relations that generalize those of
Chapter III, 7. Namely, from (2.6), (4.13), and the definition (8.3) of JA we
have
E z p ( q , t ) - 1 P p ( x ) P p (y) = E C A ( q , t) - 1 C ( q , t) - 1 JA ( x; q , t ) J). ( y; q , t ) .
p A
If we now substitute (8.19) in the right-hand side of this relation, and
compare the coefficients of pp(x)Pq(y) on either side, we shall obtain
(8.23) E cA(q,t)-lCA,(q,t)-l XpA(q,t)X;(q,t) = 5pu(p(q,t)
IAI-=n
where
(8.24 )
l(p)
{p(q,t) =zpO (1- qPi)-l(l_ tp;)-l.
i-I
An equivalent statement is
(8.25) E {p(q,t)-t XpA(q,t)X:(q,t) = 5AILCA(q,t)C(q,t).
Ipl-n
Finally, it follows from (8.16) and (8.20) that
XpA(l, 1) = E XpPx()n)
J.'
and therefore by orthogonality of the X: (Chapter I, 7)
(8.26 )
XpA(1,1) = {!
if p = (In),
othelWise.

358 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
Remark (8.27). Let V = Z[Sn] be the regular representation of Sn, and for
each partition A of n let M).. be an irreducible Sn-module with character
X).., so that in particular M(lll) is the one-dimensional module affording the
sign character 8. Then the conjecture (8.18?), together with (8.26), would
imply that for each partition J.L of n there is a bigrading of the regular
representation
V= EB Vh,k
h k J.£
,
where 0  h  n( J.L) and 0  k  n( J.L'), such that for each A the multiplic..
ity of M).. in h,k is equal to the coefficient of thqk in KAp.(q, t). By (8.14)
and (8.15) these bigradings would satisfy
Vh,k M  Vh',k  Vk,h
J.£ 'CJI (111) - P. - p.' ,
where h' = n( J.L) - hand k' = n( J.L') - k.
Recently Garsia and Haiman [G4] have put forward a conjecture in this
direction. Let A = Z[xt,.. ., xn' YI'...' Yn]' where the x's and y's are 2n
independent indeterminates. The polynomial ring A is bigraded: A =
EBh,k;.O Ah,k, where Ah,k consists of the polynomials tEA that are
homogeneous of degree h (resp. k) in the x's (resp. y's). The symmetric
group Sn acts diagonally on A:
wi ( XI' · · · , X n , Y 1 , · · · , Y n) = I (X w (1)' · · · , x w( n ), Y w (1)' · · · , y w (n »
for WE Sn and lEA, and this action respects the bigrading.
Now let J.L be a partition of n and let (PI' qI)'''.' (Pn, qn) denote the set
of pairs {(i - 1, j - 1): (i, j) E J.L} arranged in lexicographical order. We
have E Pi = n( J.L) and Eqi = n( J.L'). Let
J1 (x Y) = det(xPiy!1;) .. EAn(p.),n(p.').
J.£ ' } } I<I,J<n
Clearly, wJ.£ = 8(w)4J.£ for WE Sn. Let Hp. cA be the linear span of all
the partial derivatives of J.£(x, y) of all orders with respect to the x's and
y's. Then HJ.£ is stable under the action of Sn' and moreover
H = EB Hh,k
J.£ k J.£
h,
where Hh,k = H nAh,k is Sn..stable. Let (f'Jh,k denote the character of the
J.£ J.£ J.£
Sn-module H:,k, and for each partition A of n let
C)..J.£(q, t) = E (X).., cp:,k)thqk
h,k

8. INTEGRAL FORMS
359
where as in Chapter I, 7 the X,\ are the irreducible characters of S n.
Then Garsia and Haiman conjecture (iDe. eit.) that
C,\p.(q, t) = K,\p.(q, t-1 )tn(p.).
In particular they have verified that this is so for all pairs (A, J.L) of
partitions of n  6, and in other special cases.
2 12 3
1 2 tHE 21
1  12 t 1 13
4 31 22
4
31
22
212
14
The matrices K(q, t)', n  6
3
13
21
1 q + q2 q3
t 1 + ql q
t3 t + t2 1
212
14
1 q + q2 + q3 q2 + q4 q3 + q4 + q5 q6
t 1 + qt + q2t q + q2t q + q2 + q3t q3
t2 t + qt + qt 2 1 + q2t2 q + qt + q2 t q2
t3 t + t2 + qt3 t + qt2 1 + qt + qt2 q
t6 t3 + t4 + t5 (2 + t4 t + t2 + t3 1
5
41
32
5
41
32
312
221
213
IS
1 q + q2 + q3 + q4 q2 + q3 + q4 + q5 + q6
t 1 + qt + q 2 t + q 3 t q + q2 + q2t + q3t + q4t
t2 t + qt + qt2 + q2 t2 1 + qt + q 2 t + q 2 t 2 + q 3 t 2
t3 t + t 2 + qt 3 + q 2 t 3 t + qt + qt 2 + q 2 t 2 + q 2 t 3
t4 t2 + t3 + qt3 + qt4 ( + (2 + qt2 + qt3 + q2t4
t6 t3 + t4 + tS + qt6 t2 + 13 + t4 + qt4 + qt5
t10 t6 + t 1 + t8 + t9 t4 + tS + t6 + t7 + t8
312
15
221
213
5
41
32
312
221
213
15
q3 + q4 + 2qS + q6 + q7 q4 + qS + q6 + q 7 + q8 q6 + q7 + q8 + q9 ql0
q + q2 + q3 + q3t + q4t + qSt q2 + q3 + q4 + q4t + q5t q3 + q4 + q5 + q6 t q6
q + qt + 2q 2 t + q 3t + q 3 t2 q + q2 + q2t + q3( + q4(2 q2 + q3 + q3t + q4t q4
1 + qt + q 2 t + qt 2 + q 2 t 2 + q 3 t 3 q + qt + q2 t + q 2 t 2 + q 3 t 2 q + q2 + q3t + q3t2 q3
t + qt + 2 qt 2 + qt 3 + q2 t 3 1 + qt + qt2 + q2t2 + q2(3 q + qt + q2t + q2t2 q2
t + t2 + (3 + qt3 + qt4 + q(S ( + t2 + qt 2 + qt 3 + qt4 1 + qt + qt2 + qt3 q
t3 + t4 + 2tS + t6 + (7 t2 + (3 + t4 + t5 + (6 ( + (2 + t3 + t4 1

360 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
6
51
42
6
SI
42
412
32
321
313
23
2212
214
16
1 q + q2 + q3 + q4 + qS . q2 + q3 + 2q4 + q5 + 2q6 + q7 + q8 -
1 + qt + q 2 t + q 3 t + q 4 t q + q2 + q3 + q2t + q3t + 2q4t + q5t + q6t -
t
t2 t + qt + qt2 + q2t2 + q3t2 1 + qt + 2q2t + q3t + q2t2 + q3t2 + 2q4t2 -
t3 t + t2 + qt3 + q2t3 + q3t3 t + qt + q2t + qt2 + q2t2 + q3t2 + q2t3 + q3t3 + q4t3
t3 t2 + qt2 + q2t2 + qt3 + q2t3 t + qt + q2t + qt2 + q2t2 + q3t2 + q2t3 + q3t3 + q4t3
t4 t2 + t 3 + qt 3 + qt 4 + q2 t 4 t + t2 + 2qt2 + qt3 + 2q2t3 + q2t4 + q3t4
t6 t3 + t4 + tS + qt6 + q2 t6 t2 + t3 + qt3 + t4 + qt4 + q2t4 + qt5 + q2t5 + q2t6
t6 t4 + qt4 + tS + qt5 + qt6 t2 + t3 + qt3 + t4 + qt4 + q2t4 + qtS + q2t5 + q2t6
t7 t4 + t5 + t6 + qt6 + qt7 2t3 + t4 + qt4 + t5 + 2qt5 + qt6 + q2t7
t10 t6 + t7 + t8 + t9 + qt10 t 4 + t 5 + 2t 6 + t 7 + qt 7 + t 8 + qt8 + qt 9
tIS t10 + tll + t12 + t13 + t14 t7 + t8 + 2t9 + t10 + 2tll + t12 + t13
412
32
6
51
42
412
32
321
313
23
2212
214
16
q3 + q4 + 2q5 + 2q6 + 2q 7 + q8 + q9 q3 + q5 + q6 + q7 + q9
q + q2 + q3 + q4 + q3t + q4t + 2q5t + q6t + q7t q2 + q4 + q3t + qSt + q6t
q + qt + 2q2t + 2q3t + q4t + q3t2 + q4t2 + q5t2 q + q2t + q3t + q3t2 + q5t2
1 + qt + q2t + q3t + qt2 + q2t2 + q3t2 + q3t3 + q4t3 + q5t3 qt + q2t + q2t2 + q4t2 + q3t3
qt + q2t + q3t + qt2 + 2q2t2 + 2q3t2 + q4t2 + q3t3 1 + q2t2 + q3t2 + q4t2 + q3t3
t + qt + 2qt2 + q2t2 + qt3 + 2q2t3 + q3t3 + q3t4 t + qt2 + q2t2 + q2t3 + q3t4
t + t2 + t3 + qt3 + q2t3 + qt4 + q2t4 + qt5 + q2t5 + q3t6 qt2 + t3 + qt4 + q2t4 + q2t5
qt2 + t3 + 2qt3 + q2t3 + 2qt4 + q2t4 + qt5 + q2tS qt2 + t3 + qt3 + qt4 + q3t6
t2 + t3 + qt3 + t4 + 2qt4 + 2qt5 + qt6 + q2t6 t2 + t4 + qt4 + qt5 + q2t6
t 3 + t4 + 2 t 5 + t 6 + qt 6 + t 7 + qt 7 + qt 8 + qt 9 t 4 + t 5 + qt 6 + t 7 + qt 8
t6 + t7 + 2t8 + 2t9 + 2t1O + tll + t12 t6 + t8 + t9 + t10 + t12
321
6
51
42
412
32
321
313
23
2212
214
16
q4 + 2qS + 2q6 + 3q7 + 3q8 + 2q9 + 2ql0 + ql1
q2 + 2q3 + 2q4 + 2q5 + q6 + q4t + 2q5t + 2q6t + 2q7t + q8t
q + 2q2 + q3 + q2t + 3q3t + 3q4t + q5t + q4t2 + 2q5t2 + q6t2
q + q2 + qt + 2q2t + 2q3t + q4t + q2t2 + 2q3t2 + 2q4t2 + q5t2 + q4t3 + q5t3
q + q2 + qt + 2q2t + 2q3t + q4t + q2t2 + 2q3t2 + 2q4t2 + q5t2 + q4t3 + q5t3
1 + 3qt + q2t + qt2 + 4q2t2 + q3t2 + q2t3 + 3q3t3 + q4t4
t + qt + t2 + 2qt2 + q2t2 + 2qt3 + 2q2t3 + qt4 + 2q2t4 + q3t4 + q2t5 + q3t5
t + qt + t2 + 2qt2 + q2t2 + 2qt3 + 2q2t3 + qt4 + 2q2t4 + q3t4 + q2t5 + q3t5
t + 2t2 + qt2 + t3 + 3qt3 + 3qt4 + q2t4 + qt5 + 2q2t5 + q2t6
t2 + 2t3 + 2t4 + qt4 + 2t5 + 2qt5 + t6 + 2qt6 + 2qt7 + qt8
t4 + 2t5 + 2t6 + 3t7 + 3t8 + 2t9 + 2t10 + t11

6
51
, 42
412
32
_321
,313
i':23
2212
214
16
8. INTEGRAL FORMS
361
313
23
6
SI
42
412
32
321
313
23
2212
214
16
q6 + q7 + 2q8 + 2q9 + 2ql0 + qll + q12 q6 + q8 + q9 + ql0 + q12
q3 + q4 + 2qs + q6 + q7 + q6t + q't + q8t + q9t q4 + qS + q7 + q6t + q8t
q2 + q3 + q4 + q3t + 2q4t + 2q5t + q6t + q6t2 q2 + q4 + q4t + q5t + q6t2
q + q2 + q3 + q3t + q4t + q5t + q3t2 + q4t2 + q5t2 + q6t3 q3 + q2t + q4t + q4t2 + q5t2
q3 + q2t + 2q3t + 2q4t + q5t + q3t2 + q4t2 + q5t2 q3 + q2t + q3t + q4t + q6t3
q + qt + 2q2t + q3t + q2t2 + 2q3t2 + q3t3 + q4t3 q + q2t + q2t2 + q3t2 + q4t3
1 + qt + q2t + qt2 + q2t2 + qt3 + q2t3 + q3t3 + q3t4 + q3t5 qt + qt2 + q2t2 + q3t3 + q2t4
qt + q2t + qt2 + 2q2t2 + qt3 + 2q2t3 + q3t3 + q2t4 1 + q2t2 + q2t3 + q3t3 + q2t4
t + qt + 2qt2 + 2qt3 + q2t3 + qt4 + q2t4 + q2t5 t + qt2 + qt3 + q2t3 + q2t5
t + t2 + t3 + qt3 + t4 + qt4 + 2qt5 + qt6 + qt' t2 + qt3 + t4 + qt5 + qt6
t3 + t4 + 2t5 + 2t6 + 2t' + t8 + t9 t3 + t5 + t6 + t 7 + t9
2212
214
16
q7 + q8 + 2q9 + ql0 + 2qll + q12 + q13 ql0 + qll + q12 + q13 + q14 q15
q4 + qS + 2q6 + q' + q8 + q't + q8t + q9t q6 + q' + q8 + q9 + qlOt qlO
2q3 + q4 + qS + q4t + 2qSt + q6t + q7t2 q4 + qS + q6 + q6t + q7t q7
q2 + q3 + q4 + q3t + q4t + q5t + q4t2 + q5t2 + q6t2 q3 + q4 + q5 + q6t + q6t2 q6
q2 + q3 + q4 + q3t + q4t + q5t + q4t2 + q5t2 + q6t2 q4 + q5 + q4t + q5t + q6t q6
q + q2 + 2q2t + q3t + 2q3t2 + q4t2 + q4t3 q2 + q3 + q3t + q4t + q4t2 q4
q + qt + q2t + qt2 + q2t2 + q3t2 + q2t3 + q3t3 + q3t4 q + q2 + q3t + q3t2 + q3f3 q3
q + qt + q2t + qt2 + q2t2 + q3t2 + q2t3 + q3t3 + q3t4 q2 + q2t + q3t + q2t2 + q3t2 q3
1 + qt + 2qt2 + q2t2 + qt3 + q2t3 + 2q2t4 q + qt + q 2 t + q 2 t 2 + q 2 t 3 q2
t + t2 + qt2 + t3 + qt3 + 2qt4 + qtS + qt6 1 + qt + qt2 + qt3 + qt4 q
t2 + t3 + 2t4 + t5 + 2t6 + t' + t8 t + t2 + t3 + t4 + tS 1
Examples
1. We have
1 - fx;qj-l
E gn(X;q,t) = n 1- j-I
n>O I,} x;q
= E s).(1,q,q2, ...)S,\(X;t)
A
= E qn('\)H,\ (q) -1 S,\ (X; t)
A

362 VI SYMMETRIC FUNCTIONS.WITH TWO PARAMETERS
by Chapter I, (4.3) and Chapter I, 3, Example 2, where HA(q) == cA(q, q) is the
hook-length polynomial. Hence
I(n)(x;q,t) == (q;q)ngn(x;q,t)
q"().)(q; q)"
= E H() S.\(x;t)
I.\I-n A q
and therefore
KA(n)(q, t) == qn().)(q; q )n/H.\ (q)
== E q,(T)
T
(Chapter I, 5, Example 14); the sum is over all standard tableaux T of shape A,
and r(T) is the sum of the positive integers k such that k + 1 lies in a lower row
than k in T.
By duality (8.15) it follows that
KJ.(l")(q, t) = tn('\')(t; t )n/H). (t).
2. Let A == (r + 1,1") == (rls) in Frobenius notation. For each partition J.L of n =
r + s + 1, KAp.(q, t) is the coefficient of US in the polynomial
Il (ti-1 + qi-1U)
(i,j)
where the product is over all (i,j) E J.L with the exception of (i,j) == (1,1).
This can be proved by applying the specialization B-u.t to both sides of (8.11).
By (8.8) we have
&-u,tJII-(x; q, t) == Il (ti-1 + qi-1U)
(i, j) e JI.
and
I( p)
B-u,tS.\(x; t) = Ez;lX/ Il (1 - (_U)Pi)
P ;-1
which is the Schur function SA corresponding to the series E hnyn =
(1 + uy)/(1 - y), so that SA == 0 unless A is a hook, and SA = (1 + u)US if A = (,Is).
It follows that
(*) Il (tl-1 +qi-1u) == E K(rls)p.(q,t)(l +u)US,
(i,j)ep. r+s-n-l
which gives the result stated.
3. Consider the functions KAp.(q, t) when q == t. We have then IlL == Hp.(t)sp.(x), so
that
Hp.(t)sp.(x) = EK).p.(t,t)S.\(x;t).
A

8. INTEGRAL FORMS
363
Since (s.\(x» and (S.\{x; t» are dual bases for the scalar product < u, v)o,,, it follows
that
K).JI.(t, t) = HJI.(t) ( SA' Sp,)o,t.
Now
l( p)
( ) -  -1 A P. n {1- Pi)-t
SA' Sp, 0,1 - I.J Zp Xp Xp . t
p ,-1
=-(SA*sp,)(1,t,t2, ...)
where SA * sJI. is the internal product defined in Chapter I, 7.
If X: + 0, the polynomial n(1 - tPI) divides HJI.(t) (Stanley [824]), from which it
follows that KAJI.{t, t) e Q[t]. But also SA * sJI. is of the form Evavsv with coeffi..
cients a., EN, so that
(SA,SP.)O,I == E avtn(v)Hv(t)-l e Z[[t]].
v
Hence KAp.(t, t) e Q[t] n Z[[t]] == Z[t].
Also
X;(t,t) = XpAHA(t)/n (1- tPI) e Z[t].
4. Let A be a partition. For each square S in the positive quadrant define
cA(s) = 1_qQJ,(s)t1J,(s}+1,
C (s) == 1 - qQ,,(S)+ 1 tl,,(s)
.if seA, and c.\(s) =- c(s) == 1 otherwise. If S is any set of squares, let
cf = n c(s). n c.\(s).
sES sES
'With this notation we have by (6.24)
Jp,J(r) = E "1.\/p.I)"
A
summed over partitions A:J 1.£ such that A - J..L is a horizontal strip of length r,
:here
"1A/JI.(q,t) = (q;q)rc(q,t)/cf(q,t),
[d S is the union of the columns that contain squares sEA - 1.£.
t" In particular
(1 - qrts)J(r)/(}l/) == (1 - tS)J(rls_l) + (1 - qr)J(r_lls)
l.i r, s ;;. 1.
[5:-1t JL == (2111 - 2) and let A be a partition of n. Then
KAp.(q, t) == K.\JI. (0, t) + qtll( JI.)K.\,JI.(O, t-1).

364 VI SYMMETRIC FUNCTIONS-WITH TWO PARAMETERS
(From the last equation of Example 4 above, when r = 1 we have
(1 - tn)l == (1 - qtn-1 )1(1)1(1"-1) - (1 - q )1(1")'
By replacing (q,t) by (O,t) and then by (O,t-1), and eliminating elen-1 and e
from the three resulting equations, we obtain "
1J1.(q, t) = 1J1.(0, t} + ( _1)n qt1 +n(n -1)/2 1J1.(0, t-1),
from which the result follows by picking out the coefficient of SA (x; t) on either
side, and bearing in mind that S.\(x; t-1) == (-t)-I.\IS.\,(x; t).)
6. We have
det Kn(q, t) == Il F.\(q, t)
1'\1_ n
where
D (t) Il (1 - qa(s)tl(s)+ 1).
£). q, =
sEA
a(s» 0
For
K(q,t}' =M(J(q,t),S(t»
= M(J(q, t), P(q, t»M(P(q, t), m)M(S(t), m)-1
where M(J, P) is diagonal, with determinant n,\ c.\(q, t); M(P, m) is unitriangular,
hence has determinant 1; and (Chapter III, 6) M(S, m) has determinant
n,\ b.\(t) = n.\ c.\(O, t}. It follows that
det K,.(q, t) = Il c.\(q, t)/c.\(O, t)
I.\I-n
which is equivalent to the result stated.
7. When q = 1 we have PJI. = eJl." so that
(1) 1J1.(x; 1, t) = (t; t}JI.,e,(x).
Moreover, it folIows from Chapter I, 3, Example 2 and Chapter I, (4.3') that
(2)
e,(x) = E s.\ (x; t}tn(.\') / H.\ (t).
IAI-r
Let us write
(3)
u.\(t} == tn('\')(t; t)r/H.\(t)
for A a partition of r. By Chapter I, 5, Example 14 we have
u.\(t} = E tP(T)
T
summed over the standard tableaux T of shape A, where p(T) is the sum of the
entries k in T such that k + 1 lies in a column to the righ.t of k. In particular,
therefore, u,\ (t) is a polynomial in t with positive integral coefficients.
(4)

8. INTEGRAL FORMS
365
From (1), (2), and (3) we have
Jp.(x; 1, t) == n (E UAi(t)SAi(X; t»)
,;> 1
where the inner sum is over partitions Ai of J.Li'. It follows that
(5)
KA(l, t) == E CA2... U).1(t)UA2(t)...
(where the c's are Littlewood-Richardson coefficients), and hence that KA(l, t) is
a polynomial in t with positive integral coefficients. Dually, by (8.15), KA(q, 1) is a
polynomial in q with positive integral coefficients.
A closer examination of the equation (5), in the context of the 'algebra of
tableaux' [S7] shows that
(6)
KA(l, t) == E tP(T,)
T
summed over all standard tableaux T of shape A, where
peT, J.L) == E p( PiT)
i;> 1
and piT is the restriction of T to the ith segment of [1, n], of length p!;,
determined by the partition J.L' == (J.L/l' ,12' ...). An equivalent formulation is
'.(7)
KA(I, t) = E tc(T,)
T
'summed as before over the standard tableaux T of shape A, where
c(T, ,.,.) = E c( PiT)
i;> 1
:,d c( piT) is the charge (Chapter III, (6.5)) of the skew standard tableau PiT.
It can also be shown that if w E Sn is a transposition,
oK
A (1,1) == t( X A(l) + X A(W »n( J.L),
at
oKA
p. (1,1) = t( xA(l) - xA(w»n( J.L).
aq
::8. (a) Since
l( p) 1 - t Pi
J(n)(x;q,t) = (q;q)n E z;lpp(X) n 1- Pi'
Ipl-n ,-I q
Jt follows that
X; n) (q , t) = (q; q ) n f1 (1 - q Pi) - 1

366 VI SYMMETRIC FUNCTIONS WITH TWO PARAlv1ETERS
for all partitions p of n. Dually,
XJIP1)(q,t) = sp"(t; t)n fl (1- tPi)-l.
(b) Another special case is
X()(q,t) = fl (ti-1 - qj-1)
(i ,j)
where the product is over all (i, j) E A with the exception of (1, 1). For by (8.20) we
have
X()(q, t) = E X()KAP,(q, t)
p.
and X() is zero unless J.L is a hook (r I s), in which case it is equal to (-l)s. The
result now follows by setting u = - 1 in the formula (*) of Example 2.
(c) We have
A c (q , t ) 
X(l")(q, t) = n Li 'PT(q, t)
(1 - t) T
summed over the standard tableaux T of shape A.
9.
1,\ (1, t, t2, . . . ; q, t) = tn(A).
(Set u = tn in (8.8), and then let n -+ 00.)
10. For each partition A, let IA(x; q) denote the Schur function SA in the variables
xiQj-l(i,j  1). Then (IA(x; q» is the basis of Ap dual to the basis (S,\(x; t», and
hence
KAp,(q, t) = (IA{x; q), Ip.{x; q, t».
We can now define
K A/v, p. (q , t) = (I A/V (x; q ) , I p. (x; q, t) )
where A::> II and I AI = I J.LI + I vi (otherwise it is zero). Since
I A/v = E C;1T I1T
'If
where the c's are Littlewood-Richardson coefficients, hence are integers  0, it
follows that
KA/v,p.(q,t) = E C;lrKlrP.(q,t)
'If
which will be a polynomial with positive integer coefficients if (8.18?) is true.
Since X A/v = E.". c:", X."., it follows from (8.16) that KA/v, p.(1, 1) = X A/v (1), the
number of standard tableaux of shape A - v.

8. INTEGRAL FORMS
367
We have
KA/.", p, (t, q) = KA,/v', p.,(q, t),
K (-I -I) _ -n(p.') -n(p.)v ()
A/v, p. q , t - q t ft A '/v ' ,p. q, ( .
l1. In'this example we shall assume the truth of (8.18?).
a) Let
lA(X; q,t) = E VAp.(q,t)mp.(x)
IL"A
[ben (1- t)-/(p.)vAp.(q,t) e Z[q,t].
We have already observed that (8.18?) implies that VA", E Z[q, t]. On the other
land, we have
JA(X; q, t) = E KP.A(q, t)Sp.(X; t)
p.
I( p)
= E KJA.A(q, t) E Z;IX:pp n (1- tPI),
JA. p i-I
lod
Pp = E Lpumu
u'>p
vith coefficients Lpu E N. Hence
I( p)
vAu(q,t) sa E Kp.).(q,t) E z;IX:Lpun (1- tPi),
JA. p<u ;=1
lnd since p  u implies that I( p)  I( u), every term in the inner sum is divisible
.Y (1 - t)/(u).
b) For any three partitions A, 1-£, JI we have
(1)., Jp.l.,) E Z[q, t]
-again on the assumption that (8.18?) is true. For this it is enough to show that
:SJ,.t S",S.,) e Z[q, t], and hence it is enough to show that (S)., SI'-) E Z[q, t]. But
l( p)
(SJ,.,Sp.) = E Z;IXpAXt n (l-qPi)(l- (Pi)
p i-I
= ",(SA * Sp.)
here SA * sp. is the internal product defined in Chapter I, 7, and ((J is the
pecialization defined by cp(p,) == (1 - q'X1 - t') for r  1. We have then
(1 - qu)(l - tu)
E rp(h,)u' = (1 - )(1- )
r,> 0 U qtu

368 VI SYMMETRIC FUNCTIONS-WITH TWO PARAMETERS
so that cp(h,) e Z[q, t] for all r  0, and hence successively q;(SA) for all A and
cp(s). * Sp.) for all '\, JL, lie in Z[q,t]..
Notes and references
For other cases in which KAp.(q, t) can be shown to be a polynomial, see
[S32].
9. Another scalar product
In this section we shall work with a finite number of variables x =
(Xl'''.' Xn). We shall also assume, for simplicity of exposition, that t = qk
where k is a non-negative integer (see, however, the remarks at the end of
the section).
Let Ln = F[xl: I,..., xn:f: I] be the F-algebra of Laurent polynomials in
Xl'''.' Xn (i.e. polynomials in the Xi and xi I). If f = I(xl,..., Xn) E Ln, let
f == f(xll, .. · , x; I) and let [/]1 denote the constant term in I. Also recall
from 3 that Tq,xlf(xl,..., xn) = I(xl,..., qxi,..., xn).
(9.1) Let f, g E Ln. Then
[(Tq,x;!)gL = [(Tq,x;g)jL.
Proof. Since both sides are linear in each of 1 and g, we may assume that
f and g are monomials, and then the result is obvious. I
Now let
4= Mx; q, t) = rJ (XiX;1; q)", /( !xiX; 1; q)",
(9.2)
k-l
= n n (l-q'X;Xjl)
1+) ,-0
since t = qk. Clearly  E Ln, and is symmetric in Xl'.'" Xn. We define a
scalar product on Ln as follows:
(9.3)
, 1
<I,g)' = <I,gn = ,Lm]l.
n.
(9.4) Let f, g E An F. Then
,
<D/,g)' = <1,Dg)'.
where D is the operator defined by (3.4).

9. ANOTHER SCALAR PRODUCT
369
Proof. Let
L\+ = 0<' (XiX;1; q )00/( txjXj-1; q)oo
, J
so that L\ = ++. Then we have
n I-tx1x;1
ITq,x+= n -1 =A1(x;t)
I j-2 l-X1Xj
in the notation of 3, so that
( Al (x; t) Tq, X I I ) L\ + = Tq, x 1{ I  + )
for all IE An,p, and therefore
[ Al (x; t) (Tq, xII) g  ] 1 = [Tq, x I( I  + )g d + I 1
which by (9.1) is symmetrical in I and g. By interchanging Xl and Xi' it
follows that
[ Ai(x; t )(Tq, xii) g] 1
is symmetrical in I and g for each i = 1,2,..., n. Hence [(D/)g]l IS
symmetrical in f and g, which establishes (9.4). I
(9.5) The polynomials P).{x; q, t), where X = (Xl'.'" Xn) and leA)  n, are
pairwise orthogonal lor the scalar product (9.3).
Proof. Since DPA = CAAPA by (4.15), where
n
C =  q).;tn-i
)'A  ,
i-I
it follows from (9.4) that
C AA (PA, Pp.)' = c p.p. (PA, Pp.)' ·
,Since c AA :#= cp.p. if A:#= J.L, the result follows. I
;!Remarks. 1. It follows from (9.5) that each of the operators D: defined in
,3 is self-adjoint for the scalar product (9.3). For the PA are simultaneous
;',eigenfunctions of these operators, by (4.15). Alternatively, this result can
..be established by direct computation, using the expression (3.4), for D:.
:2. When q = t (but not othelWise) the two scalar products <I, g)n (2.20)

370 VI SYMMETRIC FUNCTIONS -WITH TWO PARAMETERS
and <I, gYn coincide. For (x; t, t) == TIi.,.j(l-x;X;l):= a8QS' where as =:
n; < j(Xi - Xj) (Chapter I, 3), aI)d therefore
1
<SA' SJl.) == r [ s,\sJl.as as] 1
n.
1
== I [ a A + s a JI. + s ] 1 == 8,\ JI. ·
n.
On the other hand, when q = t the scalar product (f, g)n is that of
Chapter I, so that (s,\, sJI.)n == 8,\JJ. by Chapter I, (4.8).
It remains to calculate the scalar product (PA, P,\)'. One form of the
answer is, with the notation of (6.14)
(9.6)
1 - qO'(S)tn -/'(s)
(PA. QA) = cn Q 1- qD'(S)+ Itn-I'(s)-l
where I(A) <; nand
(9.7)
1
cn = (1, lYn == -[]1
n!
Proof. We shall prove (9.6) by induction on IAI. Let A, J.L be partitions of
length <; n such that A:J J.L and A - JL consists of a single square s. Also
let v == JL + (In), so that v::J A and v - A is a vertical strip of length n -1.
Since en ==x1... xn we have enfn == 1 and hence from the definition (9.3) of
the scalar product .
( P,\, e 1 Q JI. )' == (e 1 en P,\ , en Q JI. )' ·
But f1en == en-I, and enQJI. == bJl.enPJI. == bJl.Pv (4.17). Hence
(1) < PA, e 1 Q JI. )' == b JI. b; 1 (en - 1 P,\ , Qv )' ·
On the other hand, by (6.24) and (9.5) we have
(2) (P,\, e1QJI.)' == cp/JI.(P,\, Q,\)',
(3) < en - 1 P,\ , Q., )' == t/J/,\ ( P., , Qv )' == .p /,\ b., ( P., , P., ) ,
and
(4) (P." P.,)' := (enpJI.' enpJI.)' := (PJI.' PJI.)' == b;1 (PI-" QJI.)'.
From (1)-(4) we obtain
(5)
'Pl/JI.(PA, Q,\)' == .p/,\ (PJI.' QJI.)'.

9. ANOTHER SCALAR PRODUCT
371
Now from (6.24) we have
t/1/A = n bl)(u)/bA(u),
O'ER
cp/p, == n bp,(u )/bA(u)
O'ER
where R is the row containing the unique square sEA - J.L. Hence
"'/A/ cp/p, == n bv (u ) /bp, ( u ).
O'eR
S Since JI == J.L + (1n) it follows that if u ERn J.L and 7 is the square immedi.
ately to the right of u, we have al)(7) == aJL(u) and 11)(7) = lp,(u) and
: therefore bv(7) = bp,(u). Hence
{. (6)
t/J/A/ cp/p, = bl) (Ul)
r'.,where Ul is the leftmost square in the row R. For this square we have
''al)(ul) == a'(s) and ll)(ut) = n - 1 -l'(s). Hence it follows from (5) and (6)
.;:/ that
 .
(PA, QA)'
(Pp., Qp,)'
1 - qO'(S)tn-I'(S)
1 - qO'(S)+ 1 tn -I'(s)- 1 '
1:;/
I ".::,
ttnd the proof of (9.6) is complete. I
:-:".,.,.
--,
tj' Another (equivalent) formula for the scalar product (PA, PA) is given in
,Example 1 below.
:..-
J:.' We shall now renormalize the scalar product (9.3), and define
t4:. .
r.;(9.8) (f, g y = c; 1 (f, g Yn
.
;C
I,for f g.E An.F' so that (l.lY = 1. The original scal.ar product (1.5) is then
[.:the hnnt of the scalar product (9.8) as n -+ 00. PrecIsely, we have
(:..
it:.
'(9.9) Let f, g E Ap and let Pn: AF --+ Ap,n be the canonical homomorphism
IXhapter I, A2). Then
Jii'
;y,;
10> ( Pnf, Png y --+ (f, g)
.
f-9s n --+ 00, where the scalar product on the right is that defined by (1.5).
I'.:
f:root It is enough to verify this when f= PA and g == QJL' If A + J.L, both
'Ialar products are zero, by (9.5) and (4.7). If on the other hand A = J.L, it
_follows from (9.6) that (PA, QA): --+ 1 as n --+ 00. I

372 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
Remark. We have assumed throughout this section that t = qk where k is
a non-negative integer. This restriction is not essential, and may be
avoided as follows. Assume that q is a complex number such that Iql < 1
so that the infinite product (z; q)co converges for all z E C, and define '
11 -
(f,gYn ==, f(z)g(z)(z; q, t) dz
n. T
where T = {z = (Zt, . . ., zn) E en: Iz;1 = 1, 1  i  n} is the n-dimensional
torus, and dz is normalized Haar measure on T. The integrand in (9.10) is
a continuous function on T, provided that It I < 1.
When t = qk, kEN, this definition agrees with (9.3), since for a Laurent
polynomial f E Ln the integral of f over T is equal to the constant term of
f.
(9.10)
Examples
1. (a) Let x z=: (Xl" . ., xn) and let
1 - tx.x:-l
d'(x;q,t)==d(X;q,t)n I J
i<j 1 - XiX;1
k
= n n (1-q'XiX;1)(1-q'-lXi1Xj)
I<J<J<n ,-1
if t == qk. The constant term in d' is [ZI] equal to
Ii [ik]
i-2 k
where the square brackets denote q-binomial coefficients (Chapter I, 2, Example
3).
(b) From Chapter III, (1.3) we have
E W(n I-txiX:) = Ii I-ti
weS" i<j 1-xixj i=21-t
(c) Deduce from (a) and (b) that the constant cn of (9.7) is given by
Cn=Ii[ikl]
i- 2 k 1
and hence that
(1-t)(I-qt-1)
L(cn) == 1 E tj-i
- q i<j
where L is the operator (6.12).

9. ANOTHER SCALAR PRODUCT
373
(d) From (c) and (9.6) we have
L( (PA, PA)) = L(cll) + (1 - qt-1) E (qO'(s)tn -I'(s) - qQ(s)t'(s)+ 1)
sEA
which by (6.15) is equal to
(1 - t)(1 - qt-1)
E q).;-Ajtj-i.
1 - q i<j
Hence
(q A/-Ajtj-i; q )oo(qA/-Aj+l tj-i; q)oo
(PA. PAY" = 1J (qAl- Ajtj-i+ I; q }..(qAi- )./+ Itj-i-I; q )..
k-1 1 - qAi- Aj+'tj-i
== n n 1 _ A/-Aj-'tj-i
,<} ,- 1 q
if t'= qk.
2. (a) Let 0 < q < 1 and let f be a function defined on the closed interval [0,1].
The q-integral of f is defined to be
00
tf(x} dqx = (1- q) E q'f(q'}
o ,-0
for all f such that the series on the right converges. Thus it is the limit as n --+ 00 of
he Riemann sums of f corresponding to the subdivisions of [0, 1] at the points
q, q2,..., q" (provided that x/(x) --+ 0 as x --+ 0). More generally, if f(x) =
[(Xl'...' Xn) is a function of n variables defined on the unit cube Cn = [0, l]n, the
q-integral of f is defined to be
j, f(x)dqx= (l_q)n E q1olf(qO)
CII aE Nil
y.rhere a==(al,...,an),lal=al + ... +an and f(qO)=f(qOl,...,qO").
(b) "t T, S be positive integers (or, more generally, positive real numbers). Then
(1)
t X,-I(qx; q }$-I dqx = fq(r }f/s}/fq(r + s}
o
=Bq(r,s)
:here rq(r) = (q; q ),-1/(1 - q )'-1 is the q-gamma function. The formula (1) is a
_qanalogue of Euler's beta integral, which is the limiting case as q --+ 1. (It is
quivalent to Chapter I, 2, Example 5, which may be rewritten in the form
.;
'"'1
E am (qm+l;q).. = (b;q}..(q;q).. .
mO (a-1bqm;q)oo (a;q)oo(a-1b;q)00
',n, f..
:e:, ':.
':f we set a = q' and b == q'+s, then this sum, multiplied by 1 - q, is just the
qintegral (1).)

374 VI SYMMETRIC FUNCTIONS WlTH TWO PARAMETERS
3. Let x = (Xb...' XII)' let t = qk and define
k-l
4*(x; q, t) IE n n (x; - q'xj)(x; - q-'xj)
t<i<j<n r-O
(1) == (-1)AqB(xt...xlI)k(n-04(x;q,t)
where A == tkn(n - 1) and B == tk(k - 1)n(n - 1). In this example we shall evalu-
ate the multiple q-integral
1
fA. = n! . PA,(x; q, t)I1,.(x; q, t) dqx
where
n
4, I (x; q, t) == 4* (x ; q, t) n x [- 1 (qx i; q ) S - 1 ,
i-I
and ,\ is a partition of length < n. (When ,\ = 0, 10 is a q-analogue of Selberg's
integral.)
(a) For this purpose we shall expand P).4* as a sum of monomials, say
P).4* = E cAfJ X (j
13
summed over 13 e Nn such that 1131 = 1 AI + kn(n - 1). Then we have
(2)
n
n!l). =- E cAfJi n X[+fJ;-l(qx;;q)s_l dqxi
fJ C"i-l
_  nn rq(r + 13;)rq(s)
- I..J c).IJ
fJ ; - 1 rq (r + s + J3i)
by Example 2(b), so that
(3) n!l). =Bq(r,s)nE cA/3(q';q)/3/(q,+s;q)/3
13
where (as in 2, Example 1) (a; q)fJ means n(a; q )fJ;.
(b) To evaluate the sum in (3) we shall apply the specialization 8u,1 (6) to the
y-variables in the Cauchy formula (4.13). In this way we obtain
(4)
 nn (UXi; q)oo
i-I 8u,t(QI£)PI£(X) = (.).
,." ;-1 xt,q 00
Let 1.£ Ia ( 1.£1' . . . , J.LII) be the partition defined by 1.£; = Ai + (n - 1)k + a (1  i  n),
where a is a positive integer to be determined later. From (4) we have
nn (UXt;q)oo
8 u, t ( Q 1£ ) < Plot- ' PI£)' = < (.)' PI£)' ·
i-I xi,q 00
(5)

9. ANOTHER SCALAR PRODUCT
375
By (4.17) the right-hand side of (5) is the constant term in
1 ( l)k+a nn (UXi1; q)oo
,PA(x;q,t)(Xt...x,,) /1- (x;q,t) (-1.)
n. i-I Xi ,q 00
which by (I) and (2) is equal to
1 n ( -1. )
,(-OAqB(Xt...x/l)"EcAllxlln UXt:q "".
n. fJ i-I (Xi ,q)oo
Now the constant term in xf+fJ'(uxi1; q/{xil; q)oo is the coefficient of xf+fJj in
(UXi; q lo/{xj; q )00' which is (Chapter I, 2, Example 5)
(u; q)o+fJ,
(q; q)o+fJl
{u;q)a {uqO;q)fJj
{q;q)a (qa+l;q)fJj.
Hence from (5) we have
n
, 1 A B (u; q ) a  (uq a; q ) f3
(6) &,."(Qj.<)( Pj.<' Pj.<) =, ( - 0 q ( . )" l.J CAli ( a+ 1. ) ·
n. q, q a f3 q, q fJ
(c) The lefthand side of (6) can be evaluated by use of Example 1 above and 6,
Example 4. We thus obtain
1 (uqa. q) n (q., q)a{uqk(l-i)., q)ul
 ' fJ ( )A -B+kn(p..) ( ) n r-
, i.J cAfJ (a + 1 ) - -1 q VA q, t
n. f3 q;q fJ i-I {q;q)p..,+k(n-O(U;q)a
here (*6, Example 4)
VA (q , t) = n ( q A; - A j t j - i ; q ) k .
l<i<j<n
Sin.
(uqk(l-i). q) :a: (uqk(l-i). q) CU. q) (uqa. q)
, 11-1 , kG-I) , a ,Aj+k(n-i)
and
(uqk(l-i); q)kG-l) = (_u)kG-Dq-C;(qu-1; q)k(i-t),
where C; == t(i - 1)k{{i - 1)k + 1), it follows that
(7) 1  {uqlJ;q)fJ A E nn {qu-l;q)kG-l)(Uqa;q)A;+k(n-O
, i..I CA/J (a+l ) = U q VA{q, t) 1
n. fJ q; q fJ i-I (q a + ; q ) A, + k(2 n - i - 1)

376 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
where the exponent of q is
n
E= -B+kn(/.L) - E Ci
i-I
= k( n(A) + aG) ) + 2k2(;).
(d) Finally, if we now set a = r + S - 1 and u = ql-s in (7), we shall obtain from (3)
and (7) the desired result:
Ffln rq(Ai + r + k(n - i))rq(s + k(i - 1))
IA == q
i-1 rq(Ai+r+s+k(2n-i-1))
x fl r/Aj - Aj + k(j - i + 1))
l<i<j<n rq(Ai - Aj + k(j - i))
where F == k(n(A) + tm(n - 1)) + ik2n(n - lXn - 2).
(e) Deduce from (d) and *6, Example 4 that
IAllo = Butt(P).)BVtt(p»)IBwtt(p»),
where u = q'tn-1, V = tn, and w = q,+st2n-2.
Notes and references
The relationship (9.9) between the two scalar products was first remarked
by Kadell, in the context of Jack's symmetric functions. Likewise, Example
3 is a q-analogue of an integral formula due to Kadell.
In the definition (9.2) of A, and in the scalar product formula of
Example 1, the structure of the root system of type An-l is clearly visible.
In fact, this aspect of the theory generalizes to other root systems: see the
last section of [M6].
10. Jack's symmetric functions
In this section we shall summarize the main properties of Jack's symmetric
functions. As we have already obselVed in 1, they are obtained from the
preceding theory by means of the specialization
q=t£r, t-+1.
Here q and t are to be thought of as real variables, and a as a positive
real number.
More generally, let
(10.1)
(q,t)-+a(1,1)

10. JACK'S SYMMETRIC FUNCTIONS
377
mean that (q,t)-"(1,1) in such a way that (1-q)/(1-t)-+a. (For
example, q == ta as above, or again q = 1 - a(I - t).) Then for any real
numbers a, b, c, d (such that ca + d =1= 0) we have
1_qDtb aa+b
-..
1-qctd ca+d
(10.2 )
as (q, t) -+ a (1,1).
(10.3) We have
(tx; q)co/(x; q)co -.. (l_x)-l/a
as (q, t) -.. a (1, 1).
This statement, and others of the same sort that will occur later, is to be
interpreted in the sense of termwise convergence of formal power series.
The coefficient of xn in (tx; q )co/(x; q)co is (Chapter I, 2, Example 5)
(t;q)n = nfl-l 1-q't
() l-q,+l
q; q n r-O
and by (10.2) this tends to the limit
n-l ra+1 n(-l/a)
fl == ( -1)
r-O (r+1)a n
which is the coefficient of xn in (1 - x)-l/a.
Again, from (10.2) we have
l(A) 1 - qAt
ZA(q,t)=ZAfl -+z a1(A)
i _ 1 1 - t A; A
as (q, t) --. a (1,1), for all partitions A. Hence the scalar product (/, g )q,t
defined by (1.5) becomes in the limit the scalar product (/, g)a on Ap
(where now F == Q(a» defined by (1.4):
(PA,Pp.)a= 8Ap.a1(A)zA.
By (10.3), the product n(x, y; q, t) defined in (2.5) is replaced by
ll(x,Y; a) = fl(l-x;Yj)-l/a.
i, j
In place of (2.7) we have

378 VI SYMMETRIC FUNCTIONS-WITH TWO PARAMETERS
(10.4) For each n  0, let (uA) and (VA) be F-bases of Ap, indexed by the
partitions of n. Then the following conditions are equivalent:
(a) (uA, V) a = 8).p. for all A, J..L;
(b) LA u.\(x)uA(Y) = II(x, Y; a).
The specialization of gn(x; q, t) (2.8) is ga)(x), defined by the generat-
ing function
E ga)(x)yn = fI (l-xiy)-I/a.
n>O i
As in f2, we define
gla)(x) = fI gl)(x)
i> 1
for any partition A = (AI' A2' ...). By specializing (2.9) and (2.11) we obtain
(10.5)
ga) = E z; la-1(A)PA
IAI-n
and
(gia), mp.) a = BAp.
so that the gia) form a basis of AF dual to the basis (mA). Hence the
ga), n  1, are algebraically independent over F = Q( a), and Ap =
F[ga),ga), ...].
For each real number {3 let lJJ denote the F-algebra endomorphism of
A F defined by
(10.6) lJJ (p,) = ( _1)'-1 {3p,
for r  1. We have wi 1 = W -1 if {3::1= 0, and lJJl is the involution (JJ of
Chapter I. From Chapter I, .14)' we have
(10.7)
W (g(a») = e
o n n
and it follows directly from the definitions (1.4) and (10.6) that (J)/3 is
self-adjoint, i.e.
(10.8)
(w{jf, g)a = (f, lJJ{jg)a
for all f, g E AFe Moreover we have
(10.9) (Wa-l f, g)o = «(J) f, g )1.
Consider next the behaviour of PA(x; q, t) as (q, t) -+ a (1,1). From (7.13')t
the coefficient of mll-(x) in PA(x; q, t) is
u).p.(q,t) = E tfJr(q, t)
T

10. JACK'S SYMMETRIC FUNCTIONS
379
summed over tableaux T of shape A and weight J.L, where .pr is given
;explicitly by (7.11 '), (6.24), and (6.20). These formulas show that c/1r(q, t) is
ta product of terms of the form (10.2), and hence has a well-defined limit
:-.p} a) as (q, t) .... a (1, 1).
':. Explicitly, the limit of bA(s; q, t) (6.20) is
(10.10) b(a)( ) _ aaA(s) + lACs) + 1
A S - ( )" () ,
aaA S + IA s + a
and the limit of c/1A/JA.(q, t) (6.24) is
(10.11)
cp}il == n ba)(s)lbla)(s)
S eRA/ - CA/
where (Ioc. cit.) CA/JA. (resp. RA/p.) is the union of the columns (resp. rows)
that intersect A - J.L. Hence (7.11')
(10.12)
r
.,,(a) - n .lll g)
'liT - "'A'(')/A(/-l)
i-I
where 0 == A(O) C A(I) C ... C A(r) == A is the sequence of partitions deter-
mined by the tableau T (so that each skew diagram AO) - A(i-1) is a
horizontal strip).
Hence the limit of UAp.(q, t) as (q, t) --. a (1,1) is
U) == E cp}a)
T
summed as above over tableaux T of shape A and weight J.L, and therefore
(10.13)
p,(a) == m +  u(a)m
A A '-' Ap. iJ.
JJ.<A
is the limit of PA(q, t) as (q, t) .... a (1,1). We have
(10.14)
(p(a) p(a» = 0
A , II- a
whenever A =1= J.L.
The Pla) are Jack's symmetric functions. They are characterized by the
properties (10.13), (10.14). From (10.10)-(10.12) it is clear that u) is a
sum of products of the form (a a + b) I(c a + d) where a, b, c, d are non-
negative integers. Also P}l) is the Schur function SA' so that ui1J is the
Kostka number KAJA.' hence is positive whenever A  J.L (Chapter I, 7,
Example 9). Hence
(10.15) We have ui} > 0 whenever A > J.L and a > o. I

380 VI SYMMETRIC FUNCTIONS W-ITH TWO PARAMETERS
Another proof of the existence of the pfa) is indicated in 4, Example 2
See also 4, Example 3(d). ·
Let (Qa» be the basis of AF dual to the basis (Pfa)}, so that
Qa) = b1a)PA(a)
where bla) = (Pia), Pia»;l E F, so that from (6.19) and (10.2) we have
b1a) = 0 b1a)(s)
se ,\
(10.16)
aa(s) + l(s} + 1
=0
SEA aa(s)+l(s}+a.
In particular we have
Pea) - e
(IT) - r'
Q(a) = g(a)
(r) r.
Also it follows from (4.14) that the pia) are well-defined at a = 0 and
a = 00, and that
P(O) - e Pel) - Q(I) - S P(oo) = m
A - A" A - A - A',\ A.
The duality theorem (5.1) becomes
(10.17)
,., Pea) = Q(a-1)
Ula A A'.
Next, let t;'JJ(a)t denote the coefficient of p,\(a) in the product PJa)pa),
so that
.fA (a) = (Q(a) p(a)p(a»
J p.v A 'p. va.
This is a rational function of a that remains finite at a = 0 and a = 00. By
(7.4) it vanishes identically unless I AI = I J.LI + I]) I and A::> J.L, A ::> )I.
The skew functions PAJ"j, Qi/ are defined as in 7 by
Q(a) =  .fA (a}Q(a) = (b(a)/b(a»)p(a)
Alp. i..J J p.v v A p. Alp. ·
II
They are zero unless A::> J..L, in which case they are homogeneous of degree
IAI-IILI. For a finite number of variables x = (x],..., xn) we have by (7.15)
(10.18)
Qli  (x 1 , · .. , X n) = 0
unless 0  Xi - Ii;  n for each i  1.
t There is a conflict here with the notation I;.,(t) of Chapter III, 3, but it should cause no
confusion.

10. JACK'S SYMMETRIC FUNCTIONS
381
Duality (10.17) generalizes to give
(10.19)
P(a) _ Q(a-1)
Wa Alp. - A'/iL'.
Next, let X be an indeterminate and let
8X: AF -+ F[X]
be the F-algebra homomorphism defined by 8X(p,) = X for all r  1, so
that 8X(PA) =X'(A}. If X is replaced by a positive integer n, then 8x(f) =
/(1,...,1) (with n l's) for f E AF. The specialization theorem (6.17) then
gives
(10.20)
n X + aa'(s) -l'(s)
8 p(a) =
X A sE A aa(s) + l(s) + 1 ·
Let
(10.21)
cA(a)= n(aa(s)+l(s)+l),
sEA
c(a)= n(aa(s)+l(s)+a)
sEA
be respectively the numerator and denominator of bla) (10.16). We have
c ( a) = a I AI C A' ( a-I ) .
Now define
(10.22)
j(a) - C (a )p(a) = C' (a )Q(a)
A - A A A A.
By comparison with (8.1) and (8.3) it is clear that
(10.23) Jla)(x) is the limit, as (q, t) -+ a (1,1), of
(1 - t)-IAIJA(x; q, t).
For the Jla), duality (10.17) and specialization (10.20) take the forms
(10.24)
(10.25)
W J(a) = aIA1J(a-1)
a A A"
8XJ}a)= n (X+a'(s)a-l'(s)).
SEA
For each partition A, let mA = uAmA denote the 'augmented' monomial
symmetric function corresponding to A, where (as in Chapter I, 6,
Example 10)
UA = n miCA)!.
i;> 1

382 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
The transition matrices M(J(a), m)n have been computed up to weight
n = 7, and suggest the conjecture
(10.261) The entries in M(J(a), m) are polynomials in a with positive integral
coefficients.
Next, let 8pA(a) denote the coefficient of Pp in J1a):
(10.27) J} a) = E 8pA( a) pp.
p
From (8.19) and (10.23) it follows that
X: (q , t) l( p)
8pA(a)= lim n O(l-tP1).
(q, t).... a (1, 1) (1 - t) Z p i - 1
(10.28)
In particular,
1
8(1")( a) = I" X(1")(1, 1)
n.
and hence by (8.26) (or also from (10.25))
(10.29) 8(1")( a) = 1
for all partitions A of n.
From (10.24) it follows that
(10.30)
o A ( a) = Ban - I( p)o A' ( a-I )
p p p
if A, p are partitions of n.
The 8pA( a) satisfy the orthogonality relations
(10.31) E zpal(P)8pA(a)8p"'(a) = AcA(a)c(a),
p
(10.32)
 c (a) -1 c' ( a ) - 1 8 A( a ) 8 ,\ (a) = B z - la -ie p)
i.." ,\ A p u pu p ·
A
For (10.31) follows from (10.26) and the orthogonality of the J1a), and
(10.32) is equivalent to (10.31).
(10.33) Remark. Since the augmented monomial symmetric functions m,\
form a Z-basis of the subring n = Z[Pl' P2' ...] of A, it would follow from
the conjecture (10.26?) that 8pA( a) E Z[ a] for all A, p, and hence that
(J}a), Ja)Ja»a E Z[a] for all A, I-L, v. Stanley [S25] makes the apparently
stronger conjecture that this scalar product should be a polynomial in a
with non-negative integer coefficients.

10. JACK'S SYMMETRIC FUNCTIONS
383
Finally, the function d defined in (9.2) is replaced by
(10.34)
A(x; a) = D (l_XiXjlf/Cl
I 'r)
where x = (Xl'...' Xn). As in 9 we define a new scalar product on An,F by
1
(f,gYn = ,1f(z)g(z)A(z; a)dz
n. T
in the notation of (9.10). From (9.5) we have
(10.35)
(10.36) The polynomials Pia)(xl,..., xn), where leA)  n, are pailWise or-
thogonal for the scalar product (10.35). I
Also from (9.6) we have, on passing to the limit as (q, t) --+ a (1,1),
) n+a'(s)a-l'(s)
(p(a) Q(a )' - n
A , A n-CnSEA n+(a'(s)+1)a-I'(s)-1
(10.37)
where
1
Cn = ,1 A(z; a)dz.
n. T
Equivalently, by 9, Example led),
n r(i-j+k)r(i-j-k+l)
(10.38) (pia), p1a» =
lc;i<jc;n r( €i - €j)r( €i - j + 1)
where k = a-I and i = Ai + ken - i), 1  i  n.
As in 9, the formula (10.37) shows that the scalar product (1.4) is the
limit as n --+ 00 of the scalar product c; I (f, g Yn = (f, g >/ (1, 1 Yn.
Examples
1. (a) Since J(» = n!anga), it follows from (10.5) that
n!
8(n)(a) = - an-1(p)
p Z
p
and hence by (10.30) that also
8;lft)(a) = Bpn!/zp.
(b) Deduce from (10.25) that
8()(a) = n (a(j - 1) - (i -1»

384 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
where the product is over the squares (i, j) E A with the exception of (1, 1), and
that
8(1"-2)( a) = n( A') a - n( A).
2. (a) Suppose that f(q, t) E Z[q, t] is such that for some positive integer m the
limit
I( q , t)
L = lim
(q,t)-'a(1,l) (l-t)m
exists. Then L E Z[a]. (We may write I in the form
f(q,t)= E ars(1-q)r(1-t)s
r,s>O
with coefficients ars E Z, from which it follows that ars = 0 if r + S < m and that
L = Lr+s-m a,s ar.)
(b) From (8.19) and (10.27) we have
XpA(q,t)
8pA(a) = U;-l lim
(q,t}-+a (1,1) (1 - t)n-l(p)
if A and p are partitions of n, where uA = nmj(A)1 Hence the conjecture (8.18?),
which implies that X:(q, t) E Z[q, t], also implies that OpA(a) E Q[ a].
(c) Let
J(a)=  V (a)m
A i..J Ap. p.
,.,.
so that, in the notation of 8, Example 11(a) we have
VAp.(q,t)
VA/a) = lim (I_t)n'
(q, t}-'a (1, 1)
Since (8.18?) implies that VA(q, t) E Z[q, t], it also implies that vAp.(a) E Z[a].
(d) If A,,.,,, v are partitions such that I AI = I ,.,,1 + I v I = n, we have
(JA,Jp.Jv)q,t
(1 - t )2n '
(J(a) J(0)1(0» = lim
A 'p. v
(q,t}-.o(1,l)
Since (8.18?) implies that (JAt Jp.Jv )q,t E Z[q, t] (8, Example 11 (b)), it also implies
that (J}o), Ja)Jo» E Z[ a].
3. (a) The coefficient of m,.,. in J(» is
c,.,. n (ap. (s) a + 1)
S e ,.,.
where ap.(s) is the arm-length at S E ,.", and cp. is the number of decompositions of
a set of n elements into disjoint subsets containing ""1' ""2' ... elements, as in
Chapter I, 2, Example 11.

10. JACK'S SYMMETRIC FUNCTIONS
385
(b) The coefficient of m(ll1) = n!en in J).(a) is equal to 1, and the coefficient of
m(21"- 2) in J).a is
an(A') - n(A) + tn(n -1).
4. (a) From 8, Example 4 it follows that
(ar + s)J(»J(\? = sJ('i}-l) + raJ(aj1IS).
Deduce that
J(f =D(1,2,...,s, -ra, -(r-1)a,..., -a)
where in general D(a1' a2, . . . , an) denotes the determinant
Pt
P2
at
PI
o
o 0
o 0
a2
Pn
Pn+1
Pn-l
Pn
Pn-2
Pn-l
PI an
P2 PI
(b) Another formula for J'j/) is the following, due to P. Hanlon. Fix a standard
tableau T of hook shape (rls) and let R, C denote respectively the row and column
stabilizers of T (so that R == Sr+ 1 and C = Ss+ 1). Then
JS'j/) = E e(v)ar+1-c(U)r/J(uv)
summed over (u, v) E R X C, where c(u) is the number of cycles in u, and I/J is as
in Chapter I, 7.
5. If A = (2r1s). is a. partition with two columns, the coefficient of mp. in ]).(0),
where,." = (2r-'ls+2,) and i  0 is
( ; ) ( a + s + i + 1) . .. ( a + s' + r).
6. Let d: A F -+ A F be the derivation defined by
dPr = 'Pr+l
for all r  1. Show that
J(» = (PI + a)n(1)
and that
Jc\? = (PI - )n(l).
7. As (q, t) -+ a (1,1), the q-integrals of 9, Examples 2 and 3 become ordinary
(Lebesgue) integrals. It follows that (with k = 1/a) the value of the integral
I>.= 1 PP/k)(x)a8(xikIlx;-I(1-x)S-ldx,
n. C" i-I

386 VI SYMMETRIC FUNCTIONS WITH TWO PARAMETERS
where C,. z:: [0,1]", x:a: (Xl"'" Xn), and dx == dx1 ... dxn, is
nn f(Ai + r + k(n - i»f(s + k(n - i»
fA - v,,(k)
i-I r(Ai+r+s+k(2n-i-l»
where
v,,(k) -= n
1 < i <j <,.
f( Ai - Aj + k(j - i + 1»
r( Ai - Aj + k(j - i» ·
(a) By making the change of variables Yi == ux; and then letting u -+ 00, deduce that
10000
,1 ...1 pl1/k)(y )a«5 (Y )2k (Yl ... Yn)r-l e-Y1- ... -Y,. dy
n. 0 0
,.
- vA(k) n r(Ai + r + k(n - i».
;-1
(b) By making the change of variables Yi = IX; in (a), multiplying by tSe-t and
integrating from 0 to 00 with respect to t, show that
1 100 100 (l/k)() 2k( )r-l(  )-a-s
, · · · PA X a «5 (x) XI. .. X II 1 + i..J Xi dx
n. 0 0
f(s) n
... f( ) vA(k) n r(Aj + r + k(n - i))
a + s i-I
where a :II: I AI + nr + n(n - l)k.
(c) In the integral of (b) above put Xi == Y;/(1 - EYj) (1  i  n), so that 1 + Ex; =
(1 - EYj)-l == u say. We have
aXil aYj == 8iju + YiU2
and therefore
det( aXil aYj) == u2 det( 8ij + Y;U)
== u,.( 1 + u E Yi) == u,.+ 1.
Deduce from (b) that
1 f. (l/k)() ( )2k( )r-l(  )S-l
, PA Y a a Y Yl .. · Y II 1 - i..J y; dy
n. D"
r(s) ,.
-r( )vA(k)nr(Aj+r+k(n-i»
a + s i-I
where a == I AI + lIT + n(n - l)k as before, and D,. is the simplex in Rn bounded by
the hyperplanes Xi - 0 (1 < i  n) and Xl + ... +xn == 1.

10. JACK'S SYMMETRIC FUNCTIONS
387
8. Let
1 11
JA,,.,,(k) == -n' /, pll/k)(x)p1/k)(x)a8(x)2k n Xi-let _x;)k-l dx,
· C" ,-1
where (as in Example 7) Cn == [0,1]", x == (Xl'...' Xn), dx == dx1 ... dxn is Lebesgue
measure, and A, J.L are partitions of length < n. Then
JA,,..<k) - VA(k)V,.(k)(f(k))n/i,Dt (Ai + ILj + r + (2n - i - j)k) k
where VA' VIJ. are as in Example 7, and (X)k == rex + k)/f(x). (See [K6] for a proof.)
Notes and references
Jack's symmetric functions were first defined by the statistician Henry Jack
in 1969 ([Jl], [J2]). He showed that when a = 1 they reduce to the Schur
functions, and conjectured that when a = 2 they should give the zonal
polynomials (Chapter VII). Later, H. O. Foulkes [F7] raised the question
of finding a combinatorial interpretation of Jack's symmetric functions.
About ten years ago the author began to investigate their properties,
taking as a starting point the differential operators of Sekiguchi [S9] (see
3, Example 3), and showed in particular [M5] that they satisfy (10.13),
(10.14), and (10.17) (duality). Shortly afterwards R. Stanley [S25] advanced
the subject further, and established the scalar product formula (10.16), the
Pieri formula, the explicit expression as a sum over tableaux, and the
specialization theorem (10.20). (Stanley worked with Jia) rather than
Pi a).)
At about the same time, K. Kadell was investigating generalizations of
Selberg's integral [SI0] and its extension by Aomoto [A6]: in this way he
was led to introduce as integrands a family of symmetric polynomials,
depending on a parameter k and indexed by partitions, which he called
Selberg polynomials, and which turned out to be precisely Jack's symmetric
functions, with parameter a = Ilk [KS]. (The integral formula generalizing
Selberg's integral is that given in Example 7 above.) In a different
direction, R. Askey [A7] had proposed a q-analogue of Selberg's integral,
and Kadell ([10], conjecture 8) was led to conjecture the existence of
'q-Selberg polynomials' which would feature in a q-analogue of the inte-
gral formula above. In the event it turned out that these q-Selberg
polynomials are our PA(x; q, t), and the q-integral formula is that of 9,
Example 3.
For another method of construction of the PA(x; q, t), see B. Srinivasan
[S22].

VII
ZONAL POoL YNOMIALS
1. Gelfand pairs and zonal spherical functions
Let G be a finite group and let A = C[ G] be the complex group algebra of
G. If teA, say f= ExeG/(x)x, we may identify I with the function
x  I(x) on G, and hence A with the space of all complex-valued functions
on G. From this point of view, the multiplication in A is convolution:
(fg)(x) = E I(y)g(z)
yz-x
and G acts on A by the rule (xIXy) = l(x-1 y). The centre of A consists
of the functions I on G such that I(xy) = I(yx) for all x, y E G, and has
the irreducible characters of G as a basis. The scalar product on A is
1
(I, g)G = IGI E I(x)g(x).
xeG
Let K be a subgroup of G and let
1
e=eK= \K\ E k,
keK
so that e2 = e. As a function on G, we have e(x) = \KI-t if x E K, and
e(x) = 0 othelWise. If Xl'. . . , X r are representatives of the left cosets xK of
K in G, the elements xie are a basis of the left A-module Ae, and the
corresponding representation 1 of G is that obtained by inducing the
trivial one-dimensional representation of K, or equivalently: it is the
permutation representation of G on G /K = {XiK: 1 <: i <: r}. Since e is
idempotent, the endomorphism ring EndA(Ae) is anti-isomorphic to eAe,
the anti-isomorphism being cp  epee). Now eAe as a subalgebra of A
consists of the functions I on G such that I(kxk') = I(x) for all x E G and
k, k' E K, that is to say the functions constant on each double coset KxK
in G. We shall denote this algebra by C(G, K).
An A -module V (or equivalently a representation of G) is multiplicity-free
if it is a direct sum of inequivalent irreducible A -modules, that is to say if
the intertwining number (V, U) = dim HomA(V, U) is 0 or 1 for each
irreducible A-module U. By Schur's lemma, V is multiplicity-free if and
only if the algebra End A(V) is commutative. Hence

1. GELFAND PAIRS AND ZONAL SPHERICAL FUNCTIONS 389
(1.1) For a subgroup K of G, the following conditions are equivalent:
(a) the induced representation 1  is multiplicity-free;
(b) the algebra C(G, K) is commutative. I
If these equivalent conditions are satisfied, the pair (G, K) is called a
Gelfand pair.
(1.2) Suppose that KxK= Kx-1K for all xEG. Then (G,K) is a Gelfand
palr.
Proof. We have f(x) = f(x-1) for all fE C(G, K) and x E G. Hence if f,
g E C(G, K),
(fg)(x) = E f(y)g(y-1x) = E g(x-1y)f(y-l)
yeG yeG
= (gf)(x-1) = (gf)(x)
which shows that C(G, K) is commutative.
Assume from now on that (G, K) is a Gelfand pair. Then with the
notation used above, Ae is a direct sum of non-isomorphic irreducible
A-modules, say
s
Ae = E9 Vi.
i...1
Let Xi be the character of Vi and let
- -
w. = X.e = ex.
I I I
(1<i<s)
where Xi is the complex conjugate of Xi" Thus
1  _ 1"_1
(1.3) wi(X) = -IKI i..J Xi(xk) = -IKI i...J Xi(X k).
keK kEK
The functions Wi are the zonal spherical functions of the Gelfand pair
(G, K).
Let d; = dim  = X;(l).
(1.4) (i) wi(l) = 1 and w;(x-1) = Wi (X) for all x E G.
(ii) W; E C(G, K) n Vi.
(iii) W.W. = S..c.lJ). where c. = IGlld..
I} I} I I' I ,
(iv) < Wi' Wj)c; = Sijdi 1.
(v) (wI'...' W,v) is an orthogonal basis of C(G, K).
(vi) The characters of the algebra C(G, K) are f (wif)(l), for 1  i < s.

390
VII ZONAL POLYNOMIALS
Proof. (i) From the definition (1.3) we have
w;(l) = (xiIK, lK >K = (X;, 1 >G
by Frobenius reciprocity. Hence wi(l) is the multiplicity of  in the
A-module Ae, hence is equal to 1. The relation w;(x-1) = w;(x) follows
from (1.3).
(ii) It is clear from (1.3) that wi(kx) = wi(xk) = wi(X) for x E G and k E K,
so that Wi E C(G, K). On the other hand, the JIj-isotypic component of A
is A Xi' hence the Vi-isotypic component of Ae is Ae Xi = A wi. Thus
A Wi = Vi and consequently Wi E Vi.
(iii) Since Xi Xj = 8ijc; Xi it follows that
- - -- 2 --  - 
w.w. = X.ex.e = X. X.e = X. X.e = u..C. X.e = u..C.W.
I) I} I ) 1 ) I} " I) I 1 ·
(iv) Since Wj(x)= Wj(x-1) we have
1 1
(Wj, wj)G =- IGI wjw/l) = IGI 8jjcjwj(1) = 8ijdi1
by (iii) and (i).
(v) It follows from (ii) that the Wi are linearly independent elements of
C(G, K). On the other hand, Schur's lemma shows that the dimension of
C(G, K) = EndA(Ae) is equal to s. Hence the Wi form a basis of C(G, K),
and they are paiIwise orthogonal by (iv).
(vi) Let 8 be a character of C(G, K). From (iii) we have 8(wi)8(wj) = 0 if
i:l= j, and 8(Wi)2 = Ci(J(Wi). Hence for some i we have (J(Wj) = Sijc; =
(wi wjXl), and hence by linearity (J(/) = (wi/Xl) for all / E C(G, K). I
Let V be an irreducible G-module (i.e. A-module) with character X,
and let VK be the subspace of vectors in V fixed by K. The dimension of
VK is the multiplicity of the trivial one-dimensional K-module in V
regarded as a K-module, so that
dim VK = (xIK, lK>K = (X, l>G
by Probenius reciprocity. Since 1 is multiplicity-free it follows that, with
the previous notation,
(1.5)
dim VK = { 
if V= Vi for some i,
othelWise.
Moreover, if V = Vi, then Wi is the unique element of the one-dimen..
sional space JliK such that wi(1) = 1, by (1.4Xi) and (ii). The zonal spherical
functions Wi are therefore distinguished generators of the irreducible
components  of the induced representation Ae = 1.

1. GELFAND PAIRS AND ZONAL SPHERICAL FUNCTIONS 391
Now let Y be an irreducible G-module such that yK =1= O. Then the
zonal spherical function w corresponding to V may be constructed as
follows:
(1.6) Let (u, v) be a positive-definite G-invariant Hermitian scalar product
on V, and let v E VK be a unit vector (i.e. (v, v) = 1). Then
w ( x) = (v, xv )
for all x E G.
Proof. Choose an orthonormal basis VI'.'" vn of V such that VI = v. For
each x E G, XVj is a linear combination of the Vi' say
n
xv. =  R..(x)v.
) £.J I} "
;-1
and x  (Rij(x)) is an irreducible matrix representation of G afforded by
V. Moreover (Vi' XVj) = Rij(x), so that in particular (v, xv) = R11(X).
Now the space U spanned by the Rij is the V-isotypic component of the
G-module A, and eUe is one-dimensional, generated by w. Since v is fixed
by K it follows that R11(xk) = R11(X) and that
Rll{kx) = (v,kxv) = (k-1v,xv) =Rl1(x)
for all k E K and x E G. Hence Rll E eUe and is therefore a scalar
multiple of w. But Rll (1) = (v, v) = 1 = w(1), hence Rl1 = w. I
A function f E A is said to be of positive type if
E f(x-Iy)g(x)g{y)
x,yeG
is real and  0 for all functions g EA.
(1.7) Let f E C(G, K). Then f is of positive type if and only if
s
f= E aiw;
;=1
with real coefficients a;  o.
Proof. If w is one of the Wi then with the notation of (1.6) we have
w( x -1 y ) = (v, X-I yv) = (xv, yv) and therefore
E w{x-1 y)g{x)g(y) = (u, u)  0
x,yeG
where U = LxEog(X)xv. Hence each W; is of positive type and therefore

392
VII ZONAL POLYNOMIALS
so also are all non-negative linear combinations of the Wi.
Conversely, if f E C( G, K) is of positive type, we may write
s
f= E aiwi
i-1
with coefficients ai E C. Taking g = Wi in the definition, (wifwi)(I) is real
and  O. But by (1.4) (iii), wifwi = ai w? = a;c;wi, so that (wifwi)(l) = aicl.
Hence ai is real and  o. I
(1.8) Let (G, K) and (G', K') be Gelfand pairs such that G' is a subgroup of
G and K' is a subgroup of K, and let W be a zonal spherical function of
(G, K). Then the restriction of W to G' is a non-negative linear combination
of the zonal spherical functions of (G', K '). .
Proof. By (1.7), wiG' is of positive type and belongs to C(G', K'). Hence
the result follows from (1.7). I
The zonal spherical functions of a Gelfand pair (G, K) may be charac-
terized in other ways:
(1.9) Let W be a complex-valued function on G. Then the following condi-
tions are equivalent:
(a) w is a zonal spherical function of (G, K).
(b) (i) w E C(G, K); (ii) w(l) = 1; (iii) for all f E C(G, K), fw = Afw for
some Af E C (depending on f).
(c) w  0 and
1
w(x)w(y) = _I IE w(xky)
K keK
for all x, y E G.
Proof. (a)  (b). Conditions (i) and (ii) follow from (1.4). As to (iii), if
f E C( G, K), say f = Ea j Wj' then f wi = Ea j Wj Wi is a scalar multiple of Wi
by (1.4Xiii).
(b)  (c). Let
1
wy(x) = IKI E w(xky).
keK
From (i) it follows that wy E C(G, K). Next, for each k E K and fE
C(G, K) we have
Afw(y) = Afw(ky) = (fw)(ky)
= E f(x-1 )w(xky)
xeG

1. GELFAND PAIRS AND ZONAL SPHERICAL FUNCTIONS 393
and hence, on averaging over K,
(1)
AtW(Y) = (fwy)(1) = IGI(wy,f*)G'
where f*(x) = f(x-1). On the other hand,
(2)
At = At w(1) = (fw)(l) = IGI( w, f*)o.
From (1) and (2) it follows that
< wy - w(y)w,f*)o = 0
for all f E C( G, K), from which we conclude that lJJy = w(y) w, Le.
I
-IKI E w(xky) = w(y)w(x)
keK
for all x, Y E G.
(c)  (a). If f, g E C(G, K) we have
(wfg)(l)= E w(xy)f(y-l)g(X-1)
x,ye G
= E w(xky )f(y-1 )g(X-l)
x,yeG
for all k E K, and hence on averaging over K we obtain
(wfg)(1) = L w(x)w(y)f(y-l)g(X-1)
x,yeG
= (wf)(l)( wg )(1).
Since w =1= 0, this shows that f (wfXl) is a character of C(G, K), and
hence w is a zonal spherical function of (G, K) by (1.4Xvi). I
Let Kt, K2, "" Ks be the double cosets of K in G, and let 1;(1  i  s)
be the functions on G defined by
J:.(x) = { l/IKI if x E Ki,
I 0 othelWise.
They form a basis of C(G, K). Hence fi is a linear combination of the
fk, say
h = E aijkfk'
k

394
VII ZONAL POLYNOMIALS
If x E Kk, then
aijk = IKlfi!j(X)
= IKI E fi(y)/j(y-1x).
yeG
Now fi(y)!j(y-1X) is zero unless y EKi nXKjl, so that
aijk = IKi nxKjll/IKI.
Since both Ki and xKj 1 are unions of cosets gK of K in G, it follows that
the structure constants aijk are integers  O. We shall use this to prove
(1.10) Let w be a zonal spherical function of (G, K), and let x E G. Then
IIUK/Klw(x) is an algebraic integer.
Proof. Let w be the character of C(G, K) corresponding to w, so that
w(f) = (fwXl) by (1.4Xvi). Then we have
w(!;)w(fj) = E aijkw(fk)
k
from which it follows that the set of s linear equations (in s unknowns
x1,...,xs)
W(fi)Xj = E aijkxk
k
has a non-trivial solution xj = w(fj). Hence
det( W(fi) 8jk - aijk) j,k = O.
On expanding the determinant, we see that we/;) is an algebraic integer,
since the aijk are integers. But if x E Ki we have
W(fi) = (J:w)(l) = 1JW(/Klw(x-1). I
Examples
1. If w is a zonal spherical function of (G, K), then I w(x)1  1 for all x E G. (Use
(1.6) and the Cauchy-Schwarz inequality.)
2. The constant function equal to 1 on G is always a zonal spherical function of
(G, K). (Use (1.5).)
3. If w is a zonal spherical function of (G, K), then so also is its complex conjugate
w. (If CJ) == ex we have ( X, 1)0 == 1, hence also ( X, 1)0 = 1.)

1. GELFAND PAIRS AND ZONAL SPHERICAL FUNCTIONS 395
In the notation of (1.4), suppose that all the characters Xi are real-valued. Then
each W; is real-valued and hence w;(x-I) = w;(x) for 1  i  s and all x E G. It
follows that f(x-I) =-= f(x) for all fE C(G, K), and hence that KxK = Kx-IK for all
x E G. This is a partial converse of (1.2).
4. Let (G, K) be a Gelfand pair, let 0 = {wI' . . . , ws} be the set of zonal spherical
functions of (G, K), and let C(O) denote the algebra of all complex-valued
functions on 0, with pointwise multiplication: (fgXw) = f(Wi)g(W;). If fE
C(G, K), the spherical transform (or Fourier transform) of f is the function Sf on (1
defined by
(Sf)(w;) = (fwi)(l) = E f(x)wi(x).
;reG
Show that S is an algebra isomorphism of C(G, K) onto C({1), with inverse S-I
given by
(1)
1
S-\p = IGI  cp(cu)djcuj
,
for (/J E CCO), in the notation of (1.4). (From (1.4Xiv) and (v) it follows that
1
f =  (f, cu)o dj CUj = IGI  (Sf)( cu)dj CUj
I I
for t E C( G, K), whence S is a linear isomorphism whose inverse S -] is given by
the formula (1). That S(fg) = (SfXSg) follows from (1.4Xvi).)
5. If (G, K) and (G', K') are Gelfand pairs with zonal spherical functions w;(l 
i  s), CIJ;(l j <; t) respectively, then (G X G', K X K') is a Gelfand pair with zonal
spherical functions Wi X wj.
6. (a) Let G be a finite group and let 0' be an automorphism of G such that
0'2 == 1. Let K = {x E G: ux ==x} and let P == {x E G; o'X =x-1}; K is a subgroup of
G, but P is not, in general. If G = KP, show that (G, K) is a Gelfand pair. (Let
XEG, say x=kp where kEK, peP. Then o'x' kp-l =kx-Ik, so that
u(KxK) = Kx-1K and hence t(O' x) = [(X-I) for all f E C(G, K). Hence for f, g E
C(G, K) we have
(fg)(x) = E f{xy-I)g(y) = E f(O'(yx-I»g(O'(y-I»
yeG yeO
= (gf)(ux-I) =gf(x)
which shows that C( G, K) is commutative.)
(b) Suppose that G == KA where K, A are subgroups of G, K nA = {I}, and the
subgroup A is abelian, normal, and of odd order. Define 0': G --+ G by u(ka) =
lea - 1, where k E K and a EA. Clearly 0' 2 is the identity. Moreover, 0' is a
homomorphism, because if x = k]a1 and y = k2az are any elements of G, where
k}, k2eK and a}, a2EA, then xy=k}a}k2a2=ka where k=k}k2EK and
a == (kilal k2)az == a2(kiIa} k2) (because A is abelian); and hence u(xy);:: lea -1 =
k}k2(k2IatIkz)aiI = (k}QI1Xk2a2"l) = O'(x)O'(y). Since A has odd order, the

396
VII ZONAL POLYNOMIALS
subgroup of G fixed by u is precisely K; also A cp = {x E G: ux =x-1}, so that
G = KP. Hence (G, K) is a Gelfand pair, by (a) above.
(c) As an example of (b), let V be'a finite-dimensional vector space over a finite
field of odd characteristic, and let Ko be any subgroup of GL(V). Since Ko acts On
V we can form the semidirect product G = V X1 Ko, whose elements are pairs
(v, k) E V X Ko, with multiplication defined by (v, kXv', k') == (v + kv', kk'). The
elements (v, 1) form a normal subgroup of G isomorphic to V, hence abelian and
of odd order, and the elements (0, k) form a subgroup K of G isomorphic to Ko.
By (b) above, (G, K) is a Gelfand pair.
7. In the situation of Example 6(b), every x E G can be uniquely written in the
form x = k(x)a(x), where k(x) E K and a(x) EA, and we have KxK = Ka(x)K
A .
The group K acts on A by conjugation, hence also on the character group A of A.
The double cosets of K in G are in one-one correspondence with the K-orbits in
A, the double coset KxK corresponding to the K-orbit of a(x).
A
For each character a E A let Wa be the function on G defined by
1  _)
wa(x) = _I I i..J a(ka(x)k ).
K keK
Show that Wa is a zonal spherical function of (G, K). (Verify that Wa satisfies the
functional equation of (1.9)(c).)
It is clear from (1) that l1Ja = l1Jf3 if a, pEA are in the same K-orbit. If S is a set
of representatives of the K-orbits in A, the wa' a E S are linearly independent
(because the characters a E A are linearly independent), and hence are all the
zonal spherical functions of (G, K).
8. If f, g E C(G,K) let f.g denote the pointwise product of f and g:(f.gXx)=
!(x)g(x). Show that the coefficients at defined by
(1)
(J).. CIJ. =  a. ClJk
I J I.J I)
k
(where the Wi are the zonal spherical functions of (G, K» are real and  O. (We
may identify G with its image in G X G under the diagonal map x t-+ (x, x). By
Example 4, the zonal spherical functions of (G X G, K X K) are ClJi X Wj' and Wi. Wj
is the restriction of Wi X wj to G. Hence the result follows from (1.8).)
9. Let G be any finite group and let Xi (1  i  r) be the irreducible characters of
G. Let G* = {(x, x): x E G} be the diagonal subgroup of G X G. The irreducible
characters of G X G are Xi X Xj: (x, y) t-+ Xi(X) Xj(Y) (1  i, j  r) and we have
( Xi X Xj' 19G)GXG = « Xi X Xj)IG*, IG* )G*
1
= IGI E Xi(X)X/X) = (Xi' ,¥j)G
xeG
which is equal to 1 if Xi = , and is zero otherwise. Hence the induced representa-
tion 19 G is multiplicity-free, with character
r
E Xi X Xi.
i= 1

1. GELFAND PAIRS AND ZONAL SPHERICAL FUNCTIONS 397
Consequently (G X G, G*) is a Gelfand pair, and the zonal spherical functions are
Wi (1  i  r), where
1
ldj(X, y) = IGI 1: Xi(gr) Xi(gy)
gEG
1
= IGI 1: Xj(X-1g-1)Xi(gy)
gEG
1
= - X.2(x-1y) = d 1X.(X-1y)
IGI I I I
where di = Xi(l). Hence w;K1 X G) = di1xi.
The functional equation (1.9Xc) for zonal spherical functions in this case
becomes the functional equation for characters:
Xi (1)  _ 1
Xi(X) Xi(Y) = IGI i..J Xi(g xgy)
geG
for all x, Y E G.
10. Let G be a finite group, A the complex group algebra of G; also let K be a
subgroup of G and cp a linear character of K (i.e. a homomorphism of K into C*).
Let
1  -1
efP = IKI i.J cp(k )k
kEK
so that e; = e'f) and the A-module Ae<p affords the induced representation cpG =
indcp. Let C<P(G, K) = e<pAefP' which is anti-isomorphic to the endomorphism
algebra EndA(AefP). As an algebra of functions on G, Cf{J(G, K) consists of the
functions feA that satisfy f(lex) = f(xk) = cp(k-1)f(x) for all k eK and x e G.
The representation cpG is multiplicity-free if and only if CfP(G, K) is commutative.
If these equivalent conditions are satisfied, as we shall assume from now on, the
triple (G, K, cp) is called Ollogical1y) a twisted Gelfand pair.
. The A-module Aef{J is a direct sum of inequivalent irreducible A-modules, say
s
Aef{J = e .
i-I
Let Xi be the character of v: and let
fP - -
Wi = Xie<p = efP Xi
or explicitly
1
wt(x) = _I I 1: X;(x-1k)cp(k-1).
K keK

398
VII ZONAL POLYNOMIALS
The functions w{P(l  i  s) are the qrspherical functions of the triple (G, K, cp). All
the reults in the text, appropriately mo.dified, remain valid in this more general
context.
(a) Show that (1.4Xi)-(vi) remain valid if Wi is replaced by wt and C(G, K) by
CfP(G, K) throughout.
(b) If V is an irreducible G-module, let
VfP = {v E V: kv = cp(k)v for all k E K}.
Then dim VfP = 1 if V:::: Vi for some i, and dim VfP = 0 otherwise. The qrspherical
function wt is the unique element of Vjrp such that wt(l) = 1.
(c) If V is an irreducible G-module such that VfP =1= 0, let (u, v) be a positive-defi..
nite G-invariant scalar product on V, as in (1.6), and let v E VfP be a unit vector.
Then the qrspherical function corresponding to V is given by w fP(x) = (v, xv) for
all x E G.
(d) The functions fE CfP(G, K) that are of positive type are the non-negative
linear combinations of the wt.
(e) Show that the following conditions are equivalent, for a complex-valued
function w on G:
( a) w is a qrspherical function for (G, K, cp).
({3) 0) wECfP(G,K); (ij) w(l) = 1; Gii) for all feCfP(G,K), we have fW=Afw
for some AI E C.
( 1') w =1= 0 an d
1
w(x)w(y) = IKI E w(xky)cp(k)
keK
for all x, y E G.
11. In continuation of Example 10, let e be a linear character of G. Then the
mapping 8: A A defined by 6(x) = e(x)x for x E G (or, in terms of functions, by
(fJfXx) = e(x)f(x)) is an automorphism of A and induces an isomorphism of
Cerp(G, K) onto CfP(G, K), where ecp is the character k....... e(k)cp(k) of K. Hence
if (G, K, cp) is a twisted Gelfand pair so also is (G, K, ecp), and the spherical
functions of (G, K, ecp) are the images under 0-1 of the spherical functions of
(G, K, cp).
In particular, if (G, K) is a Gelfand pair, and cp is a linear character of K which
extends to a linear character of G, then (G, K, cp) is a twisted Gelfand pair.
12. Let G be a finite group, K a subgroup of G. Let cp be the character of a (not
necessarily irreducible) representation of K, and let cpG = indcp. We regard cp as
a function on G, vanishing outside K. Let X be an irreducible character of G, and
let
1 1
w(x) = - E X(x-lk)cp(k-1) = - Xcp(x-1).
IKI keK IKI
(a) We have
w(1) = (xIK, CP)K= (X, cpG)G.

1. GELFAND PAIRS AND ZONAL SPHERICAL FUNCTIONS 399
(b) w(x) = w(x-1) for all x EO.
(c) w2 = CCaJ for some positive scalar C (because X, cp are positive scalar multiples
of commuting idempotents).
It follows that
1 2 C
(CaJ, w)a = IGI w (1) = IGI CaJ(l)
and hence that CaJ +- 0 if and only if X occurs in the induced representation cpG.
13. Let m, n be positive integers such that m  n, and Jet 0 = Sm+n' K = Sm X S,..
Then (Chapter I, 7)
m
ch(l) == hmhn = E s(m+"-i,i)
i-O
;,'which shows that I is multiplicity free, with character
m
E X(m+n-i,i).
i-O
I"
:::,Consequently (0, K) is a Gelfand pair, dim C(G, K) = m + 1, and the zonal
:',spherical functions are
Wi = X(m+n-i,i)eK
(Oim).
} Let E = {I, 2, 0.0' m}, F = {I, 2,..., m + n}. Under the mapping w...... wE, where
,': E G:::s Sm+n' the cosets wK of K in G are in one-one correspondence with the
In-element subsets of Fo Let
):,'
cp(w) =IE -wEI =m -IE nwEI
,
r(or w E G. The function cp lies in C(O, K) and takes the values 0,1,..., m, and the
:double cosets of K in G are the fibres K,. = cp -1(,) of, cpo In particular, Ko = K,
and IK,I/IKI = ( 7 )( ) for 0  r  m. If wj(r) denotes the value of Wj at
[Klements of K" the orthogonality relation (1.4Xiv) takes the form
r:=.
:;"', m (m) ( n ) h (m + n - i, i)
y:, E w.(,) w.(,) = . "
r:'' r , I 1 mInI '1
,,:- : r- 0 · ·
,.,,'
here h(m + n - i, i) is the product of the hook-lengths of the partition
(m + n - i, i). The numbers w;(r) can be calculated explicitly (see e.g. [Bl], p. 218):
:if
}:,:..
i::'"';:
!!T> ":"
N'W'
'" '
i.here
/:
fj.'.
(:"
dYi:'.
o.;
Qj(X)= E (-OSqi.S(;)'
.f> 0
qi. S = (:)( m + n s+ 1 - i ) / ( 7 )( ; )

400
VII ZONAL POLYNOMIALS
so that Q;(x) is a polynomial of degree i, then wi(r) = Q;(r) for 0 < i, r  m.
14. Let V be a vector space of dimension m + n over the field Fq of q elements
nd let U be a subspace of V of dimension m, where m < n. Let G = GL(V):::
GLm+n(Fq), and let K = {g E G: gV = V}, which is a maximal parabolic subgroup
of G. The induced representation 11 is the permutation representation of G on
the Grassmannian G I K = Gm, n' whose points are the m-dimensional subspaces of
v.
For each g E G let
cp(g) =m - dim(UngV)
in analogy with Example 10. Then cp E C(G, K) and takes the values 0,1,..., m,
and the double cosets of K in G are the fibres K,. = cp-l(r) of cp: in particular
Ko = K. Since cp(g) = cp(g-l) it follows that K, = K; 1 for 0  r  m and henc
(G, K) is a Gelfand pair, by (1.2).
Let k, = IK,I/IKI, so that k, is the number of m-dimensional subspaces V' of V
such that dim(U n V') = m - r. Then we have
kr = qr2[ ][;]
where [] and [;] are q-binomial coefficients (Chapter I, 2, Example 3). (Let
E = V n V' and F = V + V', so that E is an (m - r )-dimensional subspace of U,
and FlU is an r-dimensional subspace of VIV. The number of choices for E is
[m m r] = [  ], and the number of choices for F is [;]. Moreover, V'IE is a
complement of U IE in F IE; since dim(V'IE) = dim(V IE) = r, it follows that for
each choice of E and F there arc q,2 possibilities for V'.)
From Chapter IV it follows that
m
ch(I) = hm(l)hn(l) = E s(m +n -i,;)(I)
i- 0
and hence the character of 1  is E?: 0 X;, where X; is the character of G whose
characteristic is s(m + n -;,0(1). Hence the zonal spherical functions of (G, K) are
Wi = Xie K (0 < i < m). If wi(r) denotes the value of Wi at elements of K" the
orthogonality relations (1.4Xiv) take the form
 ,2 [ m] [ n ] _ 1 [ m + n ]
'-' q r r w;(r)wj(r)=d; m 5;j'
,-0
where d; is the degree of x;, which by Chapter IV, (6.7) is
di = qm+n(q) I n (qh(x) - 1)
xeA
where ,\ is the partition (m + n - i, i), and t/1,(q) = ni- l(qi - 1).

2. THE GELFAND PAIR (S2,,, H,,)
401
Notes and references
,.The theory of Gelfand pairs and zonal spherical functions was originally
developed in the context of the representation theory of locally compact
groups, and in particular Lie groups (see, for example, [H6], Chapter IV).
The prototype example is that in which G = SO(3) and K = SO(2), so that
G / K may be identified with the 2-sphere, and the zonal spherical func-
tions are the 'spherical harmonics' or Legendre polynomials (loc. cit., p.
404 ).
2. The Gelfand pair (S2n' Hn)
Let t E 82,. be the product of the transpositions (12), (34),. .. , (2n - 1,2n),
and let HlI be the centralizer of t in S2n' The group Hn is the hyperocta-
hedral group of degree n, of order 2nn!, and isomorphic to the wreath
product of S2 with S,..
To each permutation W E S2n we attach a graph f(w) with vertices
1,2,. . . ,2n and edges Bi' WBi (1  i  n), where Bi joins 2i - 1 and 2i. If
we think of the edges Bi as red and the others as blue, then each vertex of
the graph lies on one red edge and one blue edge, and hence the
components of few) are cycles of even lengths 2pt, 2P2"'" where Pt 
P2  ... · Thus w gives rise to a partition P = ( PI' P2, ...) of n, called the
coset-type of w. This terminology is justified by the following proposition:
(2.1) (i) Two permutations w, WI E SZn have the same coset-type if and only if
Wt EHnwHn.
(ii) w, w-1 have the same coset-type.
Proof. (i) The permutations w, wt have the same coset-type if and only if
their graphs are isomorphic, that is to say if and only if there is a
permutation h E S2n that preseIVes. edge-colours and maps f(w) onto
f(w1). Since h permutes the red edges Bj, it follows that h E Hn; and since
the blue edges WI Bi of f(wt) are a permutation of the blue edges hWBj of
'hr(w), the ei are a permutation of the wl1hwei, whence wl1hweHn and
consequently WI E HnwH,..
(ii) Since wf(w-1)=f(w), the graphs f(w) and f(w-1) are isomorphic,
hence determine the same partition of n. I
From (2.1Xii) and (1.2) it follows that
(2.2) (S2,., Hn) is a Gelfand pair. I

402
VII ZONAL POLYNOMIALS
From (2.1Xi), the double cosets HnwHn of H = Hn in S2n are indexed by
the partitions of n. Let
Hp = HwH
if wE S2n has coset-type p. Then we have
(2.3) IHpl = IHnI2/Z2p = IHnl2 /21(p)zp.
Proof. If w has coset-type p then
IHpl = IHwHI = IHwHw-11 = IHI.lwHw-11/IH n wHw-11
so that
(1)
IHpl = IHI2/IH n wHw-11.
Now H n wHw-1 is the group of colour-preseIVing automorphisms of
the graph r(w). On a given cycle in r(w) of length 2r, it acts as the
dihedral group of order 2r, from which it follows that
IH n wHw-11 = Z2p
which together with (1) completes the proof.
From (2.2), the induced representation 1; of S2n on the cosets of Hn is
multiplicity-free. In fact
(2.4) We have
152" =  M
H" -"W 2.\
1'\1- n
where M 2.\ is an i"educible S 2n -module co"esponding to the partition 2 A =
(2 AI' 2 A2, ...).
Proof. With the notation of Chapter I we have
ch(l:) = c(HII) = c(S2 - SII) = C(SII) 0 C(S2)
= hn 0 h2 = L S2,\
I,\I-n
by use of Chapter I, 7, Example 4 and 8, Examples 5 and 6(a).
From (2.4) and (1.5) it follows that
(2.5) Let IL be a partition of 2n and let MIA- be an i"educible S2n-module
co"esponding to /L. Then dim M:" = 1 if /L is even (i.e. J.L = 2 A), and
M:" = 0 otherwise. I

2. THE GELFAND PAIR (S2n, Hn)
403
From (2.4), the zonal spherical functions of the Gelfand pair (S2n, Hn)
are indexed by the partitions of n: they are
wA = X2Aen
where X2A is the character of M2A, and
1
en = -, , E h.
Hn heH
,.
Explicitly, since X 2A is real-valued, we have
(2.6)
1
wACS) = - E X2A(sh)
IHnl heH
n
for s E S2n. Since X2A is integer-valued, wACs) is a rational number for all
partitions A of n and all s E S 2n.
For each integer n  0, let
Cn = C(S2n' Hn)
be the space of complex-valued functions on S2n that are constant on each
double coset Hp' and let
C = e Cn.
n>O
We define a bilinear multiplication in C as follows: if fE Cm and g E Cn'
then fXg is a function on S2m XS2n cS2(m+n) (where S2m permutes the
first 2m symbols and S2n the last 2n symbols), ad we define
(2.7)
f * g = em+n(fxg)em+n.
Thus f * g is obtained from f X g by averaging over Hm + n on either side,
and lies in Cm+n.
For each partition p of n let
... ...
CPp = enwpen
where wp E Hp. Thus cpp(w) = IHpl-l if w E Hp' and cpp(w) = 0 othelWise.
Clearly the 'Pp such that I pi = n form a basis of Cn' and we have
(2.8)
'Pp * 'Pu = 'Pp u u
for any two partitions p, u.
Proof. Let I pi = m, lul = n, and let u E Hp' v E Hu. Then f(u X v) is the

404
VII ZONAL POLYNOMIALS
disjoint union of feu) and f(v), hence u X v has coset-type p U u. Now
calculate:
'Pp * 'PO' = em +n«emuem) X (enven)em +n
= em+n(em X en)(u X v)(em X en)em+n
= em+n(u X v)em+n = CPpu 0"
because em+nh = hem+n = em+n for h E Hm+n, and hence em+nCem X en):::
Cem X en)em +n = em +n. I
From (2.8) it follows that the multiplication C2.7) makes C into a
commutative and associative graded C-algebra.
For each w E 52n let p(w) denote the coset-type of w, and let
.p , ( w) = p p( w)
in analogy with Chapter I, 7, cycle-type being replaced by coset-type.
Note that .p'(w) = .p'(w-1), by (2.1Xii). Now define a C-linear mapping
ch' : C  Ac
as follows: if fE Cn' then
ch'(f) = E f(w).p'(w).
we S 2n
From this definition it is clear that
ch' ('Pp) = Pp
for all partitions p, and hence it follows from (2.8) that
(2.9) The mapping ch': C  Ac is an isomorphism of graded C-algebras.
Next, we shall define a scalar product on C: if f, g E C, say f= Ef",
g = Egn with fn' gn E Cn' we define
(f,g)= E (fn,gn)
n>O
where
(2.10)
(fn,gn)= E fn(w)gn(w)
we S 2n
(Le. the usual scalar product, multiplied by IS2nl = (2n)!). On the other
hand, we have a scalar product (u, v) on Ac, defined by
(2.11)
(pp' PO') = 8pO' Z2p = 8pO' 2/( p)zp

2. THE GELFAND PAIR (S2,., Hn)
405
as in Chapter VI, (1.3). For these scalar products, the mapping ch' is
almost an isometry:
(2.12) For each n  0, the mapping
f IHnl-1ch'(f)
is an isometry of Cn onto Ac.
Proof. Let p and u be partitions of n. Clearly ('Pp' 'Per) = 0 if p + u, and
('Pp, 'Pp) = IHpl-l = IHnl-2z2P
by (2.3). Hence
('Pp, 'Per) = IHn 1- 2 (ch' 'Pp' ch' 'Per)
for all p, u, which proves (2.12).
Now define, for each n  0 and each partition A of n,
ZA = IHnl-1 ch' wA
= 1 Hn 1-1 E W A (s ) r/1 ' (s )
S E 52"
Le.
(2.13 )
ZA = IHnl E Z2;W:PP
Ipl-n
:where w: is the value of wA at elements of the double coset Hp. The
symmetric functions ZA are called zonal polynomials.t Since the wA form a
-C-basis of C, the ZA form a C-basis of Ac.
(2.14) ZA E Z[Pl' P2, ...] for all partitions A.
Proof. We have to show that
IHnlzi.;wpA = IHnl-llHplwpA
is an integer for all A., p. It is certainly a rational number, by (2.6), and it is
an algebraic integer by (1.10). hence it is indeed an integer. I
t 'Zonal symmetric functions' would perhaps be a more logical name; but the present
terminology is universally used, and I have therefore not sought to change it.

406
VII ZONAL POLYNOMIALS
If A, J.L are partitions of n, it follows from (1.4Xiv) that
( Cd A, Cd ) == 8,\(2n) 1/ X2'\(1)
or
(2.15)
(Cd'\ Cd ) = 8Ah(2A)
where h(2A) is the product of the hook-lengths of the partition 2>... Hence
the matrix (Cdp'\) satisfies the orthogonality relations
-1 ,\  _ h(2A)
E Z2p Wp Cdp - 2 8A'
p IHnl
(2.15')
(2.15" )
 h (2 ) - 1 ,\ ,\ _ Z 2 p
i.J A Cd p Cd er - 2 8 per ,
A Illnl
from which and (2.12) we have
(2.16)
Pp = IHnl E h(2A) -1 Cdp'\ZA.
A
Hence if we define
IHnl
CA. = h(2A) ZA.
the relation (2.16) takes the form
(2.16')
Pp = E wp'\Cp
P
and in particular
(2.16" )
pf = E Cp.
p
This renormalization of the zonal polynomials is often preferred by
s ta tisticians.
Next, from (2.12) and (2.15) it follows that
(2.17)
(z,\, Z) = 8,\h(2A).
(2.18) We have
n (1-XiYj)-J/2 = E h(2A)-1 Z,\(x)Z,\(y).
iti A
Proof. By (2.17), (Z,\) and (h(2 A) - J Z,\). are dual bases of A Q for the scalar

2. THE GELFAND PAIR (S2n' Hn)
407
product (f, g), and so also are (pp) and (Z2Pp) by definition (2.11). It
follows (cf. Chapter I, (4.6)) that
E h(2A)-1 Z.\(x)Z,\(y) = E Z2Pp(X)pp(Y)
A p
n Prexy)
= exp
r>l 2r
which in turn is equal to
exp t log n (1-X;Yj)-1 = n (1-xiYj)-1/2.
By considering the hook-lengths at (i, 2j - 1) and (i,2j) in the diagram
of 2A, where (i, j) E A, it follows that
h(2A) = ht(2A)h2(2A)
where
(2.19)
hi(2A) = n (2a(s) + l(s) + i)
sEA
(i=1,2)
in the notation of Chapter VI, (6.14). From the results of Chapter VI, fi10
it follows that the Z).. satisfy the same orthogonality relations as the Jack
symmetric functions Jla) with parameter a = 2. In fact, ZA = 1}2) for all A,
as we shall now show.
For each integer n  0, let
gn = IHnl-1Z(n)
= E Z2Pp
Ipl-n
since CIJn) = 1 for all p. If now A = (AI' A2, ...) is any partition, let
g).. = g)..Jg)..2 ... · Then we have
(2.20 )
(2.21)
(g).., mp.) = 8Ap.
for all A, J-L.
Proof. The generating function for the gn is
E gn(x)yn = E Z2p1pp(x)yIPI
n>O p
n ( )-1/2
= 1-xiy
i> 1

408
VII ZONAL POLYNOMIALS
as in the proof of (2.18). It follows that
n (1 _X;Yj)-1/2 = E g,\{x)m.\{y)
i,j A
and hence that (gA)' (m.\) are dual bases for the scalar product (f, g).
(2.22) The transition matrix M(Z, m) is strictly upper triangular.
Proof. The coefficient of mp' in the expansion of Z,\ as a linear combina-
tion of monomial symmetric functions is equal to (Z,\, gJA.) by (2.21), hence
we have to show that (Z,\, g p.) = 0 unless A  J-L. From the definition of g
and from (2.9) it follows that JJ.
g = c ch' ( (J #L )
#L ,.,.
where C#L is a positive scalar, whose exact value need not concern us, and
(J #L = w( ,u1) * w( ""2) * ... . Hence by (2.12) it is enough to show that (w \ 8 p.)
= 0 unless A  J-L.
Now (J #L = en 71en, where n = I J.LI and 71 is the trivial one-dimensional
character of the subgroup S2JA. = S2JA.l X S2JA.2 X . .. of S2n. Since left and
right multiplications by en are self-adjoint for the scalar product (2.10), we
have
{W'\, (JJA.) = (en X2'\, en71en) = (en x2'\, 71)
= (en X2)(1).
Now the element X2 of the group algebra of S2n generates the X2'\.
isotypic component of the induced representation 71S2n, which, in the
notation of Chapter I, 7, is 712,.,.. Hence X2 = 0 unless X2'\ occurs in
712,.,.; and since
( X 2,\ , 712 JA. > = (s 2,\ , h 2 JA. > = K 2'\,2 JA.
(Chapter I, 7) is zero unless 2 A  2 J-L, i.e. unless A  J-L, it follows that
X 2  = 0 and hence (w'\, (),.,.) = 0 unless A  J-L. I
From (2.17) and (2.22) it follows (cf. Chapter VI, 1) that Z). is a scalar
multiple of J12), for each partition A. But by (2.13) and Chapter VI, (10.29),
the coefficient of p in each of Z,\ and J,\(2) (where n = I AD is equal to 1,
and therefore
(2.23) Z). = J12) for all partitions A.

r:- 2. THE G ELF AND PAIR (S2n' H,.)
[:
n
g- For each partition A, let c,\ denote the polynomial
409
i.
c,\(X) = n (X + 2a'(s) -l'(s))
sEA
i' (2.24)
,!
t:
!;'-
t..-
f
:
;:
= n (X - i + 2j - 1)
(i  j) E A
!;Then if Z,\(ln) denotes the value of Z,\(xI,..., xn) at Xl = ... =xn = 1, we
2have
C(2.25) Z,\ (In) = c,\ (n)
r-:by (2.23) and Chapter VI, (10.24). For an independent proof of (2.25), see
::-(4.15) below.
C(2.26) The coefficient of x'\ in Z,\ (x) is
hl(2A) = n (2a(s) + l(s) + 1).
sE ,\
This follows from (2.23) and the definition (Chapter VI, (10.22)) of Jla).
For another proof, see fi3, Example 1 below.
(2.27) If IL  A, the coefficient of x p. in Z,\ (x) is a positive integer divisible by
:':"up. = 0; __1 mi( IL)!.
Proof. The 'augmented' monomial symmetric functions mp. = up'mJl. are a
Z-basis of the subring Z[PI' P2, ...] of A (Chapter I, 6, Example 10).
Hence it follows from (2.14) that the coefficient of x p. (or of mIL) in Z). is
an integer divisible by up.; moreover it is positive by virtue of (2.23) and
Chapter VI, (10.12). I
(2.28) Let A., Ii, v be three partitions. Then (Z,h ZpZv) is an integer  0,
and is positive only if C;2v = (S2).' S2p.S2v) (Chapter I, 9) is positive.
Proof. The definition (2.11) of the scalar product shows that it is integer-
valued on the subring 7J.Pl' P2' .. .] of A. Hence it follows from (2.14) that
(Z,\, Zp.Zv) is an integer. To show that it is non-negative, it is enough by
(2.9) and (2.12) to show that (w'\, w p. * w V)  O.
Let I ILl = m, Ivl = n, IAI = p. We may assume that m + n = p, otherwise
the scalar product is certainly zero. We have
(w,\ wI'- * w") = (w'\ e/wl'- X w")ep)
= (w'\, wp. X Wv)
which up to a positive scalar factor is the coefficient of w p. X W JI in the
restriction of w'\ to S2m X S2n, and hence is non-negative by (1.8).

410
VII ZONAL POLYNOMIALS
Finally t since
(wAt wP,X Wv) == (epx2'\,(em xen)(X2P,xx2V))
== ep X2'\( X2p, X X2V)(1),
for (ZA' Zp,Zv) to be positive we must have x2'\( X2p, X X2v) ?= 0, which
implies that X2A occurs in the character obtained by inducing X2#J. X X2v
from S2m X S2n to S2p, i.e. that (S2,\' S2#J.S2v) > o. I
Remark. It seems to be an open question whether (ZA' Z#J.Zv) > 0 if and
only if c;v == (SA' Sp,Sv) > o. .
Examples
1. If A == (rls) == (r + 1,1") is a hook, then Z). is equal to the determinant
D(I, 2,..., s, - 2r, - 2r + 2,..., - 2)
in the notation of Chapter VI,  10, Example 4(a).
2. There are explicit formulas for some of the entries in the matrix (w;).
(a) w; == 1 if A == (n) or p == (In).
(b) When ,\ == (In),
Z). =n!en ==n!E &pZ;lpp
p
(Chapter I, (2.14'» and hence
wI") == (_I)n-/(p) /2n-/(p).
(c) From (2.25) it follows that
IHnlE Z2plwpAXI(p) == cA(X)
p
and hence, on equating coefficients of X on either side,
1
tJJ() = IHn-11 sQ'(2a'(s) -l'(s»
where the product omits the square (1,1) (for which a'(s) == l'(s) == 0). Hence
w() == 0 if A::> (23).
2n( A') - n( A)
,.,). -
""(21"- 2) -
n(n - 1)
where n(A) == A2 + 2A3 + ... == EseAI'(s). (Same method as in (c): equate coeffi..
cients of Xn - 1 .)
(d)

2. THE GELFAND PAIR (52n, Hn)
411
(e) Let tp: AQ -. AQ be the automorphism defined by tp(p,) = tp, for all r  1.
Use Example 1 to show that Zen -1,1) is the image under tp of
2 n - 2 ( i( n _ 1, 1) + hi i( n - 1»
where ip. r:: h( /L)sp, for any partition J-L, and deduce that
w(n -1, 1) = (2n - l)mt( p) - n
p 2n(n - 1)
where mt( p) is the number of parts of p equal to 1:
(0 Show that
W(21"- 2) = w(l") (n + l)mI( p) - 2n
P P n(n - 1) ·
(Same method as in (e).)
3. Let
11
o =:  xD +  (x. -x.)-lxD.
n i.J, I i.J , J "
;- 1 ;Jf1j
where Di a: a/ax; (Laplace-Beltrami operator). Then
DnZ,\(Xl'...' xn) = (2n(A') - n(A»Z,\(x1,..., xn)
for partitions of A of length  n. (Chapter VI, 3, Example 3(e).) The coefficients
a,\p. in
Z,\ = E aAp.mp.
,.,,<,\
may be computed recursively in terms of a'\A by means of this equation (Chapter
VI, 14, Example 3(d». '
4. Let : A --. A  A be the comultiplication defined in Chapter I, 5, Example 25.
Show that
(I, g) = (/,g)
for all I, g e A, where the scalar product on the right is that defined loco cit.
5. Let X denote the set of all graphs with 2n vertices 1,2,..., 2n and n edges, one
through each vertex. If x, y e X, the connected components of xU yare cycles of
even lengths 2 PI ;;. 2 P2;;' ... . Let p(x, y) denote the partition ( PI' P2' ...) of n so
defined.
Show that the eigenvalues of the matrix
(pp(x,y» x,yeX
are the zonal polynomials Z,\ such that I AI = n, and that the multiplicity of ZA as an
eigenvalue is X2A(1)  (2n)!/h(2A).
J

412
VII ZONAL POLYNOMIALS
6. Let B denote the sign character of S2n, and let Bn denote its restriction to H
From 1, Example 11, the induced character B;2,. is multiplicity-free, so th
(S2n' Hn, Bn) is a twisted Gelfand pair, and
B;2,. = E ex2A = E X(2A)'.
I,\I-n I.\I-n
Hence the irreducible characters that appear in B;2" are those corresponding to
partitions of 2n with all columns of even length. The e-spherical functions are
7T A' = X(2 A)'e
where e = IHnl-1Bn is the idempotent corresponding to Bn.
(a) For each fE Cn = C(S2n, Hn) let fe be the function defined by fees) = e(s)!(s)
for s E S2no Then the mapping f...... fe is an isometric isomorphism of Cn onto
C: = Ce(S2n, Hn), the algebra of functions f on S2n that satisfy f(hs) = f(sh) =
B(h)f(s) for all s E S2n and h E H,.. Under this isomorphism, the zonal spherical
function fJ) A E C n is mapped to 'TT' x E C .
(b) Let
ce= EB C:
n;.O
and define a bilinear multiplication on Ce by the rule
f * g = e+n(fXg )e +n
if f E C:n and g E C:. Then the mapping f  fe defined above extends to a graded
C-algebra isomorphism of C onto ceo The functions cppB form a C-basis of CB, and
cp: * cp: = cp; u u for any two partitions p, u.
(c) Define an embedding w ...... w* (which is not a homomorphism) of Sn into S2n as
follows. For each cycle c = (it,..., i,) in the cycle-decomposition of w E Sn, let c*
denote the 2r-cycle (2i1 - 1, 2i}, 2i2 - 1, 2iz," ., 2i, - 1,2i,), and let w* be the
product of these c*. If w has cycle-type p, then w* has cycle-type 2p and
coset-type p, and B(W*) = (-l)/(p) = (-l)ne(w). For each fE C:, let f* be the
function on Sn defined by f*(w) = f(w*), which is a class function on Sn.
Now define a C-linear mapping
chB: CB  Ac
as follows: if IE C:, then ch8(f) = (-1)nIHnI2ch(f*), here ch is the characteris-
tic map of Chapter I, 7. We have che(cp/) = Bp2/(p)pp, from which it follows that
ch B is an isomorphism of graded C-algebras.
(d) Define a scalar product (u, v)' on Ac by
(pp' Pu)' = 8pu 2 -/( P)zp.
Then f ......IHnl-lche(f) is an isometry of C: onto AntC for each n  o.
7. In continuation of Example 6, define
Z = IHnl-1 che( 'TT' A)

2. THE GELFAND PAIR (S2n, Hn)
413
if A is a partition of n. Since
1T A(w*) = B(W*) wA'(w*) = (_l)n B wA'
P P
if w E Sn has cycle-type p, it follows that
(1)
Z' I I '" - 1 A'
A = Hn '--' BpZp wpPp
p
which is the image of ZA" under the automorphism of Ac which maps each power
sum p, to (-1),-12p" and hence Pp to Bp21(p)pp for each partition p. From
Chapter VI, (10.24) it follows that
(2)
Z = 2n J}1/2)
if A is a partition of n.
(a) From (1) and (2.15') it follows that
(Z, Z)' = h(2A') SAp,.
, (b) From Chapter VI, (10.24) we have
Z(1n) = n (2n + a'(s) - 2/'(s».
.r;E A
(c) From Chapter VI, (10.22) the coefficient of x A in Z (x) is
n (a(s) + 21(s) + 2).
sEA
(d) We have (compare (2.18»
n (1 -XiYj) -2 = E h(2A,)-1 Z(x)Z(Y).
i,j A
Notes and references
The zonal polynomials were introduced by L.-K. Hua [Hl1] and A. T.
James [J4] independently, around 1960. There is a large literature devoted
to them, mostly due to statisticians: see for example [F3], Chapters 12 and
13, and [M17]; also Takemura's monograph [T1] and the bibliography
there.
The proof of (2.23) presented here is due to J. Stembridge [S29]. The
formula (2.25) was first proved by A. G. Constantine [C4], and (2.26) is due
to A. T. James [J5]. Proposition (2.28) is due to N. Bergeron and A. Garsia
[B3 ].
Example 2. These formulas are due to P. Diaconis and E. Lander [D4].
Example 3. This method of calculating the zonal polynomials is due to
- A. T. James [J6].
Example 5. I learnt this combinatorial definition of the zonal polynomi-
als from R. Stanley.

414
VII ZONAL POLYNOMIALS
The zonal polynomials ZA have been calculated up to weight I AI = 12:
see [PI].
3. The Gelfand pair (GLn(R),O(n))
Let V = Rn, with standard basis of unit vectors el"'" en' and let G =
GL(V) = GLn(R), acting on V as follows:
n
oe. = "g..e.
. )  I)'
i-I
if g = (g;j)' Let
peG) = EB pm(G)
m;>O
be the space of all polynomial functions on G, as in Chapter I, Appendix
A, fiS: pm(G) is the subspace of peG) consisting of the polynomial
functions that are homogeneous of degree m. The group G acts on peG)
by the rule
(3.1) (gp)(x)=p(xg)
for p E P( G) and g, x E G, and under this action P( G) decomposes as a
direct sum
(3.2)
peG) = EB Pp,'
JJ.
where J.L ranges over all partitions of length  n, and Pp, is the subspace of
peG) spanned by the matrix coefficients RtJ of an irreducible polynomial
representation R p, of G corresponding to the irreducible G-module Fp,(V),
in the notation of Chapter I, Appendix A. Moreover we have
(3.3)
P ::: F (V)df£
p, p.
where dp. is the dimension of Fp.(V).
Now let K = O(n) be the compact subgroup of G consisting of the
orthogonal matrices, and let ex, e be the operators on peG) defined by
(eKP){x) = f.p(kx)dk,
K
(eKP)(x)= f.p(xk)dk
K
where the integration is with respect to normalized Haar measure on the

3. THE GELFAND PAIR (GLn(R),O(n»
415
compact group K. Clearly eK (but not e) commutes with the action (3.1)
of G. Since
eKRfj(x) = f Rfj(lcx) dk
K
= (fKRr,(k)dk)R,j(X)
it follows that eKPp' C Pp. for each J-L, and a similar calculation shows that
ePp' cPp..
Next, we have eKP(kx)=eKP(x) for all kEK, by the invariance of
Haar measure, and therefore
(ekP)(x) = f eKP(lcx) dk = f eKP(x)dk = eKP(x)
K K
which shows that eK is idempotent, and likewise ex is idempotent. More-
over, the two operators eK' ex are related by
(3.4)
'-
eK - TeKT
where T is the involution of peG) defined by Tp(X) = p(x'), where x' is
the transpose of the matrix x E G.
Let
P(K\G) = E9 pm(K\G)
m;;.O
(resp.
P(G/K) = E9 pm(G/K),
m;;.O
peG, K) = E9 pm(G, K))
m;;.O
denote the subspace of P(G) consisting of the functions constant on each
coset Kx (resp. each coset xK, each double coset KxK) of K in G. In each
case the superscript m denotes the subspace of functions that are homoge-
neous of degree m. Since -1 E K we have p(x) = p( -x) for p in any of
these spaces, and hence pm(K\ G), pm(G /K), and pm(G, K) are all zero
if m is odd. Also, by (3.1), pm(G /K) is the same thing as pm(G)K.
(3.5) (i) For each partition /-L, e K Pp. and e PJ.L are subspaces of PJ.L of the
same dimension.
(ii) eK (resp. ex) projects peG) onto P(K\ G) (resp. P(G/K)).
Proof. (i) From (3.4) it follows that eK and eK, are isomorphic projections,

416
VII ZONAL POLYNOMIALS
so that their restrictions to each PI-L have the same rank, i.e. dim(eKP ) =
dim(e:rcPI-L). . JJ.
(ii) If p E eKP(G) then clearly p(kx) = p(x) for all k E K and x E G.
Conversely, if p(kx) = p(x) for k E K and x E G, we have eKP = P and
hence p E eKP(G). Hence eKP(G) = P(K\ G), and likewise e'xP(G) =
P(G/K). I
Next we have
(3.6)
EndoP(K\ G) :::: peG, K).
Proof. The proof is best expressed in terms of the Schur algebra C?5 m =
Ends Tm(v) (Chapter I, Appendix A, 8, Example 6), with G-action
m
ga = Tm(g)a (g E G, a E @5m). The mapping a t-+ Pa defined by Pa(g) =
trace(ga) is a G-isomorphism of @)m onto pm(G), and since
trace(kga) = trace(Tm(k)Tm(g)a) = trace(gaTm(k»)
it follows that
(eKPa)(g) = f trace(kga)dk
K
= trace(gaeK)
where
BK = f Tm(k)dk E @Sm,
K
Hence eKPa = PasK' and therefore pm(K\ G) = eKpm(G) is the image of
6meK under the isomorphism a  Pa. Likewise pm(G/K) (resp.
pm(G, K» is the image of EK @5m (resp. eK @5meK). NoweK is idempotent,
and therefore
Enda pm(K\G) == EndG(meK) = End6m(@5mSK)
::: 8 K @5 me K ::: pm ( G , K). I
Consider now the double cosets of K in G. We have
(3.7) If x E G, there exists a diagonal matrix  = diag( 1'.." n) such that
x E K K, and the eigenvalues of x' x are l,..., n2.
Proof. The matrix x' x is symmetric and positive definite, hence can be
brought to diagonal form by conjugating with an orthogonal matrix, say
x' x = k' yk where k E K and y is a diagonal matrix whose diagonal entries
Yi are the eigenvalues of x' x, and hence are positive real numbers. So if gi
is a square root of Yi for each i, and  = diag( 1'...' n)' we have

3. THE GELFAND PAIR (GLn(R),O(n»
417
X'X=k'2k, which shows that k1 =xk-1g-1 is orthogonal and hence
X=k1kEKK. I
From (3.7) it follows that the polynomial functions p E p2m(G, K) are
determined by their values on the diagonal matrices, and that p(x) =
p * (  12, .. . , n2) where the / are the eigenvalues of x' x, and p * is a
symmetric polynomial in n variables, homogeneous of degree m. We
conclude that
(3.8) The mapping p ...... p* is an isomorphism of p2m(G, K) onto A';:,R for
each m  O. I
Next, let I denote the space of real symmetric n X n matrices, and let
P(I) = EB pm(I)
mO
denote the space of polynomial functions on I, where pmCI) consists of
the functions that are homogeneous of degree m. The group G acts on
P(I) by
(gp)(u) =p(g'ug)
for g E G, p E P(I), and u E I.
Also let T = T2m(v) be the Zmth tensor power of V. The group G acts
diagonally on T:
g(v1 ...  v2m) =gv1 ... gv2m
and the symmetric group S2m acts on T by permuting the factors:
w ( v 1  ...  V 2 m) = Vw -1 (1)  .. ·  v w - 1 (2 m)
where g E G, W E S2m, and v1,..., v2m E V.
Let (u,v) be the standard inner product on V for which (ei,ej) = Sij'
and let cp: T -+ pm(I) denote the linear mapping defined by
m
cp(v1  ...  v2m)(u) = n (V2i-1' UV2i).
i-I
Clearly 'P commutes with the actions of G on T and on pm(I), and since
(u, u v) = (uu, v) = (v, u u) it follows that cp(ht) = cp(t) for t E T and h
in the hyperoctahedral group H=Hm. Since (ei,uej)= Uij is the (i,j)
element of u E I, it follows that cp is surjective. Hence if
TH ::-;: {t E T: ht = t for all h E H}
is the subspace of T fixed by H, we have

418
VII ZONAL POLYNOMIALS
(3.9) The mapping cp restricted to TH is a G-isomorphism o.f TH onto
pm(I,). I
From Chapter I, Appendix A, 5, we know that T decomposes under the
action of S2m X G as follows:
(3.10)
T = e TI&
JJ.
where
TI& :: Mp.  Fp. (V).
Here IL ranges over the partitions of 2m of length  n, and Mil- is an
irreducible S2m-module corresponding to JL. Hence by (2.5) we have
(3.11)
TH:: e F2A(V)
A
as G-module, where ,\ ranges over the partitions of m of length  n.
If P E P(!'), the function p defined by
p(x) = p(x'x)
is a polynomial function on G, and the mapping p  p commutes with the
actions of G on P(I,) and P(G). Moreover, it doubles degrees: if p E
pm(!,) then p e p2m(G).
(3.12) The mapping p  p defined above is a G-isomorphism of pm(I) onto
p2m(K\ G), for each m  o.
Proof. Since k' k = 1 for k E K, we have p(kx) = p(x) for all k E K and
x E G, and hence p E p2m(K\ G). Next, each positive efinite matrix u
can be written in the form u=x'x for some x E G. Since the positive
definite matrices form a non-empty open subset I + of I, a polynomial
function p E pm(I,) that vanishes on I + must vanish identically, from
which it follows that the mapping p ..... p is injective.
Now from (3.9) and (3.11) we have
pm (I,) :: e F2A(V)
,\
where ,\ runs through the partitions of m of length  n; and since eK is a
G-homomorphism it follows from (3.3) that eKPp' is isomorphic to the
direct sum of say m p. copies of F,.JV), for each partition 1-£, so that
(1)
(2)
p2m(K\G) = e eKPp' =:. e Fp.(V)mll,
JJ. p.
where JL runs through the partitions of 2m of length  n. We have

3. THE GELFAND PAIR (GLII(R),O(n»
419
already shown that pm(!,) is isomorphic to a submodule of p2m(K\ G),
and hence from (1) and (2) we have mp.;> 1 if J.L is even (i.e. J.L = 2A).
From (2) we have
E m; = dimEndG p2m(K\ G)
p,
= dim p2m(G, K) by (3.6),
which by (3.8) is equal to the number of partitions A of m of length  n. It
follows that mIL = 1 if J.L = 2A is even, and mp = 0 otherwise. Hence (1)
and (2) show that pm(!,) and p2m(K\ G) have the same dimension, and
the proof is complete. I
Remark. Let vt,..., vn E V be the columns of the matrix x E G, so that
the (i, j) element of x' x is the scalar product < Vi' vj) = v; vj of Vi and Vj.
Then (3.12) says that any polynomial function f(vI,..., vn) such that
f(kvl,..., kvn) = f(vl,..., vn) for all k E K-that is to say, any polynomial
invariant of n vectors VI'...' VII under the orthogonal group-is a polyno-
mial in the scalar products < Vi' vj). In this form it is part of the 'first main
theorem' of invariant theory for the orthogonal group ([W2], Chapter II,
9).
As a corollary of the proof of (3.12) we record
(3.13) (i) The G-module eKPp, is isomorphic to Fp,(V) if J.L is even, and is zero
otherwise.
(ii) We have
eKP(G) =P(K\G) = Ea F2A(V)
A
where A ranges over all partitions of length  n. I
Hence the G-module eKP(G) is multiplicity-free. The situation is quite
analogous to that of 1, except that the groups G, K involved are now
infinite, and the space of all functions on a finite group is replaced by the
space of polynomial functions on G. Thus (G, K) may be called a Gelfand
pair.
(3.14) Let J.L be a partition of length  n. Then
dim F (V)K = { 1 if J.L is even,
p. 0 otherwise.
Proof. From (3.3) we have
(1) dim PJl.K = dp, dim Fp,(V)K

420
VII ZONAL POLYNOMIALS
and on the other hand, from (3.5) and (3.13),
dim PILK = dim e PIL = dim e K Pp.
(2)
= {
if J..L is even,
othelWise.
The result follows from (1) and (2).
For each partition J..L of length  n, let
Pp.(G,K) =Pp. np(G,K) = (eKPp.)K.
(3.15) Pp,(G, K) IS one-dimensional if JL is even, and is zero othelWise.
Moreover
P(G,K) = EB P2A(G,K)
A
summed over all partitions A of length  n.
This follows from (3.13) and (3.14). I
If fEAR we may regard I as a function on G:
I(x) = 1(1'"'' n)
where 1"'.' gn are the eigenvalues of x E G. For example,
(3.16)
p,(x) = trace(x')
(r  1)
since the eigenvalues of x' are /, and
(3.17)
e,(x) = trace( A'x)
(r  1).
Since each IE AR is a polynomial in the e" it follows that f is a
polynomial function on G that satisfies
(3.18)
I(xy) = f(yx)
for all x, y E G (because xy and yx = y(.ty )y-l have the same eigenvalues).
In particular (Chapter I, Appendix A, 8), the character of the irre-
ducible representation R IJ. of G is the Schur function sp,:
s =Rf!-EP
IJ.  H IL
i
as a function on G. From (3.13) we have eKsp' = 0 unless J..L is an even
partition:
(3.19)
f s/xk)dk = 0
K

3. THE GELFAND PAIR (GLn(R),O(n))
421
for all x E G, unless J..L is an even partition.
Let llA = eKs2A, or explicitly
(3.20)
fiA(x) = jS2A(xk)dk= jS2A(kx)dk
K K
for any partition A of length  n.
(3.21) (i) II A (1) = 1.
(ii) ll,\ E P2A(G, K) and generates the i"educible G-submodule eKP2A of
P(K\ G).
Moreover, llA is characterized by these two properties.
Proof. (i) We have
fiA(1) = j suCk) dk = dim F2A(V)K = 1
K
by (3.14).
(ii) Since S2A E P2A, it is clear from (3.20) that llA E P2A n peG, K) =
P2A(G, K); and since llA =1= 0 (by (i) above) it follows that llA generates the
irreducible G-module eKP2A.
Conversely, any f E P2A(G, K) must be a scalar multiple of llA' by (3.15);
if also f(l) = 1, then f= llA. I
The function llA is the zonal spherical function of (G, K) associated with
the partition A.
We return now to the S2m X G-module T = T2m(v), where m  n =
dim V. From (3.10) and (3.14) we have -
TK = EB T = EB M2A
A A
as S2m-module, where A ranges over all partitions of m. From (2.4) it
follows that TK is isomorphic to the induced module 1m. Now the tensor
n
5= "e.e.
i...J I I
;= 1
is fixed by K, and hence m = 5  . ..  5 (m factors) lies in TK. Since the
subgroup of S2m that fixes m is precisely H, it follows that m
generates TK as S2m-module. Hence by (1.5) the space
THXK - (TK )H - (TH )K
2A - 2A - 2A

422
VII ZONAL POLYNOMIALS
is one-dimensional, generated by
(3.22)
lJ)m= E wA(w).wsem.
weS2m
We need now to compute the image of this element of TH under the
isomorphism cp: TH -+ pmCI) (3.9). This is given by
(3.23)
cp(wsem) = Pp
where p is the coset-type ofw.
Proof. Let fw == 'P(wsem). For any h E H we have fwh = fw' since hE/Jm =
sem; also fhw = fw' since cp(ht) = 'P(t) for all t E T. Hence fw depends only
on the double coset Hp to which w belongs. By taking w to be the product
of consecutive cycles of lengths 2 PI' 2 P2' . . . ,we reduce to the case where
w is the 2m-cycle (1,2,..., 2m). Since
[j8m = E e i  e i ...  e i  e i ,
1 1 '" '"
it follows that
w[j8m = E ej  e;  ei ...  e;  ei
'" 1 1 ",-I '"
and hence that
f w (U ) = E Ui i Ui; ... Ui i
m112 m-lm
= trace(um) = Pm(U).
We can now express the zonal spherical function 0,.\ in terms of the
zonal polynomial Z.\ defined in 2:
(3.24) We have
o.\(x) = ZA(x'x)/ZA(ln)
for all x E G.
Proof. Let cp.\ = 'P( lJ) .\[j8m), and define CpA E P( G) by CpA (X) = CPA (X' X). It
follows from (3.12) that .\ is a generator of the one-dimensional space
P2A(G, K) that is the image of TXK under the composite isomorphism
TH -+pM(I) -+p2m(K\G).
Hence, by (3.21), 0.\ is a scalar multiple of A' namely
(1)
n.\ (x) = CPA (x ' X ) / 'PA (1 n )

3. THE GELFAND PAIR (GL,.(R),O(n»
423
since 0A(I) = 1. On the other hand, it follows from (3.22) and (3.23) that
(2) ({)A = E IHplwpp = IHmlZA
p
by (2.3) and (2.12). The result now follows from (1) and (2).
Examples
1. (a) Let A. be a partition of length n and let J.L = (AI - 1, . . . , An - 1). If X E G we
have
0A(X) = f S2A(xk) dk
K
= f (detxk)2s2(xk)dk
K
= (det x)20J.£(x)
and hence if  = diag(  l' . . . , n) is a diagonal matrix we have
(1)
ZA() ZJ.£()
Z.\. (1 n ) =  1 .. · n Zit (1 n) ·
(b) Let aA denote the coefficient of A in ZA(). From (1) above we have
aA/a = ZA(1n)/Zp.(1n) = cA(n) /c(n)
by (2.25). If s is the square (i, Ai) at the right-hand end of the ith row of A, and t is
the square (i,l) at the left-hand end of the same row, we have a'(s) = a(t) and
n -/'(s) = /(t) + 1, from which it follows that
cA(n) /cp.(n) = h1(2,\) /hl(J.L)
and hence, by induction on I AI, that a A = ht(2 A) (2.26).
2. Show that if t E R and x E G,
tl2A1
f exp(t trace xk) dk = E h( ) O.\.(x).
K A 2A
(We have
m!
p'{' = E h( ) sit
1p.I-m J.L
(Chapter I, 7), hence
1. m  m! 1.
(trace xk) dk = i..J h() s(xk) dk
K p. J.L K
(1)
summed over the partitions J.L of m. The integral on the right is zero unless J.L = 2 A

424
VII ZONAL POLYNOMIALS
for some partition '\, hence the integral (1) is zero if m is odd; and if m = 2r is
even it is equal to
(2r) !
E h(ZA) O.\(x).)
IAI-r
3. For each partition JL of length  n, the irreducible G-module FJJ.(V) has a basis
indexed by the column-strict tableaux T of shape J.L, such that relative to this basis
a diagonal matrix x == diag(x1,..., xn) EGis represented by a diagonal matrix
R JJ.(x) with diagonal entries xT (Chapter I, Appendix A, 8, Example 2). Hence
sJJ.(xk) = trace R JJ.(x)R JJ.(k)
== E xTRf,T(k)
T
and therefore
0A(X) = f. s2,\(xk) dk
K
= E xTf. R}'\T(k) dk
T K '
summed over the column-strict tableaux T of shape 2,\ that can be formed with
the numbers 1,2,..., n. Each monomial xT in this sum belongs to a monomial
symmetric function mv(x l' . . . , X n)' where v  2'\, and hence 0,\ (x) is a linear
combination of these, since it is symmetric in Xl'...' xn. But O,\(x) is unaltered by
replacing any Xi by -Xi' and hence is a symmetric polynomial in xt,..., x:.
Consequently only the mv such that v is even, say v = 2J.L, will occur, and so O,\(x)
is of the form
0A(X) = E a'\JJ.m,.,.(xf,...,x:),
p.<.,A
since 2 J.L  2,\ if and only if JL  '\. This gives an independent proof of (2.22) and
hence of (2.23).
Notes and references
The material in this section is due to A. T. James [J4].
4. Integral formulas
(4.1) Let f e P(G), f =#= O. Then f is a zonal spherical function of (G, K) if
and only if
1. f(xky) dk = f(x)f(y)
K
for all x, y E G.

4. INTEGRAL FORMULAS
425
Proof. Let ,\ be a partition of length  n, and let
cpyCx) = f 'o>.(xky)dk
K
= f f s2A(xkykt) dkt dk
K K
by (3.16). Since S2A' as a function on G, lies in P2A it is clear that
'Py E P2A n peG, K) = P2A(G, K), and hence by (3.15) and (3.17) it follows
that 'Py = cOA for some c E R. Since 0A(1) = 1, we have
c=cpyCl)= f ,O>.(ky)dk=,O/y)
K
and hence
(1)
fK ,O>.(xky) dk = 'o>.(y)'o>.(x)
for all x, y E G.
Conversely, suppose that fE peG) is such that f::l= 0 and
f f(xky) dk = f(x)f(y)
K
for all x, y E G. By choosing y such that f(y) ::1= 0 we see that f(xk1) = f{x}
for all x E G and k1 E K, and likewise that f(k1y) = f(y) for all y E G and
k1 EKe Hence feP(G,K), say
f= E aAflA
A
by (3.15) and (3.21). We have then
f(x)f(y)= ff(xky)dk= Ea>.f 'o/xky)dk
K A K
and hence by (1)
E aAOA(x)f(y) = E aAflA(x)OA(Y)'
A
Since the 0" are linearly independent, it follows that aA(f - 0A) = 0 for all
A. Since not all the a A are zero, we conclude that f = .0 A for some A. I
(4.2) Let A be a partition of length  n. Then
ZA(U)ZA(T)
f Z>.(ukTk') dk = Z (1 )
K A n

426
VII ZONAL POL-YNOMIALS
for all u, T E I.
Proof. Both sides of (4.2) are polynomial functions of u and T, hence we
may assume that u and T are positive definite, say u = x' x and 'T = })".
Then UkTk' =x'xkyy'k', and hence by (3.18) and (3.24)
ZA(ukTk') = Z,\(ln)O,\ (xky).
(4.2) now follows from (4.1). I
Let dx = Oi,j-t dX;j be Lebesgue measure on G = GLn(R), and let
d v(x) = (27T) -n2/2 exp( - t trace x' x) dx,
the 'normal distribution' of the statisticians. Since trace x' x = Ei,jxt and
00
f e-/2/2 dt = (2'77' )1/2,
-00
it follows that
f dv{x) = 1.
G
The measure v is K-invariant, that is to say
dv(k]xk2) = dv(x)
for k], k2 E K. For if y = k]xk2 we have trace y'y = trace x' x and dy = dx.
(4.3) For each integer m > 0 and all u, T E I we have
f m Z.\(u )Z,\(T)
(trace UXTX') dv{x) = 2mm! E h{2A)
G IAI-m
where h(2 A) is the product of the hook-lengths of the partition 2 A.
Proof. We may write
u= k; gk],
T=k17k2
where g = diag( 1' ..., gn) and 17 = diag( 171" .., 17n) are diagonal matrices,
and k1, k2 E K. Then
trace UXTX' = trace k; klXk217k2X'
= trace Y17Y'
where y = k] xk2, and hence by K-invariance of v
f (trace IT XTX,)m d vex) = f (trace Xl1X,)m d vex)
G G

4. INTEGRAL FORMULAS
427
which is the coefficient of tmjm! in
(1)
1 exp(t trace gX17X') dv(x).
G
Now
t trace gXT/X' - i trace x'x = - t E (1 - 2tiT/j)xt.
i, j
Hence the integral (1) is equal to
n (1 - 2tiT/j) -1/2
i, j
which by (2.18) is equal to
(2) E (2t)IAIZ..\(u )ZA(T )jh(2A)
A
since Z).(U)=ZA(g) and ZA(T)=ZA(Tj). The result now follows from
equating the coefficients of tm in (1) and (2). I
(4.4) Let U, 'T E I. Then
1 Z,,(UXTX') dv(x) = Z,,(u )Z,,( d
G
for all partitions A of length  n.
Proof. Let IA(u, T) denote the integral on the left. Then we have
[,,(u,d = fx(fa Z,,(UkxTX'k')dV(X)) dk
= fa(fx Z,,(UkxTX'k')dk) dv(x)
_1 ZA(U )ZA(XTX')
- Z () dv(x)
G A In
;:y (4.2). Hence
{:(t)
ZA(U)
[,,(u, 'T) = Z,,(1I1) [,,(111' 'T)
Z..\(U)ZA(T)
- 2 IA (1 n ,In)
ZA(ln)
.,-",".1
:;\by repeating the argument.

428
VII ZONAL POLYNOMIALS
On the other hand we have from (2.16)
p;n = Zmm! . E h(2A)-1 Z,\
IAI-m
and therefore
j, (trace u XTX ,)m d v(x) = 2mm! E h(2A) -1 [>.( u, T)
G IAI-m
_ m , E Z>.(u )Z/,r) [>.Un, In)
-2 m'I>.I_m h(2A) · Z>.Un)2 ·
It now follows from (4.3) that
1,\ ( 1 n , 1 n) = Z,\ (1 n ) 2
and hence from (1) that I>..(u, T) = Z,\(u )Z,\(T).
By taking T = In in (4.4) we have
( 4.5)
fG z>.(x'ux) dv(x) = Z>.(1n)Z>.(u).
In other words, the Z,\ are eigenfunctions of the linear operator Un: An 
A n defined by
(Unf)(u) = j, f(x'ux)dv(x)
G
and the eigenvalue of Un corresponding to Z,\ is Z,\(1n).
(4.6)
As in Chapter I, Appendix A, 8, Example 5, for each partition J.L of
length  n let IL be the polynomial function on G defined by
IL(X) = n det(x(I-"t»)
,  1
for x = (Xij) E G, where xCr) = (xij)] < ;,j < r for 1  r  n. Then (loc. cit.)
the G-submodule ElL of peG) generated by Ill-' is irreducible and isomor-
phic to F",(V). We have
( 4 . 7)  IL (xb) =  I-' ( b ' x) = Ill-' ( b )  IL (x)
if bEG is upper triangular.
Now let B+ be the subgroup of G consisting of the upper triangular
matrices (bij) with positive diagonal elements bu. If x E G, Gram-Schmidt

4. INTEGRAL FORMULAS
429
orthogonalization applied to the successive columns of the matrix x shows
that
(4.8)
x=kb
with k E K and b E B+; moreover, this factorization of x is unique, since
KnB+={1}.
(4.9) We have
eK2'\(x) = c,\,\(x'x)
where c,\ is a positive constant, independent of x E G.
Proof. If x = kb as above then
eKI12). (kb) = f 112.\ (k] kb) dk]
K
= 112.\ (b) f 112.\ (k]) dk]
K
-c  (b'b) -c  (x'x)
- ,\,\ - ,\ ,\
by use of (4.7), where
c).= f 112.\(k)dk
K
is positive, since 2'\ =  is  0 on G, and is > 0 on a dense open subset
of K. I
(4.10) We have
f 11).(k' uk) dk = Z).(u )/Z).(ln)
K
for uE I.
Proof. We may assume that u is positive definite, say u=x'x (x E G),
since both sides of (4.10) are polynomial functions on I. From (4.9) it
follows that the function x  ,\ (x' x) generates the irreducible G-module
P2,\(K\ G) =::.. F2,\(V), and hence that
f 11). (k' x'xk) dk = .0). (x)
K
= Z,\(x'x)/Z,\(1,,)
by (3.24). I

430
VII ZONAL POLYNOMIALS
(4.11) We have
1 .\(x'ux)dv(x) =Z.\(u)
G
for aE I.
Proof. Since the measure JJ is K-invariant we have
1 .\(x'ux)dv(x) = f (1 .\(k'X'Uxk)dV(X») dk
G K G
= fG(fK .\(k'X'UXk)dk) dv(x)
=Z.\(ln)-l fG Z.\(x'ux)dv(x)
= ZA(U)
by Fubini's theorem, (4.10), and (4.5).
In particular (a = In)
(4.12)
Z.\(1n) = 1 .\(x'x)dv(x).
G
We shall now calculate this integral directly, using the factorization
x = kb (4.8). Let
db = n db..
.. 'J
l<}
be Lebesgue measure on B +, and let
d v(b) = (21T) - n(n + ))/4 exp( - t trace b'b) db.
The measures dx, dk, db on G, K, B+ respectively are related by
dx = cnJ18(b) dk db
where  = (n - 1, n - 2,...,1,0) and cn is a positive real number depend-
ing only on n (see [B7], Chapter VII, 3, or Example 2 below). Since
x = kb we have x I x = b' b and hence
(4.13)
d vex) = CJ18 (b) dk d v(b)
where c = (21T )-n(n-l)/4cn.

4. INTEGRAL FORMULAS
431
From (4.7), (4.12), and (4.13) it follows that
ZA (In) = cf. d). (b)2 dl) (b) d v(b)
B+
n CIO
= c' 0 (21T) -1/21 b)..+n -ie-bn/2 db..
n II II
i-I 0
(4.14)
11
== (21T) -n /2C 0 2A.+(n -i-l)/2r( Ai + t(n - i + 1».
i-I
Since Zo = 1 we obtain
ZA(ln) 11 f(A; + t(n - i + 1»
Z (1 ) = = 02A.
A n Zo(ln) ;-1 f( ten - i + 1»
n Al
=21A10 0(t(n-i+1)+j-1)
i-I j-I
or equivalently
(4.15)
ZA(ln) = 0 (n + 2a'(s) -l'(s» = cA(n)
seA
in agreement with (2.25).
Examples
1. If k = (kij) e K, let
n
wij(k) = E kri dkrj,
r-1
a differential 1-form on K.
(a) Since Erkrikrj .. 8;j' it follows that Wij + Wj; z: O.
'(b) If a is a fIXed element of K then wij(ak) = wij(k), i.e. the ClJij are left-invariant
'differential forms on K.
(c) Hence
"',.
.. .
WK= A Wij
;>j
lis a left-invariant differential form of degree !n(n - 1) == dim K (defined only up to
.ign, because we have not specified the order of the factors in the exterior
:product); moreover it is not identically zero, since ClJK(1n) = Ai> j dk;j. Hence (see
:..g. [D5l (16.24.1» WK determines a Haar measure d*k on K. Since Haar measure
-'8 unique up to a positive scalar multiple, it follows that
d*k = c dk
n

432
VII ZONAL POLYNOMIALS
for some cn > 0 depending only on n. The constant cn is calculated in Example
2(d) below.
2. (a) For x = (xij) E G let
WG(X) = 1\ dXij
i ,j
be the differential form on G (defined only up to sign, as in Example 1) that
corresponds to Lebesgue measure dx. If a E G we have
cuG(ax) = J: cuG(xa) = J: (det a)n cuG(x),
(b) For b = (bij) E B+, let
Wn+ (b) = 1\ dbij
i<j
be the differential form on B + that corresponds to Lebesgue measure db. If
a E B + we have
CUB + (ba) == J:(deta)nc5(a)-lWB+(b),
CUn+ (ab) = J: (det a)8 (a) CUB + (b).
(c) From the factorization x = kb (4.8) we have
(dx) = (dk)b + k(db)
where (dx) denotes the matrix of differentials (dxij), and likewise for (dk) and
(db). Hence
k'(dx)b-1 =k'(dk) + (db)b-1,
in which the matrix k'(dk) = (wij) (Example 1) is skew-symmetric, and the matrix
(db)b-1 is upper triangular. Deduce from (a) and (b) above that
wG = :t: (JJK A c5 cuB +
and hence that
dx = d*k. 8(b) db
= Cndc5(b) dk db.
(d) Deduce from (4.14) (with A = 0) that
Cn=VtV2'.'Vn
where vr == 21Tr/2 fr(t,) (which is the 'area' of the unit sphere in Rr: vt = 2,
V2 = 21f, v3 = 41T,...).
3. Let A be the group of diagonal matrices  = diag( 1'.'" n) with diagonal
elements i > 0, and let I + C I be the cone of positive definite symmetric
matrices. Orthogonal reduction of a real symmetric matrix shows that each u E I +
can be written in the form
(1)
u=kk'

4. INTEGRAL FORMULAS
433
with k E K and  EA. The i are the eigenvalues of u, and hence the diagonal
matrix  is unique up to conjugation by a permutation matrix. Moreover, if the gi
are all distinct, the columns of K are unique up to sign. From this it follows that
the fibre over u E I + of the mapping K X A -. I + defined by (1) is a finite set of
cardinality 2nn! = \Hn\ whenever the discriminant of u does not vanish, hence
almost everywhere on I +.
From (1) we have
(du) = (dk)k' + k(d )k' + k(dk)'
where (du) denotes the matrix of differentials (dUij)' and likewise for (dk), (d).
Hence
k'(du)k = k'(dk) + (d) + (dk)'k,
the (i, j) element of which is the differential I-form
w.. l:, + 8.. d l:, + l:. W.. = ( c. - l:.) w.. + 8.. d c.
'l ) I) I I)I l , )1 I)'
in the notation of Example 1. Hence if
w+= A duij,
i<j
n
WA = A di
i-I
(defined only up to sign, as in Example 2) we have
(2)
WI,+= :t Wx 1\ 08 WA
where as( ) = rI; <j( i - j)' It follows that
1
du= 2n.n! la6()ldk*d
(3)
Cn
= 2n.n! las()ldkd.
4. Let m  n and let X = Mmtn(R) be the vector space of all real matrices x = (Xij)
with m rows and n columns. Also let Im (resp. In) denote the space of real
symmetric matrices of size m X m (resp. n X n).
If x, Y E X we have trace(xy ') = trace(y' x), and more generally
e,(xy') = trace Ar(xy') = trace Ar(x) A'(y')
= traceA'(y') Ar(x) = e,(y'x)
for all r  1, so that
(1)
f(xy') = f(y'x)
for all fE A and x,y EX.
Now let dx = nitj dXij be Lebesgue measure on X, and let
dv(x) = (21T) -mn/2 exp( - t trace x'x) dx
be the normal distribution. Let u E Im and TE In. Then

434
VII ZONAL POLYNOMIALS
(a) We have
f r. Z,\(u )Z,\(T)
(trace u xu') dv(x) = 2',! E ()
x IAI-r h 2A
(same proof as (4.3».
(b) We have
f Z.\(uxTx')dv(x) ==Z,\(U)Z,\(T)
x
if I(A) < n. (Same proof as (4.4).) In particular,
f ZA(Xt') dv(x) == f Z,\(x'x) dv(x) == Z,\(lm)Z,\(ln).
x x
(c) Let Um,,,: Am  Am be the operator defined by
(Um IIf)(u) as f /(x'ux)dv(x)
, x
where u E Im. Then
Um,,,Z,\ :II: Z,\(ln)Z,\
for all partitions A of length  n. (Set 'T == In in (b) above.) Hence Pm,,, 0 Um,,, =
Un 0 Pm,II' where Pm,,,: Am .... An is the restriction homomorphism (Chapter I, fi2).
5. Let x, Y E G. Then
(a) i 0A(XZY) dv(z) == O,\(x)O,\(y)Z,\(ln)
G
if leA)  n. (Use (3.24), (4.1), and (4.5).)
(b) i s2" (xzy) d v(z) :a O,\(yx)Z,,(l,,)
G
if l( A) < n. (Use (3.20) and (a) above.)
6. Let J.L be a positive K-invariant measure on the cone I + of positive definite
symmetric n X n matrices, such that
f. f(u)dJ,L(u)<oo
I+
for all f E All. Define an operator E: An .... An by
(Ef)( u) == f. I( UT) dp,( T).
I+
Then each ZA such that leA)  n is an eigenfunction of E, with eigenvalue
ZA(1,,)-1f. Z,\(T)dJ,L(T).
I+

4. INTEGRAL FORMULAS
435
(Use (4.2).)
7. (a) Let dU:all: n; < j dU;j be Lebesgue measure on I. If x E G, the Jacobian of
the linear transformation u..... x' U x is the symmetric square S2(x) of x. If the
eigenvalues of x are 1"'" n' those of S2(x) are ; j (1 <e i <.j <. n), whence
det(S2x) == (det X)"+l. Hence
d(x'ux) = Idet XI"+l du
and therefore
d*u = (det u) -(n+ 1)/2 du
is a G-invariant measure on I (and on I +).
(b) Let m  n and let X = Mm,n(R), as in Example 3. The group G acts on X by
right multiplication, and we have
d(xg) = Idet glm dx
if g E G. Hence
d*x == (det x' x) -m/2 dx
is a G-invariant measure on X.
(c) Hence the linear functional
f.-+ f f(x'x) d*x
x
defines a G-invariant measure on I +. Since G-invariant measures on homoge-
neous spaces (when they exist) are unique up to a positive scalar multiple, it follows
from (a) that
(1)
ff(x'x)d*x=cm n ( f(u)d*u
x ' J1 + -
for some real number cm,n > 0 independent of f.
If we take
f( u) == (21T) -mn/2 exp( - t trace u )(det u )m/2 g( u)
in (1), we shall obtain
(2)
f g (x' x) d v (x) = ( g (u ) d JLm n ( u )
X )1+'
where
d#Lm,n(U) == c:",n exp( - t trace u )(det u )(m-n-O/2 du
and c' == (21T )-mn /2c .
m,n m,n
(d) Each matrix u E I + is uniquely of the form u =: b'b, with b E B+, and the
mapping b..... u is a diffeomorphism of B + onto I +. The Jacobian matrix

436
VII ZONAL POLYNOMIALS
(aO'ijj abk1), with rows and columns arranged in lexicographical order, is lower
triangular, so that its determinant is eq1Jal to
n (au. .jab..) = n (1 + 8. .)b..
') '} 'J " ·
i<j i<j
Hence
(3)
du == 2n(det b )!1s(b) db.
(e) By taking t(u) = exp( -", trace u) (det u)m in (1) and using (3), deduce that
n
- 2-n n
cmtn - Vm-i+1
i-I
where vr == 2", r /2 jf( tr) as in Example 2(d).
8. In this example and the following we shall use the abbreviations
etr( u ) = exp( trace u ),
I u 1 == det u
for a square (not necessarily symmetric) matrix u.
Let t == ta,'\ be the function on I + defined by
t(u) == 'uloZ.\(u)
where a  tn and 1(,\)  n.
(a) Show that
f. etr( - u )t(u) d*u == Z.\(1n)fn(a, A)
1+
where d*u == lul-(1I+1)/2 du as in Example 7, and
fn(a, A) == f. etr( - (J")I ulod,\,( u) d*u
1+
n
== ",n(n-1)/4 n r(a + A - t(i - 1».
i-I
(Use (4.10) and Example 7 above.)
(b) Let T E I + and let
g(T) == f. etr( -uT)f(u)d*u.
1+
Show that
g(T) == fn(a, A)f(T-1).
(Let T 1/2 be the positive definite square root of T. Under the change of variables
(J"'-' T1/2UT1/2 we obtain
g( T ) == 'TI-a f. etr( - u)1 u laZ.\ (UT-1) d*u.
1+
Now replace u by kuk'(keK), integrate over K and use (4.2) and (a) above.)

4. INTEGRAL FORMULAS
437
This calculation shows that the Laplace transform of the function F is
.] )4-(n+l)/2tA
the function T...... rn(a, A.) fat A (T- ).
(C) More generally, if p, T E I + show that
( etr( - pu )lulaZ.\(UT) d*u = rn(a, A)I pl-aZA( p-1'T).
JI,+
9. (a) Let a, b ;> tn and let A be a partition of length  n. Show that
J a-p b-p rn(a,A.)rn(b,O)
Z.\(UT )Iul 11n - ul du== r ( b) ZA(T)
n a+ ,A
(I)
for T e I +, where p == t(n + 1) and the integral is taken over I + n (In - I +).
(Let f(T) denote the left-hand side of (1). By replacing u by kuk', where
k e K, and then integrating over K and using (4.2), it follows that
(2)
f( T) = f(l,,)ZA (T).
Next, from Example 8(a) we have
(3) rn(a, A.)rn(b,O)ZA(ln) == J etr( -u- T)lula-PITlb-PZA(u)du dT
integrated over I +x I +. Let
Tt==U+'T,
Ut == 'Tt 1/2u'T} 1/2 ,
where TI/2 is the positive definite square root of 'Tt; then lul == lutll'T}1 and
I'TI = 11 - u11IT]I, and du]dT] = 1'T]I-pdudT. Hence the integral (3) is equal to
( etr( - Tl)ITjla+b-p!(T)) dT]
JI+
which by (2) above and Example 8(a) is equal to rn(a + b, A.)ZA(ln)!A(ln). Hence
rn(a, A.)rn(b,O)
[A(1n) = fn(a + b, ,\)
hich together with (2) completes the proof.)
.(b) Show that
J Z.\()IIU-Plln - lb-Pla8()ld
2nn! rn(a,A.)rn(b,O)
== - ZA(ln)
cn rll(a + b, A)
.where the integral is taken over  == diag(  l' . . . , n) such that °  i  1 for
:1  i  n. (Use (a) above and Example 3.)
{. This integral is the case a = 2 (i.e. k = t) of the Selberg-type integral of Chapter
;VI, 10, Example 7.

438
VII ZONAL POLYNOMIALS
10. Let A be a partition of length  n, and let m > n + 2Al. Then in the notation
of Example 7(c) we have
f. -1 Z,\(-T)
Z,\ (u 'T) d ILm n (u ) :m: ( )
1+ 'c,\ n-m+1
for T e I, where c,\ is the polynomial defined in (2.24) or (4.15). (Let f(T) denote
the integral above. By replacing u by kuk', where k e K, and then integrating
over K, it follows from (4.2) that f( 'T) = cZ,\( T), where
( Z,\(u-1) (-1
C = JI,+ Z,,(1,,) d#£m,,,(u) = JI,+ A,\(u ) d#£m,,,(u)
by (4.10). Evaluate the latter integral by setting u = bb', where b E B+.)
11. Let x and  be elements of G, where  zz: diag(  l' . . . , n) is a diagonal matrix.
Then the coefficient of f1 ... r in det(x' fx)<r) is (det x<r»2, and hence if I(A)  n
the coefficient of f'\ in f:.,\(x' fx) is f:.,\(X)2 = f:.2,\(x). From (4.10) and (2.26) it
follows that
i 42,\(X) dv(x) = h1(2A),
G
the coefficient of ,\ in Z,\( f).
Deduce that
JK A2,\(k) dk = h1(2A) /Z,\(1,,).
(Write x = kb with k E K and be B+, and use (4.13) and (4.14).)
12. Let A be a partition of length  n. Show that
h1(2 A)Z,\(x 'x)
f. 42,\ (kt xk2) dkt dk2 = 2'
Kx K Z,\(ln)
(a)
(Let f,\(x) denote the integral on the left. Clearly f'\ E peG, K), hence is a linear
combination of zonal polynomials, and we may assume that x =  = diag( 1'. · · , n)
is a diagonal matrix. Now the leading term in det(k1 fk2)(r), as a polynomial in the
i' is (det kr)Xdet kY»f1 ... fr' and hence the leading term in 42,\(kt fk2) as a
polynomial in 1' "" n is 42,\(kt)42,\(k2) 2'\. Hence f,\(x) is a scalar multiple of
Z,\(x'x), and the scalar multiple is given by Example 11.)
(b) Show that
h1(2A)Z,\(x'x)
f. Au(kxy) dk dv(y) = Z (1) ,
KxG ,\ n
i 42,\(yxz) dv(y) dv(z) = ht(2A)Z,\(x'x).
GxG
(U se (a) above and (4.5).)

4. INTEGRAL FORMULAS
439
13. Let Xl'''.' X, e G. Then
r
1 2A(YOXIYIX2... Y'-lX,y,) dv(yo)... dv(y,) = h1(2A) n ZA(XXi)
Gr+1 i-I
if l( A)  n, where Gr+ t = G X ... X G to r + 1 factors. (Induction on r: the case
r = 1 is Example 12(b). If we denote the integral above by tA(Xt,..., x,) we have
fA ( Xl' .. · , X, + t) = 1 fA (X t yx 2 , X 3 , · .. , x, + 1 ) d v (y )
G
if r  1, and the inductive step is completed by (4.5).)
14. Let f..£ be a positive measure on G such that df..£(kt) = d,u(xk) = df..£(x) for all
k eK, and such that
1 t(x) df..£(x) < 00
G
for all polynomial functions t on G. If l(A)  n, show that
1 A(x'ux)df..£(x) = u,\Z,\(u)
G
for all u E I, where
UA = Z,\(ln) -1 1 o,\(x) df..£(x).
G
(Use (4.10) and (4.2).)
15. Let ,ul' f..£2 be positive measures on G satisfying the conditions of Example 14.
Show that
1 2A (yxz) dl-Ll(y) df..£2(z) = VA 0,\ (x),
GxG
where VA = ht(2A),ut(OA)JL2(0,\)/Z,\(ln). (Replace y by k1yk3 and z by k4zk2,
where k1,..., k4 E K; integrate first with respect to k1 and k2, and then with
respect to k3 and k4, using Example 12(a) and (4.1).)
Notes and references
Most of the integral formulas in this section are to be found in Takemura's
monograph [Tl]. In particular, Proposition (4.1) is a classical criterion for
zonal spherical functions, due originally to Gelfand [G7], and (4.2), (4.3)
are due to James [J5]. Takemura (loc. cit.) essentially takes (4.6) as his
definition of the zonal polynomials (the eigenvalues ZA(1n), where A runs
through the partitions of a fixed number r, will be all distinct provided n IS
large enough). Proposition (4.10) is a particular case of a well-known
formula of Harish-Chandra (see, for example [H6], p. 418), and (4.11) is
due to Kates [K7] (see [T1], p. 34). Example 7 deals with the 'Wishart
density' (see e.g. [F3], p. 53), and Example 8 is due to Constantine [C4].

440
VII ZONAL POLYNOMIALS
5. The complex case
In this section let G = GLn(C) and let K be the unitary group U(n). We
shall show that (G, K) is a Gelfand pair in the sense of 3, and that the
associated zonal spherical functions may be identified, up to a scalar
factor, with the Schur functions s>..' where A is a partition of length  n.
A polynomial function on G is by definition the restriction to G of a
(complex..valued) polynomial function f on the space of complex n X n
matrices, regarded as a real vector space of dimension 2n2. Thus f is a
polynomial in the_2n2 algebraically independent functions Xij, X;j' where
X..(g)=g.. and X..(g)=g-.. if g=(g..)eG
I) I) I) I) IJ.
Let
P(G) = EB pm(G)
mO
be the space of all polynomial functions on G, where pm(G) is the
subspace of P(G) consisting of the p E P(G) that are homogeneous of
degree m (as polynomials in the Xij and Xij). The group G acts on P(G)
by the rule
(5.1)
(gp)(x) = p(xg)
for p E P(G) and g, x e G. If p E P(G) then the function p: x  p(x) is
also in P( G).
Let Q(G) = C[Xij: 1  i,j  n]. Then we have
P(G) - Q(G)  Q(G)
and
Q( G) = EB P>..
>..
where ,\ ranges over all partitions of length  n, and P>.. is the subspace of
Q(G) spanned by the matrix coefficients R of an irreducible polynomial
representation RA of G (Chapter I, Appendix A). Hence
(5.2)
P( G) = EB PA, JJ.
A,p.,
where '\, J..t are partitions of length  n, and PA, JJ. is the subspace of P(G)
spanned by the products RRt/.
We can now define the operators eK, e as in 3 (K now being the
unitary group), and propositions (3.3)-(3.8) will remain valid in the present

5. THE COMPLEX CASE
441
context, provided that the transposed matrix x' is replaced by the conju-
gate transposed x* = x I. Correspondingly, I is now the space of hermitian
n X n matrices, and
P(I) = EB pm(I)
mO
the space of polynomial functions on I (regarded as a real vector space of
dimension n + n(n - 1) = n2). The group G acts on P(I) by the rule
(gp)(u) =p(g*ug)
for g E G, p E P(I), and u E I.
Let V (resp. V) be a complex vector space of dimension n, with basis
e1,..., en (resp. el'''.' en). The group G acts on V and V by
n
oe. =  g..e.
0' } i...J I} I'
i -= 1
n
gej = L gijei
i= 1
if g = (gij) E G. _
The tensor space T2m(v) of 3 is here replaced by T = Tm(v)  Tm(v);
the group G acts diagonally, as before, and (in place of the symmetric
group S2m) the group Sm X Sm acts by permuting the factors in T.
Let (u,D) be the bilinear form on Vx V for which (ei,ej) = Oij. We
have then
(gu, v) = (u, g*v)
for g E G, u E V, and v E V.
Let cp: T -+ pm (I) denote the linear mapping defined by
m
cp(u1  ... um Vl  ... Vm)(u) = n (U;,UVi).
i-I
From our definitions it foHows that cp commutes with the actions of G on
T and pm(I). The place of the hyperoctahedral group Hm in 3 is taken
here by the diagonal subgroup  = A.m = {(w, w): wE Sm} of Sm X Sm' and
it is clear from the definition of cp that cp(ht) = cp(t) for t E T and h E .
If TA is the subspace of T fixed by , we have as in 3
(5.3) The mapping cp is restricted to TA is a G-isomorphism of TA onto
pm(I). I

442
VII ZONAL POLYNOMIALS
From Chapter I, Appendix A, 5 it follows that T decomposes under the
action of (Sm X Sm) X G as follws:
T = Ee TA,}L
A,p.
where
TA,}L = (MA fg) M}L) fg) (FA (V) fg) F}L(V»).
Here ,\ and J.L are partitions of m of length  n, and MA (resp. M}L) is an
irreducible Sm-module corresponding to ,\ (resp. J.L). Since (Chapter I, 7)
.
dim(M,\ fg) M}L)A = ( X 'X}L, l)sm
= ( X A, x}L) S m = 5,\ p.
it follows that
TA::: EB (FA(V) fg)(V»)
A
as G-module, where ,\ ranges after the partitions of m of length  n.
Now the G-modules F,\(V)  F,\(V), and more generally FA(V) fg) Fp.(V)
for any two partitions '\, J-L of length  n, are irreducible. For the matrix
coefficients Rt E Q(G) are linearly independent (Chapter I, Appendix A,
(8.1», and likewise the Rfl E Q(G) are linearly independent; hence the
products RRki are linearly independent functions on G, which (loc. cit.)
proves the assertion.
(5.4)
If P E P(I), the function p defined by
p(x) = p(x.x)
is a polynomial function on G, and the mapping p  p commutes with the
actions of G on P(I) and peG). Moreover, it doubles degrees: if p E
pm(I) then p E p2m(G).
(5.5) The mapping p  p defined above is a G-isomorphism of pm(I) onto
p2m(K\ G), for all m  o.
The proof is the same as that of (3.12), and just as in the real case has
the following consequences:
(5.6) (i) The G-module e K PA, p. is isomorphic to  (V) fg) FA (V) if ,\ = J-L, and
is zero otherwise.
(ii) We have
eKP(G) =P(K\G) = Ef) (FA(V) F,\(V»)
A

5. THE COMPLEX CASE
443
where A ranges over all partitions of length  n.
(5.7) Let A, p., be partitions of length  n. Then
dim( F,,(V) @ Fp,(V) t = {
if A = p."
otherwise.
(5.8) P)..,,.,.(G, K) = PA,,.,. np(G, K) is one-dimensional if A = p." and is zero
otherwise. Moreover,
peG, K) = (B PA,)..(G, K)
,\
summed over all partitions A of length  n.
In these statements, P(K\ G), peG, K) etc. are defined exactly as in the
real case (3). In particular, it follows from (5.6Xii) that the G-module
P(K\ G) is multiplicity-free, so that (G, K) is a Gelfand pair.
As in 3, we regard each symmetric function IE Ac as a polynomial
function on G: if x E G has eigenvalues g),..., gn then I(x) = I( g),..., gn).
Then the character of the irreducible G-module F)..(V)  Fp.(V) is s)..sp. E
P).., p.' and hence by (5.6) we have eK(sASp.) = 0 unless A = p.,:
(5.9) fK (s"s,)(xk) dk = 0
for all x E G, unless A = J..L. I
Let 0A = eK(s)..s)..), or explicitly
fi,,(x) = f. s,,(xk)S'/xk)dk
K
where A is a partition of length  n. As in 3 we have
(5.10)
(5.11) (i) 0A(l) = 1.
(ii) 0A E PA,A(G, K) and generates the i"educible G-submodule eKP)..,).. of
P(K\ G).
Moreover, 0).. is characterized by these two properties. I
These functions 0).. are the zonal spherical functions of the pair (G, K).
In order to compute them we return to the (Sm X Sm) X G-module T =
Tm(v)  Tm(v), where m  n.
From (5.7) we have
TK = EB T)..).. = EB (M).. M)..)
,\ )..

444
VII ZONAL POLYNOMIALS
as Sm X Sm-module. Hence TK is isomorphic to the induced module
1imxSm. Now the tensor
n
8=  e.e.EVV
L.J I I
i-1
is fixed by K, and hence
8 =
m
E
ej ...  ej  ej ...  ej
1 m 1 m
itt...,im
lies in TK. Since the subgroup of Sm X Sm that fixes 8m is precisely , it
follows that 8m generates TK as Sm X Sm - module. Hence by (1.5) the
space
T4XK = (TK )d = ("[4 )K
A,A A,A A,A
is one-dimensional, generated by wA8m where wA is the zonal spherical
function of the Gelfand pair (Sm X Sm') corresponding to the partition A.
From 1, Example 9 we have
wA8m=XA(I)-1 E XA(vw-1).(vXw)8m
v,weSm
n!
E X A ( w ) . (w XI) 8m
wESm
X A(I)
where XA is the character of MA' since (v X w)8m = (vw-1 X,,1)8mo
From the definitions of 8m and the mapping q;: T  pm() (5.3) it
follows easily that
q;«w X 1) 8m) = Pp
where p is the cycle-type of w E Sm' and hence
cp(wm) = cA E Z;lXpApp = CASA
P
in the notation of Chapter I, 7, where cA = (n!)2/xA(I). From this it
follows, as in (3.24), that
(5.12) We have
0A(X) = sA(x*x)/sA(ln)
for all x E G.
Another proof of (5.12) is given in Example 1 below.

5. THE COMPLEX CASE
445
Examples
1. (a) Let p E Q(G). If plK = 0 then p = O. (Each g E G can be written as
g = ktxk2' where k}k2 E K and x is a diagonal matrix, say x = diag(xl'''.' xn). For
fixed k1, k2 E K let q(x1,..., xn) = p(k1xk2); then q is a polynomial function of n
complex variables Xl'.'" xn which vanishes whenever Ixll = ... = Ixnl = 1. Deduce
that q = 0 and hence that p(g) = 0 for all g E G.)
(b) For each partition ,\ of length  n, the G-module F,\(V) remains irreducible as
a K-module. (Consider the matrix coefficients of F,\(V), and use (a) above.)
(c) If g...... (Rt(g» is a matrix representation of G afforded by (V), we have
(xEG,kEK)
(1)
s,\(xk) = E R(x)R(k)
i ,j
and
(2)
s,\(xk) = s).(xk) = s).(k-1x*)
= E R:S(k-1 )R:r(x*).
r,s
The matrix coefficients satisfy the orthogonality relations [S12]
f Rt(k)R;.(k-l) dk = d;lisjr
K
where d,\ = dim (V) = s).(ln)' since by (b) above the matrix representation k..-+
(Rt(k» of K is irreducible.
Deduce from (1), (2), and (3) that
(3)
!1 ,\ (x) = f (s). s).) ( xk) d k = d; 1 S). ( Xt *) .
K
2. Let f E P( G). Then f is a zonal spherical function of (G, K) if and only if f:;: 0
and
f f(xky) dk = f(x) f(y)
K
for all X, y E G. (Same proof as for (4.1).)
3. Let U, T E I. Then
f S,\(UkTk-1) dk = s).(u )s,\( T )/s).(ln).
K
In view of (5.12), this follows from Example 2.
4. Let dx be Lebesgue measure on G and let
dJl(x) = (21T) -n2 exp( - t trace x*x) dx
so that dJl(xk) = dJl(kx) = dJl(x) for k E K, and fa dJl(x) = 1.

446
VII ZONAL POLYNOMIALS
Let U, T E I. Then
1 (trace U XTX*)m d v(x) = 2m E s,,(u )s,,( T)
G A
summed over partitions A of m. (The proof follows that of (4.3).)
5. Let U, T E II Then
1 s,\(uXTx*)dv(x) = 2IA1h(A)sA(u)SA(T)
G
for all partitions A of length  n, where h( A) is the product of the hook-lengths of
A. (The proof follows the same lines as that of (4.4).)
6. Let A be a partition of length  n, and let ,\ be as in 4.
(a) Show that
eKIA(x)/2 = CAA(X*X)
for x E G, where cA = f.14A(k)/2 dk.
K
(b) If U E I, show that
f.  A ( k -IU k ) d k = SA ( U ) / SA (1 n ) ,
K
and that
1 ,\(x*ux)dv(x) = 2IA1h(A)SA(U).
G
In particular,
1 ,\(x*x)dv(x) = 21AI n (n +c(s»
G sEA
where c(s) = a'(s) -['(s) is the content of sEA. (These formulas are the counter-
parts of (4.9)-(4.12), and may be proved in the same way.)
Notes and references
James [J5] was aware that the complex zonal polynomials are essentially
the Schur functions (5.12). See also Farrell's paper [F3], and [Tl], Chapter
V.
All the integral formulas of 4 have their complex counterparts, and
Examples 2-6 are merely a sample.
6. The quaternionic case
In this section let G = GLn(H), the group of non-singular n X n matrices
over the division ring of quaternions, and let K = U(n, H) be the quater-
nionic unitary group. We shall show that (G, K) is a Gelfand pair in the

6. THE QUATERNIONIC CASE
447
sense of 3, and that the associated zonal spherical functions may be
identified, up to a scalar multiple, with the Jack symmetric functions
(Chapter VI, 10) with parameter a = !.
If x = a + bi + cj + dk E H in the usual notation for quaternions (i2 =
j2 = k2 = -1, ij = - ji = k, etc.), where a, b, c, d E R, let
(6.1) 8 (x) = (a + bi. c + d )
-C+dl a-bz
so that x  8(x) embeds H in the algebra of complex 2 X 2 matrices. More
generally, if g = (grs) E G = GLn(H), let
(6.2) 8(g) = (8(grs))1<r,s<n
so that 8 is an injective homomorphism of G into the complex general
linear group Ge = GL2n(C). ObseIVe that 8(g*) = 8(g)*, where the aster-
isks denote the conjugate transpose. We shall usually identify G with its
image 8(G) in Gc.
A polynomial function f on G is by definition the restriction to G of a
(complex-valued) polynomial function on the matrix space Mn(H), re-
garded as a real vector space of dimension 4n2. In view of (6.1) and (6.2), f
is a polynomial in the 4n2 algebraically independent functions
( 6.3 )
1
2 (X2r-1,2s-1 + X2r,2..),
1
2 (X2r- 1,2s - X2r,2s-1)'
1
2i (X2r-1,2S-1 - X2r,2)'
1
_2. (X2r-l 2s +X2r 2s-t)
l ','
for r, s = 1,2,..., n, where Xpq (1 <:p, q <: 2n) are the coordinate func-
tions on Ge (Le. Xpq(g) = gpq for g = (gpq) E Ge). '
Let P(G) denote the algebra of polynomial functions on G, generated
over C by the functions (6.3), and let P(Gc) denote the algebra C[Xpq:
1 <:p, q <: 2n] of polynomial functions on Ge.
(6.4) (i) The restriction map f  fiG is an isomorphism of P(Ge) onto P(G).
(ii) Each i"educible polynomial representation of G e remains in-educible on
restriction to G.
Proof. (i) Each f E P( G c) is uniquely expressible as a polynomial in the
4n2 functions (6.3). If fiG = 0, then f vanishes for all real values of these
4n2 functions, hence is identically zero.
(ii) Suppose that R: g  (Rij(g)) is a polynomial representation of Ge
whose restriction to G is reducible. By replacing R by an equivalent
representation, we may assume that for some r such that 1 <: r < d, where
d is the degree of R, we have R;j(g) = 0 for all (i, j) such that i > rand

448
VII ZONAL POLYNOMIALS
j <: r, and all g E G. By (i) above, it follows that this is so for all g E Gc,
and therefore R is reducible. I
In view of (6.4), we may identify P(G) and P(Gc)' As in 3, G acts on
P( G) by the rule
(gp )(x) = p(xg)
where p E P(G) and g, x E G, and under this action P(G) = P(Gc) de-
composes as a direct sum
P(G) = EB P,
J.L
where J.L ranges over all partitions of length <: 2n, and P is the subspace
of P(G) spanned by the matrix coefficients R of an irreducible polyno-
mial representation R  of G (or Gc) corresponding to the irreducible
G-module F(c2n), in the notation of Chapter I, Appendix A.
Now let
K = U(n,H) = {g E G: gg* = I}.
Since 8(K) = 8(G) n U(2n), it follows that K is a bounded closed sub-
group of G, and hence is compact. The set-up is now entirely analogous to
that of 3, and we need only pay attention to those points in the
development that differ from the real case. Propositions (3.3)-(3.8) remain
valid in the present context, provided that the transpose x' (x E G) is
replaced by x*.
Next let  denote the space of quaternionic hermitian n X n matrices
(Le. matrices u such that u * = u), and let
P() = EB pm()
m;>O
denote the space of polynomial functions on  (regarded as a real vector
space of dimension n + 2n(n - 1», where pm() consists of the functions
that are homogeneous of degree m. The group G acts on P() by
(gp)(u) =p(g*ug)
for g E G, p E P(), and u E .
Also let V = C 2n, the elements of V being regarded as column vectors
of length 2n. The group G acts on V via 8: gv = (}(g)v for g E G and
v E V. Let I = (}(jln), so that I is the diagonal sum of n copies of the
matrix ( _  6)' and let
(6.5)
(u, v) = u' Iv

6. THE QUATERNIONIC CASE
449
for u, v E V. The bilinear form so defined on V is skew-symmetric, and we
have
(6.6) (gu,v) = (u,g*v)
for all g E G. For it follows from (6.1) and (6.2) that B(g)' J = JB(g*), and
hence that
(gu, v) = u' B(g)' Iv = u'IB(g*)v
= (u,g*v).
Let T = T2m(v) be the 2mth tensor power of V. As in 3, the group G
acts diagonally on T, and the symmetric group S2m acts by permuting the
factors. Let cp: T -+ pm(!,) denote the linear mapping defined by
m
cp(v1  ...  v2m)(u) = n (V2i-1' UV2i>.
i-I
Clearly cp commutes with the actions of G on T and pm(I), and since by
(6.6) we have
(u,uv)=(uu,v)= -(v,uu)
for u E I, it follows that
cp (ht) = B (h) cp (t )
for t E T and h E H = Hm' where B is the sign character of S2m. Finally, if
e],. .., e2n is the standard basis of V = C2n and ej,..., ein is the dual basis,
so that (ei, ej> = 8ij, then (e;, ueq) is the (p, q) element of o( u), from
which it follows easily that cp is surjective. Hence if
TeH = {t E T: ht = B(h)t for all h E H}
we have (compare (3.9)):
(6.7) The mapping cp restricted to TeH is a G..isomorphism of TeH onto
pm(I). I
Under the action of S2m X Gc' T decomposes as
T = E9 T
J.I.
where
TJL =MJL F(V).
Here J.L ranges over the partitions of 2m of length  2n, and M is an
irreducible S2m-module corresponding to J.L. By (6.4Xii), Fp.(V) remains

450
VII ZONAL POLYNOMIALS
irreducible as a G-module. Since (2, Example 6) M;H is one-dimensional
if JL' is an even partition (or equivalently if JL = A U A for some partition A
of m) and is zero othelWise, it follows that
TeH = EB FAU A(V)
A
as G-module, where A ranges over the partitions of m of length  n.
(6.8)
If p E P(I), the function p defined by
flex) = p(x.x)
is a polynomial function on G, and the mapping p t-+ P commutes with the
actions of G on P(I) and peG). Moreover, it doubles degrees: if p E
pm(I,) then p E p2m(G).
(6.9) The mapping p t-+ fl defined above is a G-isomorphism of pm(I) onto
p2m(K\ G), for all m  o.
The proof is the same as that of (3.12), and just as in the real case has
the following consequences:
(6.10) (i) The G-module eKPp' is isomorphic to FJL(V) if IL' is even, and is
zero othetwise.
(ii) We have
eKP(G)=P(K\G) = EB FAUA(V)
A
where A ranges over all partitions of lengh  n.
(6.11) Let JL be a partition of length  2n. Then
K { 1 if JL' is even,
dim F:,(V) = 0
r- otheIWise.
(6.12) PIA-(G, K) is one-dimensional if JL' is even, and is zero otherwise.
Moreover,
P(G,K)= EB PAUA(G,K)
A
summed over all partitions A of length  n.
In these statements, ex, P(K\ G) etc. are defined exactly as in the real
case.
As in 3, we may regard each fE Ac as a polynomial function on Gc: if

6. THE QUATERNIONIC CASE
451
X E Gc has eigenvalues 1'...' g2n' then I(x) = I( gt,..., g2n). With this
understood, we have as in (3.19)
J. s,..(8(xk)) dk = 0
K
unless J..L' is an even partition.
(6.13)
Let A be a partition of length  n, and- define (}A E peG) by
(6.14)
,o.\(x) = J. s.\U.\(8(xk))dk
K
for x E G. Then as in (3.21) we have
(6.15) (i) {lA(l) = 1.
(ii) {l A E PA U A (G, K) and generates the i"educible G-submodule e K PA U A of
P(K\ G).
Moreover, 0A is characterized by these two properties.
The function {} A is the zonal spherical function of (G, K) associated with
the partition A.
We return now to the S2m X G-module T = T2m(v), where m  n. We
have (compare 3)
TK = Ea T,\Ku'\::: EB M,\ U A
A ,\
as S2m-module, where A ranges over all partitions of m. From 2, Example
6 it follows that TK is isomorphic to .the induced module 82m, where 8H is
the restriction to H of the sign character of S2m. As before, let et,..., e2n
be the standard basis of V = C 2n, and let ej,..., e!n be the dual basis,
such that <e;,eq)=pq for lp, q<;2n. We have eir-t= -e2r and
eir = e2r- t for 1  r  n.
The tensor
(6.16)
2n 2n
 = E e; fi!J e p = - E e p  e;
p-l p-l
is fixed by K, by virtue of (6.6), and hence m = 8  ...  S (m factors)
lies in TK. From (6.16) it follows that h m = 8(h)m for h E H, and
hence just as in the real case that fj8m generates TK as S2m-module. Hence
(2, Example 6) the space
T::K = (TAKu A)&H = (Tt{fA)K

452
VII ZONAL POLYNOMIALS
is one-dimensional, generated by
(6.17)
1Tm = E B(W)w'\'(w).w8m.
we S2m
We have now to compute cp( 'IT '\Bm) E pmCI). For this purpose we need
to interpret symmetric functions as polynomial functions on I. If a E I,
then trace( u ) E R, because the diagonal elements of u are real. More
generally, trace(u') E R for each r  1, since u' E. If now p =
( p], P2"") is any partition, we define
pp(U) = n trace(uPi)
i 1
thus defining Pp' and hence by linearity any symmetric function, as a
polynomial function on I. Since by (6.1) and (6.2) trace(8(u)) = 2 trace(u),
it follows that
(6.18)
pp( fJ(u)) = 2/(P)pp(u).
In place of (3.23) we now have
(6.19)
8(w)cp(wBm) = B 2/(P)p
P p
as functions on , where P is the coset-type (2) of W E S2m' and Bp =
( -l)m - /( p).
Proof. Let fw = B(w)cp(w8m). For any h E H we have fwh =fw' since
h8m = B(h)Bm; also fhw = fw' since cp(ht) = B(h)cp(t) for any t E T. Hence
fw depends only on the double coset Hp = HwH. As in the proof of (3.23),
we are reduced to the case where w is the 2m-cycle (1,2,..., 2m). Since
Bm = E ej  ei  ...  ei  ei
11m '"
summed over iI,..., im running independently from 1 to 2n, we have
wBm = Eei ei ei ... ei ei
mIl m-l m
and hence (since B( w) = - 1)
(1)
fw(u) = - E<ei ,uej)...(ei ,uei).
m 1 m-l m
Since
(ei,uej)= (uei,ej)= -(ej,uei)= -8(U)ji

6. THE QUATERNIONIC CASE
453
it follows from (1) that
fw(u) = (_l)m-l L lJ(u )i1,im." lJ(u )imim-1
= ( _ 1) m - 1 trace ( lJ (u ) m )
= (-1)m-12trace(um)
and hence that fw = (-1)m-12Pm' which completes the proof.
We can now express the zonal spherical functions 0A in terms of the
symmetric functions Z introduced in 2, Example 7:
(6.20) We have
o'A(X) = Z(x*x)/Z(ln)
for all x E G.
Proof. Let CPA = cp( 7T A5m), and define A E peG) by A(X) = CPA(X*X). It
follows from (6.9) that A is a generator of the one-dimensional space
PAU A(G, K) that is the image of T&HXK under the composite isomorphism
T&H --+ pm(I) --+ p2m(K\ G). Hence by (6.15)(ii) 0A is a scalar multiple of
A' namely
(1)
0, A (x) = CPA (x * x) / CPA (1 n )
since 0A(ln) = 1.
On the other hand, from (6.17) and (6.19) we have
cp =  IH Is wA'21(p)p
A 1..J p p p p
p
I 12  -} A'
= Hm 1..J SpZp WpPp
p
and therefore (2, Example 7)
(2) CPA = IHm IZ.
The result now follows from (1) and (2).
Examples
1. Let f E P(G). Then f is a zonal spherical function of (G, K) if and only if f =1= 0
and
f f(xky) dk = f(x)f(y)
K
for all x, y E G. (Same proof as for (4.1).)

454
VII ZONAL POLYNOMIALS
2. In this and the following Examples we define Pp' for each partition p, as a
function on G by the rule (6.18) .
pp(x) = 2 -/( P)pp( 8(x)).
By linearity this defines each symmetric function f E A as a polynomial function
on G, such that f(xy) = f(yx) for all x, y E G. With this definition we have
f' · - Z(U)Z(T)
ZA(ukTk ) dk - '()
K ZA In
for all u, T E I. In view of (6.20), this follows from Example 1.
3. Let dx be Lebesgue measure on G and let
d v(x) = (21T) -2n2 exp( - t trace x*x) dx,
so that d v(la) = d v(xk) = d v(x) for k E K, and fa d v(x) = 1. Let ,,1] E G be real
diagonal matrices. Then trace( 'X1]x*) is a real number for all x E G, and we have
( m Z( )Z( 1])
], (trace X1jx*) dll(x) = 2mm! E h( ') ·
G IAI-=m 2A
(The proof follows that of (4.3), using 2, Example 7(d).)
4. Let u, T E I. Then
(1)
/, Z(UXTX*) dv(x) = 2IAIZ(u )Z( T).
G
In particular,
/, Z(x*ux)dv(x) = 2IAIZ(ln)Z(u).
G
(Let IA(u, T) denote the integral on the left-hand side of (1). We may assume
without loss of generality that u and T are real diagonal matrices " 1]. Since
(2)
pi =m! E h(2A,)-lZ
IAI-m
(2, Example 7) it follows that
/, (trace ,X1]x*)m dv(x) = m! E h(2A,)-t 1).( " 1]).
G IAI-m
On the other hand, Example 2 shows that IA(" 1]) is a scalar multiple of
Z( ,)Z(1]), and hence the result follows from Example 3.)
5. (a) If x E G, then det 8(x) is real and positive. (Let x = vb where b = (bij) is
upper triangular and v is lower unitriangular. Then det O(v) = 1 and det O(b) =
ni_tlbuI2. Since elements of the form x = vb are dense in G, the result follows.)
Define det x to be the positive square root of det 8(x).

6. THE QUATERNIONIC CASE
455
(b) If x = (xij) E G (resp. Gc), let x(r) = (xij)! < i,j< r for 1  r  n (resp. 1 ,  2n).
Then 8(x(r») = 8(x)(2r) and hence
det 8(xi2r) = (det x(r»)2. ·
(c) If x E G (resp. Gc) and J.L is a partition of length  n (resp. 2n) let
(x) = n (det x(lLi»).
i 1
Then we have
AU A(8(x)) = 2A(X)
for x E G and leA)  n.
(d) Show (as in (4.9)) that
eK2A(X) = CAA(X.X)
where C A is a positive constant, and deduce that
f. A(k.uk) dk = Z(u )/Z(ln)
K
for u E I.
6. (a) Let A be a partition of length  n. Show that
fG A(x"ux)dv(x) = 2IAIZ(u)
for u E I. (Use Examples Sed) and 4.) In particular,
(1)
Z(ln) = 2-IA11 A(X.X) dv(x).
G
(b) Let B+ be the subgroup of G consisting of upper triangular matrices (bij) with
positive real diagonal elements bii. Each x e G factorizes uniquely as x = kb,
where k eK and b EB+, and we have
(2)
dx = cn6(b) dk db,
where dk is normalized Haar measure on K, db is Lebesgue measure on B+, and
5 = (4n - 1,4n - 5,...,3), and cn is a positive constant. By evaluating the integral
(1) show that
Z(ln) = n (2n + a'(s) - 21'(s))
sE ,\
in agreement with 2, Example 7(b).
(c) Show that the constant cn in (2) is given by
Cn = V4Vg · · · v4n
where Vr = 21Tr/2/r(t,) as in 4, Example 2(d). (Use (2) to evaluate fa dv(x) = 1.)

456
VII ZONAL POLYNOMIALS
Notes and references
As far as I am aware, the quaternionic analogue of James's theory (3),
and in particular the identification of the quaternionic zonal polynomials
with the Jack symmetric functions with parameter a = t, announced in
[M5], has not been worked out in print before. See however [016], where
the three cases (real, complex, quaternionic) are handled simultaneously.
As in the complex case, all the integral formulas of 4 have their
quaternionic analogues, and Examples 1-6 provide a sample of these.

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Notation
Chapter I
leA)
I AI
9'n
.9
m i( A)
A'
n(A)
(a I /3)
A=>I-L
A-J.L
A + I-L
AUI-L
AI-L
AXI-L
A'";J;I-L
Rij (i <j)
Sn
8
hex)
c(x)
'Pr(t)
A*
.-
A
A- I-L
h(A)
c,,(X)
A"
A
AA
x"
length of a partition A: I, 1
weight ( == sum of parts) of a partition A: I, 1
the set of partitions of n: I, 1
the set of all partitions: I, 1
multiplicity of i as a part of A: I, 1
conjugate of a par(to)n A: I, 1
EO -1).\; = E ; : I, 1
Frobenius' notation for a partition: I, 1
Ai '";J; ILi for all i '";J; 1: I, 1
skew diagram: I, 1
partition with parts Ai + ILi: I, 1
partition whose parts are those of A and of I-L: I, 1
partition with parts Ai I-Li: I, 1
partition with parts mine Ai' ILj): I, 1
Al + ... + Ai  ILl + ... + lLi for all i '";J; 1: I, 1
raising operator: I, 1
symmetric group on n symbols: I, 1
(n - 1, n - 2, . .. , 1, 0): I, 1
hook length at x E A: I, 1, Example 1
content of x E A: I, 1, Example 3
(1 - tX1 - t2)... (1 - tr): I, 1, Example 3
p-quotient of A: I, 1, Example 8
p-core of A: I, 1, Example 8
A, J.L have the same p-core: I, 1, Example 8
product of the hook-lengths of A: I, 1, Example 10
content polynomial of A: I, 1, Example 11
Z[ Xl' .. . , X 11 ]5 n: I, 2
ring of symmetric functions: I, 2
AA (A a commutative ring): I, 2
01 02 · I 2
Xl X2 .... ,
monomial symmetric function generated by x ": I, 2
r th elementary symmetric function: I, 2
E ertr = n{1 + Xit): I, 2
e A1 e "2 · .. : I, 2
rth complete symmetric function: I, 2
E hrtr == n(l - Xit)-l: , 2
h,\ 1 h "1. · .. : I, 2
involution of A which interchanges er and hr: I, 2
mJt,
er
E(t)
e"
hr
H(t)
h"
co

468
NOTATION
SA
Z"
[: ]
C(G)
SeW)
'forgotten' symmetric function w(m )): I, 2
Tth power sum: 1, 2
E Prtr-1: 1,2
P).lP).2 ... : I, 2
(-1)1;\1-/().): I, 2
n .mi(A) ( \)f. I 2
;;.11 .mi/\..,
q-binomial coefficient 'Pn{q) / 'Pr{q) 'Pn-r{q): 1,2,
Example 3
cycle indicator of a subgroup G of S n: I, 2, Example 9
sign of w E S n: I, 3
skew-symmetric polynomial generated by x a: I, 3
Schur function: I, 3
fA
Pr
pet)
PA
aa
SA
[:]
HA(q)
( f)
SJ,.(x/y)
SA/p,
,\
Cp.v
.K,\ - p., v
f1-

M(u,v)
K
L
,\ R IL
X'\
Xp'\
f*g
M,\
fog
F,\
generalized q-binomial coefficient: I, 3, Example 1
hook polynomial nsE,\(l - qh(s»): I, 3, Example 2
generalized binomial coefficient: I, 3, Example 4
'supersymmetric' Schur function: I, 3, Example 23
skew Schur function: I, 5
coefficient of s,\ in slJ-sJI: 1,5
(SA/p., hv): I, 5
adjoint of multiplication by f E A: I, 5, Example 3
diagonal map A -+ A  A: I, 5, Example 25
transition matrix: I, 6
Kostka matrix M(s, m): I, 6
M(p, m): I, 6
A is a refinement of J.L: I, 6
irreducible character of S n: I, 7
value of X,\ at elements of cycle-type p: I, 7
internal product of f, g E A: I, 7
Specht module: I, 7, Example 15
plethysm (f, g E A): I, 8
irreducible polynomial functor: I, Appendix A, 5
Chapter II
o
P
k
[(M)
a,\{q)
G;JI(o)
a discrete valuation ring: II, 1
maximal ideal of a: II, 1.
residue field a /p: II, 1
length of a finite a-module M: II, 1
number of automorphisms of a module of type A: II, 1
number of submodules N of an a-module M of type A,
such that N has type II and M / N has type J.L: II, 2
Hall algebra of 0: II, 2
Hall polynomials: II, 4
H(o)
gs(t), g;v(t)

NOTATION
469
Chapter III
RA(Xh..., Xn; t) symmetric polynomials defined in III, 1
vm(t) 'Pm(t)/(1 - t)m: III, 1
UA(t) n;;.. 0 Uml(A)(t): III, 1
PA(x; t) Hall-Littlewood symmetric function: III, 2
q r(x; t) (1 - t )p(r)(x; t) (r  1): III, 2
bA(t) n;;"1 'Pm/{A(t): III, 2
QA(X; t) bA(t)PA(x; t)): III, 2
qA(X; t) qAl(X; t)qAz(X; t)... : III, 2
f;v(t) coefficient of PA in Pp.Pv: III, 3
ZA(t) ZA n(1- tAi)-l: 111,4
SA(X; t) det(qAi-i+j(x; t)): 111,4
QA/P.' PA/P. skew Hall-Littlewood functions: III, 5
'PA/P.(t), tJlA/P.(t), 'PT(t), tJlT(t): III, 5
K(t) transition matrix M(s, P): III, 6
XpA(t) coefficient of PA(x; t) in pp(x): III, 7
Q;(q) qn(A)XpA(q-1) (Green's polynomials): 111,7
r Z[qh q2' q3'" .]: III, 8
Pf(A) Pfaffian of a skew-symmetric matrix A: III, 8
Chapter W
k
q
f
F
kn
M
Mn
Nn,m
finite field: IV, 1
number of elements in k: IV, 1
algebraic closure of k: IV, 1
Frobenius automorphism x.... xq: IV, 1
extension of k of degree n contained in k: IV, 1
multiplicative group of k: IV, 1
multiplicative group of k n: IV, 1
norm homomorphism Mn -+'Mm (where m divides n):
IV, 1
A
lim Mn: IV, 1
-+
subgroup of L fixed by F": IV, 1
pairing between L nand M,,: IV, 1
set of F-orbits in M, identified with the set of monic
irreducible polynomials (other than t) in k[t]: IV, 1
degree of f E <1>: IV, 1
OLn(k): IV, 2
partition-valued function on <1>: IV, 2
EfEcI» d(f) 1,..(f)l: IV, 2
conjugacy class parametrized by,..: IV, 2
qd(f): IV, 2
n fEci» a".(f)(qf) = order of centralizer of any element of
c....: IV, 2
induction product of class functions: IV, 3
characteristic function of c....: IV, 3
L
Ln
< , x)"
<I>
d(f)
On
,..
IIJJ.II
c...
qf
a,..
Ul 0 . .. 0 U,
7T....

470
NOTATION
An
A
Rn
R
B
space of class functions on G n: IV, 3
EBn > 0 An: IV, 3 .
Z-module generated by the characters of Gn: IV, 3
EBn> 0 Rn: IV, 3
polynomial algebra over C generated by independent
variables en(f) (n  1, f E <1»: IV, 4
qjn(A)PA(Xf; qj 1): IV, 4
aA(qf)PA(f): IV, 4
n feci> Pp.(f)(f): IV, 4
nfecl> Qp.(f)(f) = ap.Pp.: IV, 4
characteristic map: IV, 4
Pn/d(f) (x Ef, n a multiple of d = d(f)): IV, 4
(_l)n-l Lxe M <, x )nPn(X): IV, 4
n
set of F-orbits in L: IV, 4
number of elements of ep E e: IV, 4
Prd() (E ep, d = deep)): IV, 4
Schur function in the <p-variables: IV, 4
partition-valued function on 0: IV, 4
Ee8 d(cp)IA(ep)l: IV, 4
ne8 S'A()( ep): IV, 4
Z-submodule of B generated by the SA: IV, 4
irreducible character of Gn: IV, 6
value of XA at the class c,..: IV, 6
PA (f)
QA(f)
..
P,..
Q,..
ch
Pn(x)
Pn( )
o
deep)
Pre ep)
SA(ep)
A
IIAII
SA
s
xA
X:
Chapter V
F
lal
o
p
k
q
'1T
V
G
G+
K
L( G, K) (resp.
L(G+, K))
!*g
H(G, K) (resp.
H(G+, K))
'1TA
CA
P
CIJ
w(f), I( w)
non-archimedean local field: V, 1
absolute value of a E F: V, 1
ring of integers of F: V, 1
maximal ideal of 0: V, 1
(finite) residue field 0 Ip: V, 1
number of elements in k: V, 1
generator of p: V, 1
normalized valuation on F*: V, 1
GLn(F): V, 2
G nMn(o): V, 2
GLn(o): V, 2
space of complex-valued continuous functions
of compact support on K\ G I K (resp. K\ G+ IK): V, 2
convolution product in L(G, K): V, 2
Ilecke ring of (G, K) (resp. (G+, K)): V, 2
diag( '1T A) , . .. , 1T An): V, 2
characteristic function of K 11' AK: V, 2
t(n - 1, n - 3, .. . , 1 - n): V, 2
spherical function on G relative to K: V, 3
Fourier transform of f E L(G, K) by w: V, 3.

WS
G+
m
Tm
T(X)
(S, W)
Chapter VI
F
z)..(q, t)
(a; q)oo
II(x, Y; q, t)
gn(x; q, t)
g)..(x; q, t)
WU, v
T
U,Xt
Dn(X; q, t)
D'
n
A;(x; t)
Dn(X; a)
DO
n
P)..(x; q, t)
b)..(q, t)
Q)..(x; q, t)
Wa
,
CP)../p.' 'P)../p.'
t/!)../p., "'/p.
a(s), a'(s)
l(s), /'(s)
BU,I
b)..(s;q,t)
fv(q, t)
P)../P.(q, t), Q)../p.(q, t)
c)..(q, t)
c(q, t)
j)..(x; q, t)
K)..,.(q, t)
Pptx; t)
X;(q, t)
Ln
I
[/]1
(x; q, t)
(q, t) a (1,1)
ga)(x)
NOTATION
471
spherical function with parameter s = (Sl'...' sn): V, 3
{x E G+: v(det x) = m}: V, 4
characteristic function of G: V, 4
Hecke series Em Tmxm: V, 4
zeta function defined by w: V, 4
the field Q( q, t): VI, 2
zAn(l-q)..t)/(l-t)..t): VI, 2
no(1 - aq'): VI, 2
ni,j(a;Yj; q)oo/(xiYj; q)oo: VI, 2
coefficient of yn in n(a iY; q >OO/(xiy; q >00: VI, 2
n g)..t(x; q, t): VI, 2
the endomorphism of Ap defined by
p,  ( -1),-lp,(1 - u') /(1 - v') (r  1): VI, 2
I( x l' · · · , X n)  I( Xl' · . · , UX;, · · · , X n): VI, 3
see VI, (3.2)
coefficient of X' in Dn(X; q, t): VI, 3
nji(ai -xj)/(xi -Xj): VI, 3
see VI, 3, Example 3
Laplace- Beltrami operator: VI, 3, Example 3(e)
symmetric functions defined by VI, (4.7)
(P).., p)..) -1: IV, 4
b)..(q, t)P)..(x; q, t): IV, 4
the automorphism of Ap defined by
p,  (-l),-la,p,: IV, 5, Example 3
coefficients in Pieri formulas: VI, (6.24)
arm length and colength of  E A: VI, 6
leg length and colength of sEA: VI, 6
specialization p,  (1 - u')/(1 - t'): VI, 6
(1 - qQ(s)tl(s)+ 1)/(1 - qQ(S)+ 1 tl(s»: VI, 6
coefficient of P).. in Pp,Pv: VI, 7
skew functions: VI, 7
n (1 - qQ(S)tl(S)+ 1». VI 8
S e ).. . ,
n (1 - qQ(S)+ 1 tl(s». VI 8
S e ).. . ,
c)..(q, t)P,,(x; q, t): VI, 8
coefficient of S)..(x; t) in Jp,(x; q, t): VI, 8
pp(x)n(l - tPt): VI, 8
coefficient of z; 1pp(X; t) in J)..(x; q, t): VI, 8
F[x1:1: 1,..., x: 1]: VI, 9
f(XI!,...' x; 1) if f = /(X1'''., xn): VI, 9
constant term in fE Ln: VI, 9
ni  j(X;Xj-1; q )oo/(a;xj 1; q )00: VI, 9
(q, t) -+ (1,1) and (1 - q)/(1 - t) -+ a: VI, 10
coefficient of yn in n(1-xiy)-1/o: VI, 10

472
ga)(x)
bla)(s)
.1, (0)
't').1 
p(a)
).
b(a)
).
Qia)
pea) Q(a)
)./p.' )./p.
t;v(a)
ex
c).(a)
c (a)
J(a)
).
9p).( a )
Chapter VII
C(G,K)
1
VK
Hn
few)
Hp
w).
W).
P
Z).
Z'
).
G
K
dk
peG)
P(K\ G), P( G / K),
P(G,K)
I
I+
P(I)
T
0,,\
dx
dv(x)
x(r)
dp.
B+
rn(a, A)
NOTATION
ng1)(x): VI, 10
(a(s)a + l(s) + 1)/(a(s)a + l(s) + a): VI, 10
limit of «/!).Ip,(q, t) as (q, t) -+ a (1,1): VI, 10
Jack symmetric function: VI, 10
nSEA bla)(s): VI, 10
bla)Pla): VI, 10
skew Jack functions: VI, 10
coefficient of pea) in p(a)p(a), VI 10
). IL JI . ,
F-algebra homomorphism AF -+ F[X] defined by
p, ...... X (r  1): VI, 10
nSE).(a(s)a + l(s) + 1): VI, 10
nse).(a(s)a + l(s) + a): VI, 10
c).(a)p}a): VI, 10
coefficient of Pp in J).(a): VI, 10
space of functions on G constant on the double cosets of
K: VII, 1 .. .
I l i
permutation representation of G on G / K: VII, 1
subspace of vectors v E V fixed by K: VII, 1
hyperoctahedral subgroup of S 2n: VII, 2
graph attached to W E S 2n: VII, 2
double coset of Hn in S2": VII, 2
zonal spherical function of (S2n, H"): VII, 2
value of w). at permutations of coset-type p: VII, 2
zonal polynomial, equal to 1}2): VII, 2
21).IJ11/2): VII, 2, Example 7
GLn(R): VII, 3
O(n): VII, 3
normalized Haar measure on K: VII, 3
space of polynomial functions on G: VII, 3
subspaces of P( G): VII, 3
space of real symmetric n X n matrices: VII, 3
cone of positive definite symmetric n X n matrices: VII, 3
space of polynomial functions on I: VII, 3
2mth tensor power of V = Rn: VII, 3
zonal spherical function of (G, K) indexed by A: VII, 3
Lebesgue measure on G: VII, 4
(21T )-n2 /2 exp( - t trace x' x) dx: VII, 4
the r X r matrix (Xij)l < i,j < r (x E G): VII, 4
polynomial function on G defined by x ...... O(det X(IL'i»),
J.L a partition of length  n: VII, 4
subgroup of G consisting of upper triangular matrices
with positive diagonal elements: VII, 4
1Tn(n-l)/4 n7-l rea + Ai - t(i -1)): VII, 4, Example 8

INDEX
abacus: I, 1, Ex. 8
Adams operations: I, 2
addition of partitions: I, 1
antisymmetrization: I, 3
arm length, colength: VI, 6
array: II, Appendix
augmented monomial symmetric function: I,
6, Ex. 9
augmented Schur function: I, 7, Ex. 17
basic characters of GL,.(k): IV, 6, Ex. 1
Bell polynomials: I, 2, Ex. 11
bitableau: I, 5, Ex. 23
border of a partition: I, 1
border strip: I, 1
Cauchts determinant: I, 4, Ex. 6
character polynomial: I, 7, Ex. 14
characteristic map: I, 7 and IV, 4
characters of S,.: I, 7
charge of a word or tableau: lIlt 6
column-strict plane partition: It 5, Ex. 13
column-strict tableau: I, 1
complement of a partition: I, 1, Ex. 17
complete symmetric functions: I, 2
completely symmetric plane partition: I, 5,
Ex. 18
composition (= plethysm): I, 8 and I,
Appendix A, 6
conjugate diagram: I, 1
conjugate partition: I, 1
connected components of a skew diagram:
I, 1
content: I, 1, Ex. 3
content polynomial: I, It Ex. 11
coset-type of a permutation: VII, 2
cotype: lIt 1
cycle indicator: I, 2, Ex. 9
cycle-product: I, Appendix B, 3
cycle-type of a permutation: I, 7
cyclic o-module: I, 2, Ex. 9
cyclically symmetric plane partitions: I, 5t
Ex. 18
diagonal map: I, 5, Ex. 25
diagram of a partition: I, 1
Dickson's theorem: I, 2, Ex. 27
dominance partial order on partitions: I, 1
domino: I, 6, Ex. 5
domino tableau: I, 6, Ex. 7
domino tabloid: I, 6, Ex. 5
double diagram of a strict partition: I, 1,
Ex. 9
double strip: III, 8, Ex. 11
doubly stochastic matrix: I, 1, Ex. 13
dual of a finite o-module: II, 1
elementary o-module: II, 1
elementary symmetric functions: I, 2
even partition: I, 5, Ex. 5 and VII, 2
exterior powers: I, 2
finite o-module: II, 1
flag manifold: II, 3, Ex.
forgotten symmetric functions: I, 2
Fourier transform: V, 3 and VII, 1, Ex. 4
Probenius automorphism: IV, 1
Probenius notation for partitions: I, 1
fundamental theorem on symmetric
functions: I, 2
Gale- Ryser theorem: I, 7, Ex. 9
Gaussian polynomials: I, 2, Ex. 3
Gelfand pair: VII, 1
generalized binomial coefficients: I, 3,
Ex. 4 .
generalized tableau: I, 8, Ex. 8
Greents polynomials: lIlt 7
Hall algebra: II, 2
Hall polynomial: lIt 4
Hall- Littlewood functions: III, 2
Hecke ring: Vt 2 and V, 5
Hecke series: V, 4 and V, 5
height of a border strip: I, 1
homogeneous polynomial functor: I,
Appendix A, 1
hook-length: I; 1, Ex 1 and Ex. 14
hook polynomial: I, 3, Ex. 2
horizontal strip: I, 1
hyperoctahedral group: VII, 2
induction product: I, Appendix A, 6 and
IV, 3
internal product: I, 7

474
Jack's symmetric functions: VI, 10
Jacobi-Trudi formula: I, 3
Jordan canonical form: IV, 2, Ex. 1
Kostka numbers: I, 6
Laplace- Beltrami operator: VI, 3, Ex. 3 and
VII, 2, Ex. 3
lattice permutation: I, 9
lattice property: III, 8
leg length, colength: VI, 6
length of a finite o-module: II, 1
length of a partition: I, 1
linearization of a polynomial functor: I,
Appendix A, 3
Littlewood- Richardson rule: I, 9
local field: V, 1
LR -sequence: II, 3
marked shifted tableau: III, 8
matrix coefficients of a representation: I,
Appendix A, 8
maximal tori: IV, 2, Ex. 4
metric tensor: VI, 1, Ex.
modified cycle-type: I, 7, Ex. 24
monomial symmetric functions: I, 2
Muirhead's inequalities: I, 2, Ex. 18
multiplication of partitions: I, 1
multiplicative family: VI, 1, Ex.
multiplicity-free representation: VII, 1
Murnaghan-Nakayama rule: I, 7, Ex. 5
Nakayama's conjecture: I, 7, Ex. 19
natural ordering of partitions: I, 1
Newton's formulas: I, 2
normal distribution: VII, 4
odd partition: III, 8
orthogonality relations: III, 7 and VI, 8
parts of a partition: I, 1
parts of a plane partition: I, 5, Ex. 13
p-bar core: I, 1, Ex. 9
p-core of a partition: I, 1, Ex. 8
Pfaffian: III, 8
Pieri's formula: I, 5 and VI, 6
plane partition: I, 5, Ex. 13-19
plethysm: I, 8 and I, Appendix A, 6
Polya's theorem: I, 2, Ex. 9 '
polynomial functor: I, Appendix A, 1
polynomial representation: I, Appendix
A,8
positive type: VII, 1
power sums: I, 2
INDEX
p-quotient of a partition: I, 1, Ex. 8
p-reidue of a partition: I, 1, Ex. 8
p-tableau: I, 5, Ex. 2
q-binomial coefficient: I, 2, Ex. 3
quadratic reciprocity: I, 3, Ex. 17
raising operator: I, 1
refinement of a partition: I, 6
reverse lexicographical order on partitions:
I, 1
ribbon: I, 1
rim of a partition: I, 1
ring of integers of a local field: V, 1
ring of symmetric functions: I, 2
row-strict array: II, Appendix
scalar product: I, 4; III, 4; VI, 1
Schur algebra: I, Appendix A, 8, Ex. 6
Schur functions: I, 3
Schur operation: I, 3
Schur Q-functions: III, 8
Sergeev- Pragacz formula: I, 3, Ex. 24
shape of a tableau: I, 1
shifted diagram: I, 1, Ex. 9
shifted standard tableau: III, 8, Ex. 12
shifted tableau: III, 8
skew diagram: I, 1
skew Hall-Littlewood functions: III, 5
skew hook: I, 1
skew Schur functions: I, 5
Specht module: I, 7, Ex. 15
special border strip: I, 6, Ex. 4
spherical functions: V, 3
spherical transform: VII, 1, Ex. 4
staircase partition: III, 8, Ex. 3
standard tableau: I, 1
Stirling numbers: I, 2, Ex. 11
strict partition: I, 1, Ex. 9
strictly lower (upper) (unO triangular: I, 6
symmetric function: I, 2
symmetric polynomial: I, 2
symmetric powers: I, 2
symmetrical plane partitions: I, 5, Ex. 15
tableau: I, 1
transition matrix: I, 6
twisted Gelfand pair: VII, 1, Ex. 10
type of a finite o-module: II, 1
unimodal polynomial: I, 3, Ex. 1 and I, 8,
Ex. 4

Vandermonde determinant: I, 3
vertical strip: I, 1
weight of a marked shifted tableau: III, 8
weight of a partition: I, 1
weight of a plane partition: I, 5, Ex. 13
weight of a tableau: I, 1
Weyt's identity: I, 5, Ex. 9
INDEX
475
Wishart density: VII, 4, Ex. 7
wreath product: I, Appendix A, 6 and I,
Appendix B
zeta function: V, 4, and V, 5
zonal polynomials: VII, 2
zonal spherical functions: VII, 1