Polynomial Representations of GLn

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 830 J.A. Green Pol...

Author: James A. Green | James A. Green | K. Erdmann | Manfred Schocker

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

830

J.A. Green

Polynomial Representations of GLn 2nd corrected and augmented edition

with an Appendix on Schensted Correspondence and Littelmann Paths by K. Erdmann, J.A. Green and M. Schocker

ABC

Author and co-authors for the appendix James A. Green 19 Long Close Oxford OX2 9SG United Kingdom e-mail: [email protected]

Manfred Schocker Department of Mathematics University of Wales Swansea Singleton Park, Swansea SA2 8PP United Kingdom e-mail: [email protected]

Karin Erdmann Mathematical Institute University of Oxford 24-29 St Giles Oxford OX1 3LB United Kingdom e-mail: [email protected]

Library of Congress Control Number: 2006934862 Mathematics Subject Classification (2000): Primary: 20C30, 20G05, 20G15, 16S50, 17B99, 05E10 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-46944-3 Springer Berlin Heidelberg New York ISBN-13 978-3-540-46944-5 Springer Berlin Heidelberg New York DOI 10.1007/3-540-46944-3 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 ° The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the authors using a Springer LATEX package Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper

SPIN: 11008118

VA41/3100/SPi

543210

Preface to the second edition

This second edition of “Polynomial representations of GLn (K)” consists of two parts. The ﬁrst part is a corrected version of the original text, formatted in LATEX, and retaining the original numbering of sections, equations, etc. The second is an Appendix, which is largely independent of the ﬁrst part, but which leads to an algebra L(n, r), deﬁned by P. Littelmann, which is analogous to the Schur algebra S(n, r). It is hoped that, in the future, there will be a structure theory of L(n, r) rather like that which underlies the construction of Kac-Moody Lie algebras. We use two operators which act on “words”. The ﬁrst of these is due to C. Schensted (1961). The second is due to Littelmann, and goes back to a 1938 paper by G. de B. Robinson on the representations of a ﬁnite symmetric group. Littelmann’s operators form the basis of his elegant and powerful “path model” of the representation theory of classical groups. In our Appendix we use Littelmann’s theory only in its simplest case, i.e. for GLn . Essential to my plan was to establish two basic facts connecting the operations of Schensted and Littelmann. To these “facts”, or rather conjectures, I gave the names Theorem A and Proposition B. Many examples suggested that these conjectures are true, and not particularly deep. But I could not prove either of them. This work was therefore stalled, until I sought the help of my colleagues Karin Erdmann and Manfred Schocker. They accepted the challenge, and within a few weeks produced proofs of both conjectures. Their proofs constitute the heart of the Appendix, and make it possible to begin a comparison of the Littelmann algebra L(n, r) with the Schur algebra S(n, r). Karin and Manfred have made this Appendix possible, and have written large parts of the text. It has been a happy experience for me to work with them. A few weeks before the ﬁnal manuscript of the Appendix was ready, we heard that A. Lascoux, B. Leclerc and J.-Y. Thibon have published a work

VI

Preface

on “The plactic monoid”, which contains results equivalent to Theorem A and Proposition B. Their methods are rather diﬀerent from ours, and they prove also many important facts which do not come into our Appendix. We give a brief summary of this work in §D.11. Oxford, August 2006

Sandy (J. A.) Green

Contents

Polynomial representations of GLn 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Polynomial Representations of GLn (K): The Schur algebra 2.1 Notation, etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The categories MK (n), MK (n, r) . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Schur algebra SK (n, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The map e : KΓ → SK (n, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Modular theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The module E ⊗r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Contravariant duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 AK (n, r) as KΓ-bimodule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 13 14 16 17 19 21

3

Weights and Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Weight spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Some properties of weight spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Irreducible modules in MK (n, r) . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 23 24 26 28

4

The modules Dλ,K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 λ-tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Bideterminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Deﬁnition of Dλ,K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 The basis theorem for Dλ,K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The Carter-Lusztig lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Some consequences of the basis theorem . . . . . . . . . . . . . . . . . . . . 4.8 James’s construction of Dλ,K . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 33 34 35 36 37 39 40

VIII

Contents

5

The Carter-Lusztig modules Vλ,K . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Deﬁnition of Vλ,K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Vλ,K is Carter-Lusztig’s “Weyl module” . . . . . . . . . . . . . . . . . . . . 5.3 The Carter-Lusztig basis for Vλ,K . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Some consequences of the basis theorem . . . . . . . . . . . . . . . . . . . . 5.5 Contravariant forms on Vλ,K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Z-forms of Vλ,K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 43 45 47 48 50

6

Representation theory of the symmetric group . . . . . . . . . . . . . 6.1 The functor f : MK (n, r) → mod KG(r) (r ≤ n) . . . . . . . . . . . . . 6.2 General theory of the functor f : mod S → mod eSe . . . . . . . . . . 6.3 Application I. Specht modules and their duals . . . . . . . . . . . . . . . 6.4 Application II. Irreducible KG(r)-modules, char K = p . . . . . . . 6.5 Application III. The functor f : MK (N, r) → MK (n, r) (N ≥ n) 6.6 Application IV. Some theorems on decomposition numbers . . .

53 53 55 57 60 65 67

Appendix: Schensted correspondence and Littelmann paths A

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 The Robinson-Schensted algorithm . . . . . . . . . . . . . . . . . . . . . . . . A.3 The operators e˜c , f˜c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 What is to be done . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 74 75 78

B

The Schensted Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Notations for tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 The map Sch : I(n, r) → T (n, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Inserting a letter into a tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Examples of the Schensted process . . . . . . . . . . . . . . . . . . . . . . . . . B.5 Proof that (µ, U, V ) ← x1 belongs to T (n, r) . . . . . . . . . . . . . . . . B.6 The inverse Schensted process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.7 The ladder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 81 82 85 88 89 92

C

Schensted and Littelmann operators . . . . . . . . . . . . . . . . . . . . . . . 95 C.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 C.2 Unwinding a tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 C.3 Knuth’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 C.4 The “if” part of Knuth’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 107 C.5 Littelmann operators on tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . 114 C.6 The proof of Proposition B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

D

Theorem A and some of its consequences . . . . . . . . . . . . . . . . . . 121 D.1 Ingredients for the proof of Theorem A . . . . . . . . . . . . . . . . . . . . . 121 D.2 Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 D.3 Properties of the operator C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Contents

IX

D.4 The Littelmann algebra L(n, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 D.5 The modules MQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 D.6 The λ-rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 D.7 Canonical maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 D.8 The algebra structure of L(n, r) . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 D.9 The character of Mλ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 D.10 The Littlewood–Richardson Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 140 D.11 Lascoux, Leclerc and Thibon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 E

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 E.1 Schensted’s decomposition of I(3, 3) . . . . . . . . . . . . . . . . . . . . . . . 147 E.2 The Littelmann graph I(3, 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Index of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

§I.

Introduction Issai Schur determined the polynomial replesentations of the complex

general linear group in 1901.

GLn(¢)

in his doctoral dissertation [S], published

This remarkable work contained many very original ideas, developed

with superb algebraic skill.

Schur showed that these representations are

completely reducible, that each irreducible one is "homogeneous" of some degree

r ~ 0

(see 2.2), and that the equivalence types of irreducible

polynomial representations of

GLn(~) ,

of fixed homogeneous degree

in one-one correspondence with the partitions not more than

n

parts.

see 3.5).

~I

in

n

kn)

of

are

r

into

Moreover Schur showed that the character of an

irreducible representation of type function

k = (~i'''"

r,

k

is given by a certain symmetric

variables (since described as a "Schur function";

An essential part of Schur's technique was to set up a correspond-

ence between representations of

GL (~) n

of fixed homogeneous degree

and representations of the finite symmetric group

G(r)

on

r

r,

symbols,

and through this correspondence to apply C. Frobenius's discovery of the characters of

G(r)

[F, 1900].

This pioneering achievement of Schur was one of the main inspirations for Hermann Weyl's monumental researches on the representation theory of semi-simple Lie groups [We, 1925, 1926].

Of course Weyl's methods,

based on the representation theory of the Lie algebra of the Lie group and the possibility of integrating over a compact form of

F,

F,

were very

different from the purely algebraic methods of Schur~s dissertation; in particular Weyl's general theory contained nothing to correspond to the symmetric group GLn(~),

G(r).

In 1927 Schur published another paper IS'] on

which has deservedly become a classic.

"dual" actions of

GLn(¢)

(see 2.6) to rederive

and

G(r)

on the

In this he exploited the

~ th

tensor-space

E~r

all the results of his 1901 dissertation in a new

and very economical way.

Weyl publicized the method of Schur's 1927 paper,

with its attractive use of the "double centralizer property",in his influential book "The Classical Groups" [We',1939]. in Chapters 3(B) and

4

of that book has become the

polynomial representations of

GL (~) n

standard treatment of

(and, incidentally, of

representation theory of the symmetric groups this explains the comparative neglect

In fact the exposition

G(r)),

Alfred Young's

and perhaps

of Schur's work of 1901.

I think

this neglect is a pity, because the methods of this earlier work are in some ways very much in keeping with the present-day ideas on representations of algebraic groups. in part

It is the purpose of these lectures to give some accounts,

based on the ideas of Schur's 1901 dissertation, of the polynomial

~epresentatlons of the general linear groups infinite field of arbitrary characteristic.

GLn(K),

where

K

is an

Our treatment will be "elementary" use algebraic interesting

in the sense that we shall not

group theory in our main discussion.

But it might be

to indicate here some general ideas from the representation

theory of algebraic

groups

inverse is not important

(or of algebraic

in this context),

semigroups,

since the group

which are relevant

to

our

work. Let

F

be any semigroup

associative multiplication) field.

A representation

space over

K)

~(IF) = ~V'

•

is a map

for all

identity map on

V.)

: KF-+ EndK(V);

(i.e.

is a set, equipped with an

with identity element of

F

on a

K-space

• : F ~EndK(V )

g, g' E F.

KF

T

~,

and let

V

(i.e.

which satisfies

(For any set

We can extend

here

F

V,

K

a vector T(gg')

= ~(g)~(g'),

we denote by

~V

linearly to give a map of

is the semlgroup-algebra

of

be any

F

the

K-algebras

over

K,

whose

We can make

KF

act

elements are all formal linear combinations

=

whose support on

V

by

Z g~F

supp K = {g ~ F : K

Kv = ~(~)(v) (V,~),

(V',~')

is, by definition,

bijective

representation KF-modules;

or simply

V. a

~'(g)f = f~(g)

is called a

a right

# 0}

g

A

K-map

and thereby get a left KF-module,

f : V ~ V'

for all

g E r.

Kr-modules

(i.e, A

f

(V,~),

is a linear map)

KF-map which is

or an equivalence

One has analogous

KF-module

is finite.

KF-map between such

KF-isomorphism,

~, ~'.

~ K, g

(~ ~ K~, v ~ V),

denoted

which satisfies

K Kgg,

definitions

between the

for right

can be regarded as a pair

(V,~)

where

T

: F ~ EndK(V)

i.e.

~(gg')

is an a n t i - r e p r e s e n t a t i o n

= ~(g')~(g)'

The set

KF

g ~ f(g)f'(g),

element If

~

s ~ F

of

KF

and

are defined

defined

takes each

f ~ KF

K-algebra

Lsf ,

Ls, R s

maps

homomorphism)

a representation

F

rifht

Thus

KF

KF-module

that if

s ~ F

on

and

(using

L).

f ~ KF

takes

that these actions

f ~ K F. f ® f'

: F ~K

KF

while

F.

for all

The identity 1K

of

f

of

K.

by

s

g ~ F.

In particular,

L : s ~ L

K-algebra L s, R s

both

R : s ~ Rs

gives

gives an anti-represent-

s

KF- module

(using

R)

actions by

and a o,

so

we write

and

f o s = L f. s (sof) o t = s o(fot) K F ® K F ~ K FxF

(®

to the function m a p p i n g s, t ~ F.

structure

on

and

F

for all means

F x F ~ K

s, t E F

~)

which

by

This linear map is injective

, and we use it to identify

8 : K F -* K F×F

to

given by

We denote both module

com~ute:

(f, f' ~ K F)

The semigroup

of

into itself and is a

K F ~ K F.

The~e is a linear map

(s,t) ~ f(s)f'(t),

g

is defined

It is easy to check that

K F,

s o f = R f s Notice

ff'

("point")

can be made into a left

and

e.g.

K-algebra,

to the identity element

Rsf

V,

g, g' E F.

R f : g ~ f(gs), s

EndK(KF ) .

ation.

K-space

is s commutative

"pointwise",

g ~ F

belong to the space of

f : F ~K

on the

then the left and r i g h t translates

to be the maps

Each of the operators

F

for all

for every element

L f : g ~ f(sg), s

map (i.e.

~ ~V

of all maps

with algebra operations take

~(IF)

of

KF @ KF

with a subspace of

gives rise to two maps

~ : K F -~ K~

K FxF.

as follows: Both

A,e

if are

f £ K F,

then

&f : (s,t) -~ f(st),

K-algebra maps.

and

~(f) = f(IF)-

We shall say that an element

f ~ K F is

finitary, or is a representative function, if it satisfies any one of the conditions

FI, F2, F3

<see e.g.

[H, Ch. 2]).

FI.

The left

below:

these three conditions are in fact equivalent

KF-submodule

KFof

generated by

f

is finite-

dimensional. F2.

The right

KF-submodule

foK/"

generated by

f

is finite-

dimensional. F3.

Af ~ K F ® K F.

(h

This means that there exist elements

fh' fh E K F

runs over some finite index set) such that

(Is)

Af =

[ih fh ® fh' "

This equation is equivalent to the system of equations

(ib)

f(st) =

~ h

fh(s)f{(t),

all

s, t 6 F.

It is also equivalent to each of the following systems

(ic)

tof =

Z f~(t)fh, h

all

t ( F,

fos =

~ fh(s)f~, h

all

s 6 F.

or (id)

The set

F = F(K F)

of all finitary functions

f: F ~ K

is a

K-bialgebra (see [Sw] for the definitions of coalgebras and bialgebras). It is a

K-subalgebra of

£F ~ F ~ F

K ~,

(this means that if

and is also closed to f

A

in the sense that

is finitary, then the functions

fh' fh F ,

in (la) can be chosen to be themselves finitary).

equipped ~ith the maps

these two structures on fact that

A

and

c

A : F ÷ F ~ F , ~ : F + K ,

F ,

of algebra and coalgebra,

are both

K - a l g e b r a maps

F i n i t a r y functions on

F

d i m e n s i o n a l r e p r e s e n t a t i o n s of on a f i n i t e - d i m e n s i o n a l V ,

z(g)v b = gv b =

Suppose

V °

rab(g)

~ K .

E rab(g)v a , a eB

The functions

c o e f f i c i e n t functions of

If

K-coalgebra;

are linked by the ~,

T

p. i ~ ) .

is a r e p r e s e n t a t i o n of

{vb : b e B)

T ,

or of the

KF-module

for

rab

is a

F

K-basis of

V .

g c F , b e B ;

: F ~ K (a,b g B)

or of the

of these functions is a subspace of T ,

is a

we have equations

(le)

here

K-space

appear as coefficient functions of finite-

F .

K-space

(see

The

KF

KF-module

are called

V = (V,T)

.

The

K-span

called the coefficient space (1)

We denote this space by

cf(V) =

of

E K.rab a,b

;

it is e l e m e n t a r y to verify that it is independent of the choice of the basis {vb} . i.e.

The m a t r i x

R = (tab)

R(gg') = R(g)R(g')

Kronecker delta).

gives a m a t r i x r e p r e s e n t a t i o n of

, R(IF) = (~ab)

for all

tab

(if)

~ r ac ~ rcb , S~rab) ' = 6ab ' eEB

The m a t r i x

,

R = (rab)

C = cf(V)

all

a, b c B .

is sometimes called an "invariant m a t r i x "

are finitary~ hence that

~,

p.140].

it follows that all the coefficient functions cf(V)

is a subspace of

is a s_ubcoalgebra of

As a m a t t e r of fact, every finitary f u n c t i o n space of some f i n i t e - d i m e n s i o n a l

(I)

is the

viz°

From the first equations

shows that

(~ab

These conditions translate into conditions on the

coefficients

Arab =

g, g' c F

F ,

KF-module

F ,

F = F(K F) . i.e.

f : F ÷ K V ;

that

But AC~

rab

(If)

also

C ~ C

lies in the coefficient

for this purpose

In [HI, this is called the ~'space of r e p r e s e n t a t i v e functions" of T, or

V.

we could take

V = KFof

(see

FI).

It is for this reason that finitary

functions are sometimes called "representative functions". If K-space), left

S

is any

mod (S)

S-modules.

shall denote the category of all finite-dimensional Similarly

dimensional right over

K

F(KF), the

K-algebra (possibly of infinite dimension as

mod'(S)

S-modules.

An algebraic representation theory of

could be defined as follows: i.e.

A

is a

K-subspace of

"A-representation theory" of

full subcategory

mOdA(KF )

dimensional left

KF-modules

f : V ~ V' just the

of

F,

first choose a subcoalgebra F(~)

V

such that V, V'

satisfying

~A c A ® A.

cf(V) c A,

of finite-dimensional right way.

The assumption

if

KF-module

cf(V) ~ A ;

A-rational left

V

fh' fhT from

appearing in (ic) (id)

(la)

implies that if

KF-moduleso

A

It is A-rational

mod~(KF)

A-rational in the same f ~ A,

then the functions

can themselves be chosen to belong to

follows that

is

then mOdA(KF)

We can define the category

KF-modules which are

gA c A ~ A

Then

(The morphisms

clear that submodules, quotient-modules and finite direct sums of A-rational.

of

of this category are, by definition,

"rational", or more precisely "A-rational",

modules, are themselves

A

whose objects are all finite-

In some contexts we say that a

is the category of finite-dimensional

F

is defined to be the study of the

mod (KF),

between two objects Ki'-maps.)

is the category of all finite-

is a left and right

A.

KF-submodule of

Then KF ;

also by quite elementary calculations that any finite-dimensional left (or right) (or

KF-submodule

mod~(KF))o

V

of

A

belongs to the category

mOdA(KF)

Examples i.

Let

closed field

K

F

be an affine algebraic group over an algebraically

(see for example ! H, p.21]), and

regular functions on mOdA(KF)

F

(A

A = K[F]

is often called the affin~ ring of

is the category of rational (finite-dimensional)

the usual sense of algebraic group theory~ sub~oalgebra of

F(KF),

remarks apply when

F

In this case,

A

is an affine algebraic semigroupo F(~)

= K F.

2.

If we take

F).

Then

KF-modu!es in is not orlly a

but a sub-blalgebra (see [Se, p. 46]).

finite semigroup, then of course mOdA(Kr) = mod (KF).

the ring of

The same

Let

F

A = K F,

be a then

The (left and right)

K~-module structures on

A,

are dual to tile (right and left) "regular"

KF-module structures on

K

given by multiplication:

we may identify

(KF)

Let

= HomK(KF, K)f

integer, and

3.

F = CL (K)

K

K.

polynomial functions

f : F ~ K

be an infinite field,

We could take

A

(see 2.1).

R = (tab)

F

2.2) by taking

mention that semigroup

g E F

~(n)

Mn(K )

r

M (K), n

F.

matrices

V = (V,T)

in

see 2.2) are

K-bases

(including {Vb}

of

The study of such represent-

the space of polynomial functions in the

n

2

coefficients of a

(see 2.1 for a precise formulation.)

Finally we might

can also be regarded as the affine ring of the algebraic of a l l

n x n

matrices (singular or not) over

we may regard polynomial representations of ations of

n x n

We get another category (denoted

A = ~(n,r),

which are homogeneous of degree

general element

~(n),

obtained by using

ations is the subject of these lectures. in

a positive

the ring of all

The objects

are called polynomial representations of

~(n,r)

n

KF-modules, and the associated representations

the matrix representations

on

~(n),

(we shall later denote this category by

called polynomial

V)

with the dual space

the group of all non-singulsr

with coefficients in

mOdA(K~)

KF

and conversely.

CLn(K),

K,

so that

as rational represent-

Now suppose once more that identity

IF ,

and that

A

F

is an aribtrary

is a subcoalgebra

all flnitary

functions on

to the maps

A : A ~ A ~ A, s : A ~ K.

com(A)

of all right

dimensional which is

F.

A-eomodules;

K-space,

= RV

A

of the space

A-module;

[Se, p. 38] or [Sw, p. 30]). as follows:

if

an object

V

of

the identities

Our category

(le).

relative

com(A)

is a finite-

T : V ~ V ® A

(T ~ ~A)T = (~V ~ A)~ , A-comodule

better references

V E mOdA(KF),

and write down the equations

of

So we may consider the category

(see [G, p. 138], where a right

referred to as a left

F(K F)

is itself a coalgebra,

together with a "structure map"

K-linear and satisfies

(IV ® s)T

com(A),

Then

semigroup with

are [H, p. ].6],

mOdA(KF)

take any

is equivalent

K-basis

Now define

is perversely

{Vb}

of

~ : V -+ V ~ A

to V

to be the

K-linear map given by equations

(Ig)

Y(Vb)

--

% a~B

It is easy to check that using (if) we see that Conversely

given an

the elements

rab

hold, so we may use It is evident that

v a ® rab

T T

satisfies

A ; (le)

such that these morphisms

of the basis

the comodule

(V, T),

{vh}.

Moreover

identities just given.

use equations

to define the left

of(V) c A .

f : V ~ V~

b ~ B.

(Ig)

to define

the comodule identities now show that

can be regarded as a right of morphism

for

is independent

A-comodule of

,

So every

A-comodule,

in

com(A)

KF-module

A-rational,

KF-maps in

KF-module

The definition

(see the references

are the same as

V = (V, ~).

left

and conversely.

(if)

cited)

mOdA(KF).

is

10

This formal transition trivial,

from

but it is nevertheless

20 begin with,

KF-modu]es

worth making,

the basic representation

(~e should here work in the category are possibly infinite-dimensional) discovered (see

[G]).

A-comodules

is rather

from several points of view.

theory of arbitrary

Mod(A),

A-comodules

whose objects

V = (V, T)

follows to a surprising extent the pattern

by R. Brauer and C. Nesbitt

[B],

to

for finite-dimensional

Included here is the possibility

algebra

of a modular

theory,

which we shall discuss below. Next,

the

an important

A-comodule

for the

K-algebra

is the dual of the coalgebra tile product (1)

~D

= Hon~(A,

structure

to be the map of

(~)(f)

(la).

b~longs ~ ~, = ([ o~

A

on A

A,

K).

K

can be regarded as

The algebra structure on

i.e.

into

profit by

if

~,D ~ A ,

we define

which takes the element

to

(lh)

see

also permits us to

fact, namely that every right A-comodule

a left module

f ~ A

interpretation

V,

to v

=

Z ~(fh)~(f~), h

The identity element of com(A),

® ~) (v),

we make for

~v b

=

(see

s : A-* K.

K -module

(1) This product

V = (V, T)

Working in terms of a basis

(ig))

% ~(rab)V a , a~B

for

b E B.

Therefore we have three kinds of matrix representation our original

If

A -module by the rule

into an

~ ~ A , v 6 V.

this rule becomes

(]i)

V

is

A

V = (V, ~),

relstlve

is often called "convolution"

associated

to the basis

{Vb}:

with

{vb}

A

(i)

the representation

g ÷ (rab(g))

whose elements are functions on

F,

of

F ;

(il)

the matrix

satisfying equations

(If), and which

can be thought of as a kind of representation of the coalgebra (iii)

the representation

equetions g ~ F all

(li).

let

We can recover

eg : A ~ K

f 6 A.

Then

e

E A ,

g

K-algebra map

(iii)

with If

e , A

(i)

from

and the map

for

So

modA(KF )

dimensional algebra

r),

for

satisfies may be extended linearly

(i).

is finite-dimenslonal,

A = ~(n,

for each

and if we compoee the representation

and

then it is quite elementary to show mod(A )

mOdA(KF) by composition with the map

Jn the case

e

finally

given by

eg(f) = f(g),

i.e.

e :F + A

are equivalent; this amounts

to showing that every finite-dimensional left in

A ,

A ;

(ill7 very easily: g" ,

g, g' ~ I'.

e : KF ~ A ,

we recover

that the categories

of the algebra

be "evaluation at

e eg, = egg, , elF = s , to a

£ ~ (~(rab))

R = (tab)

e.

A -module

V

yields a module

Schur exploited this fact

end could thereby work with the finite-

AK(n~ r)

= SK(n, r)

(which I have called the

"Sehur algebra" in these lectures; see 2.3, 2.4), instead of with the i1~flnite-dimensicnal and irrelevantly complicated group-algebra If

A

is infinite-dimensional,

V ~ mOdA(KF) is dense in O~ group

F

it is often useful to regard modules

~s modules over some "dense" subalgebra A

~en

if, for every A = K[F]

0 # a 6 A,

by(F)

of

there is some

F

(see

K,

of

A*

~ ~ S

[CFS, §6]).

In ease

F

such that

hy(F)

S

the

is simply-

mOdA(KF)

sets up an equivalence of eategorles (J. Sullivan; see

Moreover in that case

(S

one may take for

connected and semisimple, the correspondence between mod(S)

S

is the affine ring of a connected algebraic

over an algebraically closed field

"hyperalgebra"

KF.

can be identified with an algebra

and [CPS, 6~8]). UK

12

constructed out of the complex semisimple Lie algebra associated with the root system of algebra

UK

F

(W. Haboush:

[CPS, 6.5, 6.6] o r [ H a ,

1.3]).

This

(which is sometimes defined to be the hyperalgebra of

F)

has an explicit basis with sufficiently good multiplicative properties to make it immensely valuable in studying the rational representations of In an important paper

[CL]

F.

R, Carter and G. Lusztig have used the

hyperalgebra -- rather than the Schur algebra -- to investigate the polynomial representations of

F = GLn(K).

Carte1-Lusztig use the idea, which is derived from C. Chevalley's fundamental paper

[Ch]

family of all groups

on split semisimple algebraic groups, that the

GLn(K )

(n

fixed,

of commutative rings) is "defined over

K Z".

varying over some class

K

This makes possible a '~odular

theory" for tile polynomial representations of these groups, which in its essentials corresponds to R. Brauer's modular representation theory for finite groups. as follows. the class

We can give a sufficiently general setting for such a theory

Suppose we have a family K

of all infinite fields,

K~-subcoalgebra of are satisfied. ZI. i.e.

The (a)

O~basis.

F(KFK).

(Q

AQ = (AA, ~Q, ~Q)

AQ , and

Z2~

For each

K ~ K

(here

~

®v~ ,

means

is a semigroup and

K

A~K

in is a

Suppose also that the following two conditions

is a laLtice in

{av.l of

FK

where for each

denotes the rational field.)

Q-coalgebra AZ

{FK, ~K} ,

AQ ,

(b)

there is a and

"e~ten~ion of scalars").

AZ ® K

contains a

which means

z-form

AZ = ~v Zav

&Q(A Z) _c A Z @ A Z ,

for some

~:Q(~Z) --c Z.

K-coalgebra isomorphism is made into a

AZ ,

aK : A Z ® K + A Z

K-coa]gebra by

13

In this case we say that the f~mily means of

Let

n : ~ g -+ EndcE

semisimple Lie algebra dimension n, and let or [St, p. 17 ]). defined by SLn(K ).

~ EZ

be an "admissible lattice" in

its

K ~ K

let

FK

For each pair

(~,v)

showed in over

Z.

[Ch]

of

(see also

The relevant

generated by the

cQ

isomorphisms, .tnd take

(see [Bo, p. A-5]

be the Chevalley group over

K cpKk @' c ~

gc K = L fly X cK

is a

we take this to be

[Bo,

]#thatthe

§4 AZ

the maps

of eK

A0

g = (g~v)

,

we deduce that

family

~..

Chevalley

{FK, AK}

is defined

is just the subring of

are

c~v~t ® 1K ~ cK

K-algebra (as well as

K-coalgebra)

K,

and the family

defined by the

(FK, ~ )

{FK, AK}

Z-bialgebra

AZ .

[Se, p. 46].)

Example 5. K E K. r ~ 0

Z",

A0

for all

is an affine algebraic group defined over is an "affine group scheme over

K

K-subcoalgebra (hence

F(KFK);

Z-form ;

E

of finite

(From the standpoint of algebraic group theory, each pair

See

by

define the coefficient function

From the equations

K-subbialgebra)

E

elements can be regarded as matrices

K-subalgebra generated by all the

even a

Z

be a faithful representation of a complex

over a complex vector space

For each

IY, E Z ;

cK : g ~ gl~v " ~v the

is defined over

AZ .

Example 4.

in

{FK, ~ }

Fix the positive integer

For

AK

(see 2.1).

the fa~aily Az(n, r) {F K, ~ ( n ,

we may take either

and let

~(n),

or

FK = GLn(K)

AK(n,r)

for each

for some fixed

It is completely elementnry to verify that in each case

{FK' AK}

is defined over

are described in 2.5. r)}.

n,

Z;

the relevant

Z-forms

Az(n) ,

in these lectures, we study the family

/4

The first essential of the modular representation family

{FK, AK}

reduction. K E K. which

which is defined over

We shall ~ i t e

Then an object FQ

acts.

equations like

(lj)

If

VQ

Z,

~

for the category

in

MQ

mod~(KFK) ,

is a finlte-dimensional

{Vb, Q : b E B}

N Z r]b(g)v a = " aEB

Here the functions

is a

Q-basis of

rQ ab

,

for

for any

Q-space on

VQ ,

we have

belong to

AQ

and satisfy equations like

'

a subset

Z-form (or admissible lattice) of

VQ

V Z = ~ ZVb, Q

for some

if

VZ (a)

Q-basis

All the coefficient functions

b E B,

g ~ -0 'r

We make the following definition:

(b)

is the process of modular

(le)

gvb, n~ =

~.~hich means

theory of any

rQ ab '

of VZ

VQ

(if)

is called a

is a lattice in

r iVb,Q}

of

VQ ,

VQ ,

and

relative to this basis, lie in

AZ . Another way of expressing condition ~n

AQ-COmodule by means of a map

like

(ig).

Then

(b)

VQ

into

using equations

to

YQ(v z) i V z e A z •

Now suppose that (~

is to convert

~Q : VQ ~ VQ ® AQ ,

is equivalent

(b')

(b)

means

rabK

=

®Z )

(if).

We can make the

into an object of

~K (rQab ® 1 K)

~K : AZ ® K ~ ~

K E K.

of

AK '

~

,

as follows.

usimg the

postulated in Z2.

So we may define an action of

K-coalgebra K rab

These FK

K-space

on

VK

VK = V Z @ K

Define elements isomorphism

satisfy equations like by equations

15

(ik)

Z a6B

gvb, K =

Here

r[~(g)Va,K=u '

Vb, K = Va, Q ® 1 K ,

converted,

via the

A general z-form

Z-form

into

V Z , V~ ,...

VK = VZ ® K ,

V K , V K' ,...

Different

may give non-isomorphic

MK ,

but another general theorem

form to Brauer and Nesbitt)

have the same composition

the notion of decomposition

is

possesses at least one

[G, (2.2d), p. 159]).

VQ

in

VQ

is called modular reduction.

VQ 6 MQ

or

of the same

V~ = V~ @ K ....

(due i~ its original

VK

that every

([Se, Lemma 2, p. 43]

g E F K , b ~ B,

The p~oeess by which

b 6 B.

VZ ,

theor~n guarantees

VZ

Z-forms

for

for

says that all these modules

factor multiplicities;

numbers can be defined

from this

([Se, p. 44] or

[G, (2.5a), p. 162]).

In these lectures we take and study

KF-modules

a fixed homogeneity algebra

SK(n,r )

of

SK(n,r )

V = (V,T)

degree r

where

K

I, above).

and it is shown how

when the latter is given its natural structure

group

G(r)

then every module

V

in

~(n,r)

Schur's multiplication

V

elements

.

in

By definition, homogeneous

SK(n,r)

of degree

r

theorem

SK(n,r)

~(n,r)

An alternative

can

description

as module for the synmletric (2.6e):

if

char K = O ,

in

(see (2.3b)) provides

MK(n,r ) .

are easily expressed Weights

the character of

in

is completely reducible.

method for calculating with modules

~

Schur's

rule for

spaces" of such a module

, for

algebra of the r-th tensor space

E ~r ,

This has as corollary

MK(n,r)

In chapter 2 the Schur

KF-modules

SK(n,r)-modules , and conversely.

is that it is the endomorphism

.

is an infinite field,

which belong to the category

(see Example

is defined,

be regarded as left

F = GL (K) n

V

and characters

For example, in

n

variables

the "weight-

terms of certain idempotent

are discussed

is a symmetric polynomial

in a set of

an effective

over

XI,...,X n .

in chapter 3. Z ,

which is

16

In 3.5 is reproduced

the argument by which Schur showed that the isomorphism

classes of irreducible modules with the partitions

in

% = (%1,...,%n)

Of course Schur considered only minor modification

K ;

yet been determined

of

for an arbitrary

we write this

are in one-one correspondence r

only the case

of an irreducible module of type of

~(n,r)

~%,p

% For

except in special

into not more than

K = ¢ ,

n

parts.

but his argument requires

infinite field

K .

The character

depends only on the characteristic p # O

these characters

cases.

For

~%,p

p = 0 ,

p

have not

Schur showed in

IS] that they are the symmetric functions now known as "Schur functions".

A

proof of this is given at the end of 3.5 - our proof uses some identities involving symmetric functions which can be found, Macdonald's

recent book

for example,

in I.G.

[M].

In chapters 4 and 5, I have departed widely from Schur's dissertation. These chapters K ,

are concerned with the construction,

of two modules

"explicit"

has a unique irreducible

{F%,K}, parts,

V~, K

in our category

in the sense of the "contravariant"

has a unique minimal

then

and

as

X

submodule,

factor module;

F%, K ~ VX, K ~ D%, K .

in

They are

They are dual

this is denoted

% of

gives a full set of irreducible modules

.

duality described

which is isomorphic

ranges over all partitions

% and for each

MK(n,r)

in the sense that a basis can be given for each.

to each other, V%, K

D~, K

for each

to r

F%, K .

F ,K .

D%, K

The set

into not more than

P~(n,r)

But for char K = p # 0 ,

in 2.7.

.

If

knowledge

n

char K = O , of the

F%, K

is still very incomplete.

The history of the modules indebted

to J. Towber

D%, K

and

V%, K

is interesting,

(see IT]) for much of the following

is generated by certain determinantal

expressions

and I am

information.

(here denoted

D%, K

(T%:Ti)),

17

whose significance in 1892.

as "primary covariants" was noted by J. Deruyts

Although Schur refers to two later papers of Deruyts,

no sign in [S] that he appreciated set of irreducible modules in "standard"

(T~:T i) ,

observation that the equivalently that the by G. Higman

M~(n,r)

.

The discovery of the basis of the [Y, 1902].

The

generate a Z-form V~, K

D%, Z

(and the Z-form

called these "Weyl modules",

M. Clausen

V%, K .

G.D. James describes,

are isomorphic to the D%, K .

They

Towber

~]

showed that

([DKR]) to construct both modules in his book

~a],

some

Chapter 6 r e t u r ~ t o

theory of D%, K

KF-modules which

His construction is quite different from those D%, K

in the sense of algebraic group theory.

Schur's disseration.

procedure by which he constructed G(r) .

D%, K

and more

we show in 4.8 that it yields the important and deep fact that

is an "induced" module,

group

1974].

[CI] has used recent combinatorial

G.-C. Rota and his collaborators

above;

were constructed,

are dual to each other - his framework is "functorial"

general than ours.

and

~L,

D%,Q - was made

and their construction was based on methods used

in the theory of semisimple algebraic groups. V%, K

in

V%,Z)

independently of all this, by R. Carter and G. Lusztig

and

The

can be constructed over an arbitrary field - or

(T%:Ti)

~Hi, 1965].

there is

that Deruyts had really given a complete

seems to go back to A. Young D%, K

[D!,

KF-modules

I have "reversed" the elegant from modules for the symmetric

This provides an interesting illumination of some recent work

of James on the modular representation

theory

of the symmetric groups.

1 Introduction

Issai Schur determined the polynomial representations of the complex general linear group GLn (C) in his doctoral dissertation [47], published in 1901. This remarkable work contained many very original ideas, developed with superb algebraic skill. Schur showed that these representations are completely reducible, that each irreducible one is “homogeneous” of some degree r ≥ 0 (see 2.2), and that the equivalence types of irreducible polynomial representations of GLn (C), of ﬁxed homogeneous degree r, are in one-one correspondence with the partitions λ = (λ1 , . . . , λn ) of r into not more than n parts. Moreover Schur showed that the character of an irreducible representation of type λ is given by a certain symmetric function Sλ in n variables (since described as “Schur function”; see 3.5). An essential part of Schur’s technique was to set up a correspondence between representations of GLn (C) of ﬁxed homogeneous degree r, and representations of the ﬁnite symmetric group G(r) on r symbols, and through this correspondence to apply G. Frobenius’ discovery of the characters of G(r) (see [17]). This pioneering achievement of Schur was one of the main inspirations for Hermann Weyl’s monumental researches on the representation theory of semi-simple Lie groups [54]. Of course Weyl’s methods, based on the representation theory of the Lie algebra of the Lie group Γ, and the possibility of integrating over a compact form of Γ, were very diﬀerent from the purely algebraic methods of Schur’s dissertation; in particular Weyl’s general theory contained nothing to correspond to the symmetric group G(r). In 1927 Schur published another paper [48] on GLn (C), which has deservedly become a classic. In this he exploited the “dual” actions of GLn (C) and G(r) on rth tensor space E ⊗r (see 2.6) to rederive all the results of his 1901 dissertation in a new and very economical way. Weyl publicized the method of Schur’s 1927 paper, with its attractive use of the “double centralizer property”, in his inﬂuential book “The Classical Groups” [55]. In fact the exposition in Chapters 3B and 4 of that book has become a standard treatment of polynomial representations of GLn (C) (and, incidentally, of Alfred Young’s representation theory of the symmetric group G(r)), and perhaps this explains the comparative neglect of

2

1 Introduction

Schur’s work of 1901. I think this neglect is a pity, because the methods of this earlier work are in some ways very much in keeping with the present-day ideas on representations of algebraic groups. It is the purpose of these lectures to give some accounts, in part based on the ideas of Schur’s 1901 dissertation, of the polynomial representations of the general linear groups GLn (K), where K is an inﬁnite ﬁeld of arbitrary characteristic. Our treatment will be “elementary” in the sense that we shall not use algebraic group theory in our main discussion. But it might be interesting to indicate here some general ideas from the representation theory of algebraic groups (or algebraic semigroups, since the group inverse is not important in this context), which are relevant to our work. Let Γ be any semigroup (i.e. Γ is a set, equipped with an associative multiplication) with identity 1Γ , and let K be any ﬁeld. A representation τ of Γ on a K-space V (i.e. a vector space over K) is a map τ : Γ → EndK (V ) which satisﬁes τ (gg ) = τ (g)τ (g ), τ (1Γ ) = IV , for all g, g ∈ Γ. (For any set V , we denote by IV the identity map on V .) We can extend τ linearly to give a map of K-algebras τ : KΓ → EndK (V ); here KΓ is the semigroup-algebra of Γ over K, whose elements are all formal linear combinations κ= κg g, κg ∈ K, g∈Γ

whose support supp κ = { g ∈ Γ : κg = 0 } is ﬁnite. We can make KΓ act on V by κv = τ (κ)(v) (κ ∈ KΓ, v ∈ V ), and thereby get a left KΓ-module, denoted (V, τ ), or simply V . A KΓ-map between such KΓ-modules (V, τ ), (V , τ ) is, by deﬁnition, a K-map f : V → V (i.e. f is a linear map) which satisﬁes τ (g)f = f τ (g) for all g ∈ Γ. A KΓ-map which is bijective is a KΓisomorphism, or an equivalence between the representations τ , τ . One has analogous deﬁnitions for right KΓ-modules; a right KΓ-module can be regarded as a pair (V, τ ) where τ : Γ → EndK (V ) is an anti-representation of Γ on the K-space V , i.e. τ (gg ) = τ (g )τ (g) for all g, g ∈ Γ, τ (1Γ ) = IV . The set K Γ of all maps Γ → K is a commutative K-algebra, with algebra operations deﬁned “pointwise”, e.g. f f is deﬁned to take g → f (g)f (g), for every element (“point”) g of Γ. The identity element 1 of K Γ takes each g ∈ Γ to the identity element 1K of K. If s ∈ Γ and f ∈ K Γ , then the left and right translates of f by s are deﬁned to be the maps Ls f, Rs f : Γ → K given by Ls f : g → f (sg),

Rs f : g → f (gs),

g ∈ Γ.

Each of the operators Ls , Rs maps K Γ into itself and is a K-algebra map (i.e. K-algebra homomorphism) K Γ → K Γ . In particular, Ls , Rs both belong to the space EndK (K Γ ). It is easy to check that R : s → Rs gives a representation of Γ on K Γ , while L : s → Ls gives an anti-representation. Thus K Γ can be made into a left KΓ-module (using R) and a right KΓ-module (using L). We denote both module actions by ◦, so that if s ∈ Γ and f ∈ K Γ we write s ◦ f = Rs f

and

f ◦ s = Ls f.

1 Introduction

3

Notice that these actions commute: (s ◦ f ) ◦ t = s ◦ (f ◦ t) for all s, t ∈ Γ and f ∈ K Γ . There is a linear map K Γ ⊗ K Γ → K Γ×Γ (⊗ means ⊗K ) which takes f ⊗ f (f, f ∈ K Γ ) to the function mapping Γ × Γ → K by (s, t) → f (s)f (t), for all s, t ∈ Γ. This linear map is injective, and we use it to identify K Γ ⊗ K Γ with a subspace of K Γ×Γ . The semigroup structure on Γ gives rise to two maps ∆ : K Γ → K Γ×Γ

and

ε : K Γ → K,

as follows: if f ∈ K Γ , then ∆f : (s, t) → f (st), and ε(f ) = f (1Γ ). Both ∆, ε are K-algebra maps. We shall say that an element f ∈ K Γ is ﬁnitary, or is a representative function, if it satisﬁes any one of the conditions F1, F2, F3 below: these three conditions are in fact equivalent (see e.g. [24, Chapter 2]). F1. The left KΓ-submodule KΓ ◦ f generated by f is ﬁnite-dimensional. F2. The right KΓ-submodule f ◦ KΓ generated by f is ﬁnite-dimensional. F3. ∆f ∈ K Γ ⊗ K Γ . This means that there exist elements fh , fh ∈ K Γ (where h runs over some ﬁnite index set) such that fh ⊗ fh . (1a) ∆f = h

This equation is equivalent to the system of equations fh (s)fh (t), all s, t ∈ Γ. (1b) f (st) = h

It is also equivalent to each of the following systems (1c) t◦f = fh (t)fh , all t ∈ Γ, h

or (1d)

f ◦s=

fh (s)fh , all s ∈ Γ.

h

The set F = F (K Γ ) of all ﬁnitary functions f : Γ → K is a K-bialgebra (see [51] for the deﬁnitions of coalgebras and bialgebras). It is a K-subalgebra of K Γ , and is also closed to ∆ in the sense that ∆F ⊆ F ⊗ F (this means that if f is ﬁnitary, the functions fh , fh in (1a) can be chosen to be themselves ﬁnitary). The K-space F , equipped with the maps ∆ : F → F ⊗F , ε : F → K, is a K-coalgebra; these two structures on F , of algebra and coalgebra, are linked by the fact that ∆ and ε are both K-algebra maps (see [24, p. 15]). Finitary functions on Γ appear as coeﬃcient functions of ﬁnite-dimensional representations of Γ. Suppose τ is a representation of Γ on a ﬁnitedimensional K-space V . If { vb : b ∈ B } is a K-basis of V , we have equations (1e) τ (g)vb = gvb = rab (g)va , for g ∈ Γ, b ∈ B; a∈B

4

1 Introduction

here rab (g) ∈ K. The functions rab : Γ → K (a, b ∈ B) are called coeﬃcient functions of τ , or of the KΓ-module V = (V, τ ). The K-span of these functions 1 is a subspace of K Γ called the coeﬃcient space of τ , or of the KΓ-module V . K · r We denote this space by cf(V ) = ab ; it is elementary to verify a,b that it is independent of the choice of the basis {vb }. The matrix R = (rab ) gives a matrix representation of Γ, i.e. R(gg ) = R(g)R(g ), R(1Γ ) = (δab ) for all g, g ∈ Γ (δab is the Kronecker delta). These conditions translate into conditions on the coeﬃcients rab , viz. (1f ) ∆rab = rac ⊗ rcb , ε(rab ) = δab , all a, b ∈ B. c∈B

The matrix R = (rab ) is sometimes called an “invariant matrix” [20, p. 140]. From the ﬁrst equations it follows that all the coeﬃcient functions rab are ﬁnitary, hence that cf(V ) is a subspace of F = F (K Γ ). But (1f) also shows that C = cf(V ) is a subcoalgebra of F , i.e. that ∆C ⊆ C ⊗ C. As a matter of fact, every ﬁnitary function f : Γ → K lies in the coeﬃcient space of some ﬁnite-dimensional KΓ-module V ; for this purpose we could take V = KΓ ◦ f (see F1). It is for this reason that ﬁnitary functions are sometimes called “representative functions”. If S is any K-algebra (possibly of inﬁnite dimension as K-space), mod(S) shall denote the category of all ﬁnite-dimensional left S-modules. Similarly, mod (S) is the category of all ﬁnite-dimensional right S-modules. An algebraic representation theory of Γ over K could be deﬁned as follows: ﬁrst choose a subcoalgebra A of F (K Γ ), i.e. A is a K-subspace of F (K Γ ) satisfying ∆A ⊆ A ⊗ A. Then “A-representation theory” of Γ, is deﬁned to be the study of the full subcategory modA (KΓ) of mod(KΓ), whose objects are all ﬁnite-dimensional left KΓ-modules V such that cf(V ) ⊆ A. (The morphisms f : V → V between two objects V , V of this category are, by deﬁnition, just the KΓ-maps.) In some contexts we say that a KΓ-module V is “rational”, or more precisely “A-rational”, if cf(V ) ⊆ A; then modA (KΓ) is the category of ﬁnite-dimensional A-rational left KΓ-modules. It is clear that submodules, quotient-modules and ﬁnite direct sums of A-rational modules, are themselves A-rational. We can deﬁne the category modA (KΓ) of ﬁnitedimensional right KΓ-modules which are A-rational in the same way. The assumption ∆A ⊆ A ⊗ A implies that if f ∈ A, then the functions fh , fh appearing in (1a) can themselves be chosen to belong to A. Then from (1c), (1d) follows that A is a left and right KΓ-submodule of K Γ ; also by quite elementary calculations that any ﬁnite-dimensional left (or right) KΓ-submodule V of A belongs to the category modA (KΓ) (or modA (KΓ)). Examples. 1. Let Γ be an aﬃne algebraic group over an algebraically closed ﬁeld K (see for example [24, p. 21]), and A = K[Γ] the ring of regular functions on Γ 1

In [24], this is called the “space of representative functions” of τ , or V .

1 Introduction

5

(A is often called the aﬃne ring of Γ). Then modA (KΓ) is the category of rational (ﬁnite-dimensional) KΓ-modules in the usual sense of algebraic group theory. In this case, A is not only a subcoalgebra of F (K Γ ), but a subbialgebra (see [49, p. 46]). The same remarks apply when Γ is an aﬃne algebraic semigroup. 2. Let Γ be a ﬁnite semigroup, then of course F (K Γ ) = K Γ . If we take A = K Γ, then modA (KΓ) = mod(KΓ). The (left and right) KΓ-module structures on A, are dual to the (right and left) “regular” KΓ-module structures on K given by multiplication: we may identify K Γ with the dual space (KΓ)∗ = HomK (KΓ, K). 3. Let K be an inﬁnite ﬁeld, n a positive integer, and Γ = GLn (K), the group of all non-singular n × n matrices with coeﬃcients in K. We could take A = AK (n), the ring of all polynomial functions f : Γ → K (see 2.1). The objects (V, τ ) in modA (KΓ) (we shall later denote this category by MK (n), see 2.2) are called polynomial KΓ-modules, and the associated representations (including the matrix representations R = (rab ) obtained by using the K-bases {vb } of V ) are called polynomial representations of Γ. The study of such representations is the subject of these lectures. We get another category (denoted by MK (n, r) in 2.2) by taking A = AK (n, r), the space of polynomial functions on Γ which are homogeneous of degree r in the n2 coeﬃcients of a general element g ∈ Γ (see 2.1 for a precise formulation). Finally we might mention that AK (n) can also be regarded as the aﬃne ring of the algebraic semigroup Mn (K) of all n×n matrices (singular or not) over K, so that we may regard polynomial representations of GLn (K), as rational representations of Mn (K), and conversely. Now suppose once more that Γ is an arbitrary semigroup with identity 1Γ , and that A is a subcoalgebra of the space F (K Γ ) of all ﬁnitary functions on Γ. Then A is itself a coalgebra, relative to the maps ∆ : A → A ⊗ A and ε : A → K. So we may consider the category com(A) of all right A-comodules; an object V of com(A) is a ﬁnite-dimensional K-space, together with a “structure map” γ : V → V ⊗ A which is K-linear and satisﬁes the identities (γ ⊗ IA )γ = (IV ⊗ ∆)γ, (IV ⊗ ε)γ = IV (see [20, p. 138], where a right A-comodule is perversely referred to as a left A-comodule; better references are [24, p. 16], [49, p. 38] or [51, p. 30]). Our category modA (KΓ) is equivalent to com(A), as follows: if V ∈ modA (KΓ), take any K-basis {vb } of V and write down the equations (1e). Now deﬁne γ : V → V ⊗ A to be the K-linear map given by equations (1g) γ(vb ) = va ⊗ rab , for b ∈ B. a∈B

It is easy to check that γ is independent of the basis {vb }. Moreover using (1f) we see that γ satisﬁes the comodule identities just given. Conversely given an A-comodule (V, γ), use equations (1g) to deﬁne the elements rab of A; the comodule identities now show that (1f) hold, so we may use (1e) to deﬁne the

6

1 Introduction

left KΓ-module V = (V, τ ). It is evident that cf(V ) ⊆ A. So every A-rational, left KΓ-module can be regarded as a right A-comodule, and conversely. The deﬁnition of morphism f : V → V in com(A) (see the references cited) is such that these morphisms are the same as KΓ-maps in modA (KΓ). This formal transition from KΓ-modules to A-comodules is rather trivial, but it is nevertheless worth making, from several points of view. To begin with, the basic representation theory of arbitrary A-comodules (we should here work in the category Com(A), whose objects V = (V, γ) are possibly inﬁnite-dimensional) follows to a surprising extent the pattern discovered by R. Brauer and C. Nesbitt for ﬁnite-dimensional algebras (see [5, 20]). Included here is the possibility of a modular theory, which we shall discuss below. Next, the A-comodule interpretation also permits us to proﬁt by an important fact, namely that every right A-comodule can be regarded as a left module for the K-algebra A∗ = HomK (A, K). The algebra structure in A∗ is the dual of the coalgebra structure on A, i.e. if ξ, η ∈ A∗ , we deﬁne the product2 ξη to be the map of A into K which takes the element f ∈ A to (1h) ξη(f ) = ξ(fh )η(fh ), h

see (1a). The identity element of A∗ is ε : A → K. If V = (V, γ) belongs to com(A), we make V into an A∗ -module by the rule ξv = (IV ⊗ ξ)(γ(v)), for ξ ∈ A∗ , v ∈ V . Working in terms of a basis {vb } of V , this rule becomes (see (1g)) (1i) ξvb = ξ(rab )va , for b ∈ B. a∈B

Therefore we have three kinds of matrix representation associated with our original KΓ-module V = (V, τ ), relative to the basis {vb }: (i) the representation g → (rab (g)) of Γ; (ii) the matrix R = (rab ) whose elements are functions on Γ, satisfying equations (1f), and which can be thought of as a kind of representation of the coalgebra A; (iii) the representation ξ → (ξ(rab )) of the algebra A∗ , given by equations (1i). We can recover (i) from (iii) very easily: for each g ∈ Γ let eg : A → K be “evaluation at g”, i.e. eg (f ) = f (g), for all f ∈ A. Then eg ∈ A∗ , and the map e : Γ → A∗ satisﬁes eg eg = egg , e1Γ = ε, for g, g ∈ Γ. So e may be extended linearly to a K-algebra map e : KΓ → A∗ , and if we compose the representation (iii) with e, we recover (i). If A is ﬁnite-dimensional, then it is quite elementary to show that the two categories modA (KΓ) and mod(A∗ ) are equivalent; this amounts to showing that every ﬁnite-dimensional left A∗ -module V yields a module in modA (KΓ) by composition with the map e. Schur exploited this fact in 2

This product is often called “convolution”.

1 Introduction

7

the case A = AK (n, r), and could thereby work with the ﬁnite-dimensional algebra AK (n, r)∗ = SK (n, r) (which I have called the “Schur algebra” in these lectures, see 2.3, 2.4), instead of with the inﬁnite-dimensional and irrelevantly complicated group algebra KΓ. If A is inﬁnite-dimensional, it is useful in many cases to regard modules V ∈ modA (KΓ) as modules over some “dense” subalgebra S of A∗ (S is dense in A∗ if, for every 0 = a ∈ A, there is some ξ ∈ S such that ξ(a) = 0). When A = K[Γ] is the aﬃne ring of a connected algebraic group Γ over an algebraically closed ﬁeld K, one may take S the “hyperalgebra” hy(Γ) of Γ (see [9, §6]). In case Γ is simply-connected and semisimple, the correspondence between modA (KΓ) and mod(S) sets up an equivalence of categories (J. Sullivan; see [9, 6.8]). Moreover in that case hy(Γ) can be identiﬁed with an algebra UK constructed out of the complex semisimple Lie algebra associated with the root system of Γ (W. Haboush; [9, 6.5, 6.6] or [22, 1.3]). This algebra UK (which is sometimes deﬁned to be the hyperalgebra of Γ) has an explicit basis with suﬃciently good multiplicative properties to make it immensely valuable in studying the rational representations of Γ. In an important paper [6] R. Carter and G. Lusztig have used the hyperalgebra— rather than the Schur algebra—to investigate the polynomial representations of Γ = GLn (K). Carter-Lusztig use the idea, which is derived from C. Chevalley’s fundamental paper [7] on split semisimple algebraic groups, that the family of all groups GLn (K) (n ﬁxed, K varying over some class K of commutative rings) is “deﬁned over Z”. This makes possible a “modular theory” for the polynomial representations of these groups, which in its essentials corresponds to R. Brauer’s modular representation theory for ﬁnite groups. We can give a suﬃciently general setting for such a theory as follows. Suppose we have a family {ΓK , AK }, where for each K in the class K of all inﬁnite ﬁelds, ΓK is a semigroup and AK is a K-subcoalgebra of F (K ΓK ). Suppose also that the following two conditions are satisﬁed. (Q denotes the rational ﬁeld.) ) contains a Z-form AZ , i.e. (a) AZ is Z1. The Q-coalgebra AQ = (AQ , ∆Q , εQ a lattice in AQ , which means AZ = ν Zaν for some Q-basis {aν } of AQ , and (b) ∆Q (AZ ) ⊆ AZ ⊗ AZ , εQ (AZ ) ⊆ Z. Z2. For each K ∈ K there is a K-coalgebra isomorphism αK : AZ ⊗ K → AK (here ⊗ means ⊗Z , and AZ ⊗K is made into a K-coalgebra by “extension of scalars”). In this case we say that the family {ΓK , AK } is deﬁned over Z by means of AZ . Examples. 4. Let π : GC → EndC E be a faithful representation of a complex semisimple Lie algebra GC over a complex vector space E of ﬁnite dimension n, and let EZ be an “admissible lattice” in E (see [4, p. A-5] or [50, p. 17]). For

8

1 Introduction

each K ∈ K let ΓK be the Chevalley group over K deﬁned by π, EZ ; its elements can be regarded as matrices g = (gµν ) in SLn (K). For each pair (µ, ν) deﬁne the coeﬃcient function cK µν : g → gµν . From the equa K K K tions ∆cµν = λ cµλ ⊗cλν , we deduce that the K-subalgebra generated by ΓK ); all the cK µν is a K-subcoalgebra (hence even a K-subbialgebra) of F (K we take this to be AK . Chevalley showed in [7] (see also [4, §4]) that the family {ΓK , AK } is deﬁned over Z. The relevant Z-form AZ of AQ is just the subring of AQ generated by the cQ µν ; the maps αK are K-algebra (as K well as K-coalgebra) isomorphisms, and take cQ µν ⊗ 1K → cµν for all µ, ν. (From the standpoint of algebraic group theory, each pair (ΓK , AK ) is an aﬃne algebraic group deﬁned over K, and the family {ΓK , AK } is an “aﬃne group scheme over Z”, deﬁned by the Z-bialgebra AZ . See [49, p. 46].) 5. Fix a positive integer n, and let ΓK = GLn (K) for each K ∈ K. For AK we may take either AK (n), or AK (n, r) for some ﬁxed r ≥ 0 (see 2.1). It is completely elementary to verify that in each case the family {ΓK , AK } is deﬁned over Z; the relevant Z-forms AZ (n), AZ (n, r) are described in 2.5. In these lectures, we study the family {ΓK , AK (n, r)}. The ﬁrst essential of the modular representation theory of any family {ΓK , AK } which is deﬁned over Z, is the process of modular reduction. We shall write MK for the category modAK (KΓK ), for any K ∈ K. Then an object VQ in MQ is a ﬁnite-dimensional Q-space on which ΓQ acts. If { vb,Q : b ∈ B } is a Q-basis of VQ , we have equations like (1e) Q rab (g)va,Q , for g ∈ ΓQ , b ∈ B. (1j) gvb,Q = a∈B Q belong to AQ , and satisfy equations like (1f). We make Here the functions rab the following deﬁnition: a subset VZ of VQ is called a Z-form (or admissible lattice) of VQ if (a) VZ is a lattice in VQ , which means VZ = b Zvb,Q for some Q-basis {vb,Q } of VQ , and Q (b) All the coeﬃcient functions rab , relative to this basis, lie in AZ .

Another way of expressing condition (b) is to convert VQ into an AQ -comodule by means of the map γQ : VQ → VQ ⊗ AQ , using equations like (1g). Then (b) is equivalent to (b’) γQ (VZ ) ⊆ VZ ⊗ AZ . Now suppose that K ∈ K. We can make the K-space VK = VZ ⊗K (here ⊗ Q K = αK (rab ⊗ 1K ) ∈ AK , means ⊗Z ) into an object of MK , as follows. Deﬁne rab using the K-coalgebra isomorphism αK : AZ ⊗ K → AK postulated in Z2. K These rab satisfy equations like (1f). So we may deﬁne an action of ΓK on VK by equations

1 Introduction

(1k)

gvb,K =

9

K rab (g)va,K , for g ∈ ΓK , b ∈ B.

a∈B

Here vb,K = vb,Q ⊗ 1K , for b ∈ B. The process by which VQ is converted, via the Z-form VZ , into VK is called modular reduction. A general theorem guarantees that each VQ ∈ MQ possesses at least one Z-form VZ (see [49, Lemma 2, p. 43] or [20, (2.2d), p. 159]). Diﬀerent Z-forms VZ , VZ , . . . of the same VQ may give non-isomorphic VK = VZ ⊗ K, VK = VZ ⊗ K, . . . in MK , but another general theorem (due in its original form to Brauer and Nesbitt) says that all these modules VK , VK , . . . have the same composition factor multiplicities; from this the notion of decomposition numbers can be deﬁned (see [49, p. 44] or [20, (2.5a), p. 162]). In these lectures we take Γ = GLn (K), where K is an inﬁnite ﬁeld, and study KΓ-modules V = (V, τ ), which belong to the category MK (n, r), for a ﬁxed homogeneity degree r (see Example 3, above). In chapter 2 the Schur algebra SK (n, r) is deﬁned, and it is shown how KΓ-modules in MK (n, r) can be regarded as left SK (n, r)-modules, and conversely. An alternative description of SK (n, r) is that it is the endomorphism algebra of the rth tensor space E ⊗r , when the latter is given its natural structure as a module for the symmetric group G(r). This has as corollary Schur’s theorem (2.6e): if char K = 0, then every module V in MK (n, r) is completely reducible. Schur’s multiplication rule for SK (n, r) (see (2.3b)) provides an eﬀective method for calculating with modules in MK (n, r). For example, the “weight spaces” of such a module V are easily expressed in terms of certain idempotent elements ξa in SK (n, r). Weights and characters are discussed in chapter 3. By deﬁnition, the character of V is a symmetric polynomial over Z, which is homogeneous of degree r in a set of n variables X1 , . . . , Xn . In 3.5 is reproduced the argument by which Schur showed that the isomorphism classes of irreducible modules in MK (n, r) are in one-one correspondence with the partitions λ = (λ1 , . . . , λn ) of r into not more than n parts. Of course Schur considered only the case K = C, but his argument requires only minor modiﬁcation for an arbitrary inﬁnite ﬁeld K. The character of an irreducible module of type λ depends only on the characteristic p of K; we write this φλ,p . For p = 0 these characters have not yet been determined except in special cases. For p = 0, Schur showed in [47] that they are the symmetric functions now known as “Schur functions”. A proof of this is given at then end of 3.5— our proof uses some identities involving symmetric functions which can be found, for example, in I. G. Macdonald’s recent book [39]. In chapters 4 and 5, I have departed widely from Schur’s dissertation. These chapters are concerned with the construction, for each λ and for each K, of two modules Dλ,K and Vλ,K in our category MK (n, r). They are “explicit” in the sense that a basis can be given for each. They are dual to each other, in the sense of the “contravariant” duality described in 2.7. Vλ,K has a unique irreducible factor module; this is denoted Fλ,K . Dλ,K has a unique minimal submodule, which is isomorphic to Fλ,K . The set {Fλ,K }, as λ ranges over all partitions λ of r into not more than n parts, gives a full set of

10

1 Introduction

irreducible modules in MK (n, r). If char K = 0, then Fλ,K ∼ = Vλ,K ∼ = Dλ,K . But for char K = p = 0, knowledge of Fλ,K is still very incomplete. The history of the modules Dλ,K and Vλ,K is interesting, and I am indebted to J. Towber (see [52]) for much of the following information. Dλ,K is generated by certain determinantal expressions (here denoted (Tl : Ti )), whose signiﬁcance as “primary covariants” was noted by J. Deruyts [13], in 1892. Although Schur refers to two later papers of Deruyts, there is no sign in [47] that he appreciated that Deruyts had really given a complete set of irreducible modules in MC (n, r). The discovery of the basis of the “standard” (Tl : Ti ), seems to go back to A. Young [58, 1902]. The observation that the Dλ,K can be constructed over an arbitrary ﬁeld—or equivalently that the (Tl : Ti ) generate a Z-form Dλ,Z in Dλ,Q —was made by G. Higman [23, 1965]. The Vλ,K (and the Z-form Vλ,Z ) were constructed, independently of all this, by R. Carter and G. Lusztig [6, 1974]. They called these “Weyl modules”, and their construction was based on methods used in the theory of semisimple algebraic groups. Towber [52] showed that Dλ,K and Vλ,K are dual to each other—his framework is “functorial” and more general than ours. M. Clausen [8] has used recent combinatorial theory of G.-C. Rota and his collaborators [15] to construct both modules Dλ,K and Vλ,K . G. D. James describes, in his book [27], some KΓ-modules which are isomorphic to the Dλ,K . His construction is quite diﬀerent from those above; we show in 4.8 that it yields the important and deep fact that Dλ,K is an “induced” module, in the sense of algebraic group theory. Chapter 6 returns to Schur’s dissertation. I have “reversed” the elegant procedure by which he constructed KΓ-modules from modules for the symmetric group G(r). This provides an interesting illumination of some recent work of James on the modular representation theory of the symmetric group.

§2.

Polynomial

2.1

Notation, Let

representations

of

GLn(K):

The Schur algebra

etc.

n

be a positive

integer,

K

an infinite field, and

F = GL (K) n

the group of all non-singular of elements of ates to each the of

n = {l,...,n}, g ~ F

K-subalgebra A

its

of

the

c

matrices over

let

c

KF

generated

the polynomial

For each pair

D,v

be the function which associg~v"

Denote by

independent over

over K

A

c v~,v

functions on

regarded as the algebra of all polynomials

c v(~,~ ~

K.

by the functions

are algebraically

~v

~ ~

~,v)-coefficient

are, by definition,

infinite,

n × n

in

Since

K,

so that

n

the elements

~ n);

F.

2

~K(n)

or

is

K A

can be

"indeterminates"

n_).

For each

r > 0

we denote by

of the elements expressible

~(n,r)

the subspace of

A

as polynomials which are homogeneous

consisting of degree

r

(n2+r-l) in the

c v.

particular

Then

AK(n,r)

~(n,0)

= K-IA,

1 A : g -+ iK(g ~ F).

The

(2.1b)

has finite dimension where

1A

K-algebra

A = AK(n ) =

If integers

n,r

set of all functions

i : r ~ n.

vector or "multi-index" group on the set on the right on

(both

A

has the standard

Z" r~0

by

i~

K-space;

in

function

grading

AK(n,r).

are given, we write

Such a function

i = (il,.°.,i r)

r = {l,...,r} I(n,r)

~ i)

denotes

as

r the constant

G(r)

for the

is usually written as a

with values

is denoted

I(n,r)

or

= (i (i) .... ,i (r)) ,

i~ ~ _n" G.

The symmetric

It acts naturally

so that

~ ~ G(r)

~9

acts as a "place-permutation" on the set

I(n,r)

× l(n,r)

that the elements j = i~ and

for some

g = j~

i,J

as

= (i~,j~).

Similarly

(i,j) ~

AK(n,r)

in bijeetive

The categories The maps

& : KF~

with

K r×F c

) =

the

IF

A(Cp,q)

= 1

P,q

or

k = i~

that

AK(n,r)

is

is not uniquely

if and only if monomials

G(r)-orbits

of

(i,j) ~

(2.1b), I(n,r)

determined (k,~).

and these are

× I(n,r).

(n2+r-l) . r -

is

~ g : KF ~ K

O,

=

(see introduction)

behave

as

(~,v 6 n_):

k6nZ

c k ~' ckv

of

F.

for any "multi-indices"

5

notice

(i,j)

,

from the rule for multiplying

for the unit element

(2.2b)

that

that

MK(n) , MK(n,r)

&(c

These follow

the pair ci, j = Ck, ~

of these orbits

on the functions

(2.2a)

i.e.

... c i . , r Jr

the set of distinct

correspondence

2.2

deduce,

in fact

K-basis

the number

follows

Of course

(2.1b);

Thus

to indice

by the monomials

i,j f I(n,r),

has as

means

act also

i ~ j

G(r)-orbit,

(k,g)

of the use of this notation,

K-module,

by the monomial

G(r)

We write

are in the same

ci, j = c . . ci2J2 i131

for all

We make

, ~ G(r).

(2.1b)

Here

i ~ I(n,r).

(i,j)~

I(n,r)

~ ~ G(r).

As an example spanned,

by

of

for some

on each

Z s61

Since p,q

g(c v) =

6Gv.

two matrices,

&,c

are both multiplicative

6 I(n,r)

of length

Cp, s ~ c

according

r ~ I,

6 s,q'

as

and from the formula

p = q

g(Cp,q) or

p

=

~ q.

P,q

we

20

These formulae show that bialgebra) of (for

r = 0

MK(n,r)

F(~),

A = ~(n)

is a sub-coalgebra (hence also a sub-

and that each

this is because

~(n,r)

Al A = 1A ~ IA).

for the categories

modAK(n)(kT)

is the category of finite-dimensional representations of

F = GLn(K);

and

(left)

~K(n,r)

is a sub-coalgebra of

AK(n)

We shall write

and

ME(n)

mOdAK(n,r)(F$).

Thus

MK(n)

~Y-modules which afford "polynomial"

the subcategory consisting of those

affording representations in which all the coefficients are polynomials homogeneous of degree IS, p. 5] in case

r

in the

K = ~,

c v.

By an argument first given by Schur

but valid for any infinite field

K

of any character-

istic, we have (2.2c) V =

Theorem

~ re0

V r,

Each

KF-module

where for each

V (~(n)

r e 0

v '

cf(V r) E AK(n,r) ,

i,e.

has a direct sum decomposition is a sub-module of

V

with

r

Vr ¢ MK(n'r)"

In other words each polynomial representation of

F

is equivalent to

a direct sum of homogeneous ones. Remark A

In fact (2.2c) follows from n general theorem on

is any coalgebra which is a direct sum

(p then

ranging over an index set V =

%~

Vp,

where for each

maximum sub-comodule of Vp (com(Ap)

P).

V

A

of sub-coalgebras

p

This theorem says that if p ( P

such that

[G, p. 156, (io6C).

A = Z~ P

A-comodules, where

the space

cf(Vp) E Ap,

Vp i.e.

V = (,~)

~ com(A),

is the unique such that

The proof given there does not depend on

the assumption that the sub-coalgebra summands

R

P

are minimal].

A

21 2.3

The Schur algebra

SK(n,r )

Theorem (2.2c) shows that each indecomposable module homogeneous,

i.e

V ~ MK(n,r)

confine our attention fixed, and define

for some

r > O.

to homogeneous modules.

SK(n,r )

As

K-space,

S K ~ , r ) has basis

{ei, j : i,J E l(n,r)} of

SK(n,r )

of

i

ci, j

if and only if

r

r ~ 0

be

~(n~r).

: i,j 6 I(n,r)}

For

dual to the basis

i,j 6 I(n,r),

~i,j

is the element

given by

~i'j(Cp'q) As with the

From now on, let

= HOmK(~(n,r) , K).

{~i,j

~(n,r).

is

This means that we may as well

to be the dual space of

SK(n,r) = ~ ( n , r )

V E ~(n)

=

if if

(i,j) ~ (p,q) (i,j) ~ (p,q)

all

•

p,q ~ I ( n , r ) .

we have an e~uality rule to take into account:

(i,j) ~ (k,g).

The dimension of

SK(n,r)

is

~i,j = ~k,~

of course

/ = dim AK(n,r ) . Since

algebra.

AK(n,r )

We saw in

is a coalgebra,

its dual

§i that the product

is defined as follows:

if

c ~ ~(n,r)

&(c) =

~

SK(n,r)

of elements

~,~

of

SK(n,r)

and if

Z ct @ ct t

where the sum is finite and the

ct, c t ~ AK(n,r),

(2.3a)

= ~ ~(ct)~(e ~),

(~n)(c)

is an associative

then

t The unit element of for all

c E AK(n,r).

SK(n,r)

will be denoted

a;

it is given by

s(c) = c(l r)

22

Applying (2.3a) to a basis element

of

c = Cp,q

AK(n ,r),

we ge<

(see (2.2b)) ( ~ ) (Cp, q) = s~ I(n,r) Specializing to the case where SK(n,r),

~ (c p, s)~ (c s,q ).

= ~i,j' ~ = ~k,g

we deduce a

Multiplication rule for

(2.3b)

SK(n,r ).

~i,j~k,Z

=

% {Z(i,j,k,g,p,q). IK} ~p,q, P,q

where the sum is over a set of re~r_esentatives of

are basis elements of

l(n,r) × l(n,r),

(p,q)

of the

G (r)-orbits

and

Z(i,J,k,~,p,q_)_ = Card{s 6 I(n,r)

: (i,J) ~ (p,s)

and

(k,~) ~ (s,q)}.

This multiplication rule (rather differently expressed) in due to Schur [S, p. 20]. (2.3c) (i)

Some special cases are worth noticing.

For any

i,j,k,~ 6 l(n,r)

~i,jSk,g = 0

i,i i,j

unless

i,J

j ~ k,

and

s, p,q k ~ s, From

with hence

distinct

~i,i

~i,j~k,g ~ 0,

(i,J) ~ (p,s)

and

then by (2.3b) there must

(k,g) ~ (s,q).

This implies

j ~ s

j ~ k.

(2.3c) follows that

Of course if

and

i,j j,j "

For example, (i) holds because if exist

there hold

i ~ J,

then

~2 :'i,i =

(i,i) ~ (J,j)

and

~i,i~j,j = 0

if

and hence

~i,i = <J,j"

But the

~i,i'

i # J.

form a set of mutually orthogonal idempotents, and their sum

is the unit element

e

of

SK(n,r).

This last equation,

23

sum over a set of representives of the

(2.3d) of

L = Z ~i,i, i l(n,r)

C (r)-orbits

c

is proved by evaluating both sides at all the basis elements Of great importance in the modular theory for for fixed

n,r,

the scheme or family of algebras

in the following sense. K ~i,j

elements Sz(n,r)

of

of

Let us use a superscript

SK(n,r).

SQ(n,r),

K

is "defined ever

is a

Z-order in

K-algebras

Z",

to denote the basis Z-submodule

SQ(n,r).

is

And for any field

Sz(n,r) @ K ~ SK(n,r)

which takes

K

~ ,j ® IK ~ ~i,j

The map

e : KF ~ SK(n,r)

For each

g E F

we define the element

eg(C) = c(g) (c E AK(n,r)). for all

g,g' E F;

the map

g ~ e

of

SK(n,r)

Q ~i,j(i,j E I(n,r)),

which is generated by the

there is an isomorphism of

each

2.4

is the fact that,

n

It is clear from (2.3b) that the

multiplicatively closed - - i t K,

GL

~(n,r) .

of

P,q

g

also

eg E SK(n,r )

by

It is clear from (2.3a) and (i) that

eI =

s by the definition of

linearly we get a map

~.

e : KF ~ SK(n,r)

egeg, = egg,

So if we extend which a morphism

K-algebras. Any function

f : KF ~ K. <=Z~g

(2.4a)

g

f E KF

has a unique extension to a linear map

With this convention,

EKF,

is "evaluation at

e(~)

' c -+ c(K),

the image under <";

all

e

of an element

i.e.

e E ~(n,r).

Propositions (2.4b,c) give the most important facts about

e.

24

(2.4b)

Proposition

(ii)

Let

Then

f ( AK(n,r )

Proof

(i)

exist

some

(i)

Y = Ker e,

e

is surjective.

and let

f

be any element

if and only if

If lme were a proper 0 # c E ~(n,r)

of

F K

f(Y) = 0. subspace

such that

of

SK(n,r)

eg(C)

= ~(n,r)

= c(g) = 0

,

there would

for all

g ( F,

a

contradiction. (ii)

If

f (AK(n,r)

by (2o4a).

So

K I' such that

and

f(Y) = 0. f(Y) = 0.

K E Y,

we have

Now suppose By

(i)

e(K) = 0

conversely

there

and hence

that

f

f(~) = 0,

is any element

of

is an exact sequence

0 ~ Y ~ KF ~ SK(n,r)

~ 0, *

from which it is clear y(e(~))

= f(<)

for all

there exists = e(<),

that there

c (1~(n,r)

then, we have

K E KF.

(2.4c)

Proposition

y (SK(n,r)

y(<) = <(c)

f(K) = e(K)(c)

and the proof of (2.4b)

some

By the natural

such that

f = c,

Let

exists

= c(<),

such that

isomorphism

SK(n,r)

~ ~(n,r),

for all

< (SK(n,r).

Put

all

K ~ KF.

Therefore

is complete.

V E mod(KF).

Then

V (~(n,r)

if and only if

YV = 0. Proof

Let

{v b}

by the action of if

rab(Y ) = 0

saying

be a basis KF

But of course the proof

of

V,

on this basis

for all

that all the

of

rab

a,b.

(rab)

(see (i)).

the invariant Clearly

By the last proposition,

lie in

AK(n,r) ,

this is the condition (2.4c)

and

is complete.

for

V

YV = 0

afforded

if and only

this equivalent

that is, that to belong

matrix

to

cf(V)

to

~ AK(n,r).

MK(n,r),

and so

25

There propositions are equivalent,

and in a very elementary

can be transformed

Kv

= e(K)v,

to relate the action on actions determine

V

of submodule,

all

[S,

on a module

V ~ ~(n,r),

F

action of

SK(n,r)

on a basis

We might mention

and

SK(n,r).

it holds for all

etc. coincide

on

Since both

V,

the

in the two categories.

that the action of

which is given by (2.4d),

{Vb}

of

% ~(rab)V a a

For it's clear that

outlined V

SK(n,r)

= AK(n,r)

is the same as that which

in the Introduction.

is given by equations

all

~ ~ SK(n,r) ,

(2.4d) holds, whenever

K f K

If the

(I~),

then the

and

K = g

Vb ~ B

and

v = v b.

By linearity

v ~ V.

theory

R. Brauer's

theory of modular representations

by Brauer himself

[B] and by Nakayama

More recently Serre [Se]

affine algebraic

groups,

GL

K

F K = GLn(K).)

n

(see also

of finite groups was

[N], to finite dimensional

[G]) extended

or rather to affine algebraic

group scheme

"reduction"

KF

is given by

~v b =

the group

in either category

was one of the main techniques used by Schur in his

p. 21].

action of

algebras.

V

mod(SK(n,r))

v E V

of the two algebras

is obtained by the general procedure

extended,

~ ~ KF,

module homomorphism,

dissertation

M0dular

way; an object

and

the same algebra of linear transformations

This category equivalence

2.5

MK(n,r )

into an object of the other, using the rule

(2.4d)

concepts

show that the categories

is a functor, which associates We can describe

or "decomposition"

the theory to

group schemes.

to each commutative

the characteristic

process as follows.

(The ring

modular

26 Let

Az
be the subsets of

consisting of those polynomials in the Z.

These are

"Z-forms" of Q-basis

{c~,j}

a(Az(n,r) 5 Z

(see (2.2b)).

of

i,j ~ l(n,r).

The

of all elements

Z-order

Now let

VQ

taking

such that

be any object of

which (i)

is the

Z-span of some

closed to the action of defined by rab

lie in

{vb}

Az(n,r).

That every

If

K,

there is a

c ,j ~ iK ~ ci, j

K-coalgebra

for all

~(Az(n,r) ~ Z.

M(n,r); we shall regard By a

Z-form of

Q-basls

{vb}

RQ = (rab)

where

(VQ,~) VQ

in

is the

of

VQ

as

VQ

is meant a subset

V,

and (ii) i~

is the invariant matrix

Az(n,r)-comodule determined hy

MQ(n,r)

contains at least one

Z-form,

, p. 256, §6] or [Se, p. 43, lemme 2] or [G , p. 158, (2.2c)].

If we now take any infinite field

hence as a

A~(n,r) ~ AZ(~,r) ~ ~(n,r),

Still another formulation of (ii) is that

QFQ-mOdule

follows from [B

VK = VZ ® K

is the

(see (I e )), then condition (li) just says that all the

T(V Z) ! V Z ® Az(n,r), VQ.

Sz(n,r ),

Az(n,r)

which we defined in (2.3), is the set

SQ(n,r)-module when this is convenient. VZ

and

For any infinite field

Sz(n,r)

~ ~ So(n,r)

for example

AQ(n,r),

Az(n,r) ® K ~ AK(n,r)

respectively,

whose coefficients all lie in

AQ(n), AQ(n,r);

Z-span of the

isomorphism

c

AQ(n), AQ(n,r)

K,

it is clear that the

can be regarded as a left module for ~K-mOdule in

M~(n,r).

K-space

SK(n,r ) ~ Sz(n,r) ~ K,

The transition from

VQ

to

VK

is

particularly easy to express in terms of invariant matrices; the invariant matrix

RK

defined by the

K-basis

{vb ® 1K}

of

VK,

(tab) = RQ

is the Invariant matrix defined by the basis

case where

K

has finite characteristic

the coefficient of

RQ.

is

(tab ~ IK) , where

{vb}

of

VQ.

In the

p, this amount to "reducing mod p"

27

Our notation in the preceding discussion conceals a disadvantage: there are in general many different VQ E MQ(n,r),

Z-forms

and the corresponding

may be not all isomorphic.

Vz,V~,...

KFK-mOdules

of a given

~Q-mOdule

V K = V Z ® K, V~ = V~ ~ K,...

However one of the classical results of modular

theory, deducible from [B , p. 257, (8)] or [Se, p. 44, theorhme 2] or [G, p. 162, (2.5a)], says that, for any type of simple E MK(n,r),

the multiplicity

depends only on case that d ~k'

VQ, i.e.

VQ = V

~ ( V K)

of

Lk

as composition factor in

is the same for all

is a simple

KFK-mOdule

Z-forms

VZ

of

VQ.

VK

In the

QFQ-mOdule, this multiplicity is often written

and referred to as a decomposition number for the modular reduction

MQ(n,r) ~ MK(n,r).

2.6

The module

E~r

Fix our infinite field E = E K = K.e I @ ... • K-e n {e

v

: v E n}

on which

=

gev

K,

and write

be an

n-dimensional

E

Z e ~E~ g~v ~

is an object of

Now let E<%r = E ~ . . ®

r ~ 0 E

Let

K-space with a basis

F acts "naturally":

=

% ~En

c v(g)e v ,

Since the corresponding invariant matrix is KF-module

F = F K = GLn(K ).

all

g E F, v E ~.

C = (c),

we see that the

AK(n,I).

be given, and then

in the usual way

(®

F

acts on the

here means

~).

{ei = eil ® "'" ~ eir : i E l(n,r)},

and relative to this the action of

F

is given by

r-fold tensor power Eer

has

K-basis

28

gej = ge

®.. ~

ge. ]r

Jl =

Z i(I(n,r)

The invariant matrix is E~r ( M K ( n , r ) . as an

(2.6a)

Sej =

Z i( I (n,r)

ci, j(g)ei,

all

. gilJl

g ~ F,

(ei,j) = C x ... × C,

According

SK(n,r)-module,

=

to

what

ei "" girJr

j ( l(n,r)

and this shows that

was said in (2~4),

E®r

can be regarded

by the rule

Z <(ci,j)ei, i(I(n,r)

all

< (SK(n,r) ,

j (l(n,r).

In a very famous paper [S'] which appeared in 1927, Schur rederived all the results of his 1901 dissertation ~r

[S] by an analysis of this module

Although his method gives a complete answer only when

char K = O,

it

is still valuable for fields of finite characteristic.

We make the s}~mmetric

group

act on the right of

Ef~r

G(r),

and hence also its group-algebra

KG(r),

by

(2.6b)

ei~

= ei~ ,

all

6 G(r).

i (l(n,r),

It is clear that this action commutes with that of with that of all

SK(n,r);

~ (SK(n,r),

KF,

we can verify from (2.6a) that

x ( Eo~r

and

~ (G(r).

or (what is the same) (~x)~

= ~(x~r)

We have however a stronger

statement. (2.6c)

Theorem (Schur)

representation afforded by the (i)

Im @

(ii)

Ker @ = 0.

Hence

= EndKG(r )(E ~ ) ,

SK(n,r ) ~ EndKG(r ) (E@r)

Let

~ : SK(n,r) ~ EndK(E~)

SK(n,r)-module and

E ~r.

Then

be the

for

2g

Proof

Each element

basis

{ei}

of

E®r.

of course, and the EndKG(r)(~ 'r)

0 E EndK(E Here

i,j

Ti, j ~ K.

Tin,j ~

the set

)

has matrix, say

(Ti,j)

for all

has a

lies in

Ti, j = i

and all

~p,q

of

SK(n,r )

I × I,

or

0

0

namely if

co

according as

(p,q).

~0 is such an orbit,

to be that

e E EndK(E~r)

whose

(i,j) ~ ~0 or not. (p,q) E I × I,

~r

is represented on

G(r)-orbit containing

~ ~ G(r).

K-basis in one-to-one correspondence with

G(r)-orbits On

has

i,j ~ I

it follows very readily from (2.6a) that, for any

~0 is the

e

From (2.6b) follows at once that

Ti,j,

EndKG(r)(E~r )

~ of all

element

I = I(n,r) ,

run independently over the set

define the corresponding basis element matrix

relative to the

(Ti,j) ,

if and only if

(2.6d)

Consequently

~r

by

Therefore

the basis

~(~p,q) = ~0 '

~

Now

where

induces an isomorphism

SK(n,r) -~ EndKG(r ) (E~r) , and this proves the theorem. Remark

The proof of (2.6c) shows that

sentation by the algebra of all condition having ~i,i

(2.6d).

Ti, j = 1

nr × nr

The basis element or

0

SK(n,r)

matrices

~p,q

according as

has a faithful matrix repre(Ti, j)

which satisfy

is represented by the matrix

(i,j) ~ (p,q)

or not.

The idempotents

are represented by diagonal matrices, and the "orthogonal" decomposition

(2.3d) is easy to deduce from this. (2.6e) then Proof

Corollary to (2.6c) (Sehur [S, S']). SK(n,r )

is semisimple.

Hence every

Under the given conditions on

is semisimple K-G(r)-module,

(since

char K

If

V ~ MK(n,r)

char K,

does mot divide

and in particular

E~r,

char K = 0,

or if char K = p > r,

is completely reducible.

the group algebra IG(r) I = r!).

K-G(r)

Therefore every

is completely reducible.

But the

endomomorphism algebra of a completely reducible module is semisimple,

so by

30

(2.6c), and

SK(n,r)

mod SK(n,r)

is semisimple.

is clearly "defined over is simply a version of

{~r}

Z"

,

r

fixed but with

Z.

GL -module, n

We say that the family

a

of

in the category

VQ,

{V K}

and for each

~(n,r).

varying,

GL

being regarded

n

See [Se, p. 46]).

Suppose that for each infinite field

VZ

K

in the sense of the following definition (which

V K E MK(n,r).

and

with

the definition of

as affine group scheme over

Z-form

~(n,r)

now completes the proof of (2.6e).

The family of modules

Definition

The equivalence of the categories

K

K

we have a

is defined over

an isomorphism

More exactly we say

{VK}

KFK-mOdule Z

if there is

6 K : VZ @ K ~ VK

is

Z-defined by

VZ

{6K}.

Example I.

Take

of

VQ

(we write

of

EK, E®r K ),

MK(n,r), ively.

for all

is defined over

Definition

The module

Suppose that

K

the

K

Z i~l(n,r)

Z.e i

is a

Z-form

for the basis elements

6K : V Z ~ K ~ V K

is an isomorphism in

taking ~(n,r).

So

Z. {VK} , {WK} Z,

Suppose we have for each

We say that the family

K-map

i E l(n,r),

both defined over

and for each

VZ =

e~,K, ei,K = eil,K ® "'" ® eir,K

and for each

e i ® 1 K ~ ei,K, {~K r}

V K = E~ r.

{eK}

By K

are both families of modules in VZ

and

{6K} ,

a morphism

is defined over

Z

GQ~ K

VK

>

OK - - >

and

{~K},

eK : V K ~ W K

the diagram shown commutes VZ ® K

WZ

WZ ® K

WK

if

eQ

maps

in

respectM~(n,r).

VZ

into

Wz,

31

Example 2. the

r th

Define the

r th

monomial

Dr, K

of a given

Z-form

of degree

in the variables

r

Dr, Z

The isomorphism

for all

i E I(n,r).

defined over

Z

in

by

becomes a

{Dr,K}

is defined over

belong

e I = el,Q,..., en = en,Q, takes

which have coefficients

e(i ) ,Q ~ 1K ~+ e(i ) ,K

V

In this sectlo~ we keep = HomK(V,K)

of a

(v E V).

K

one defines

a left

action

However the

to our category

which is still in

with this

{O K } is

KF-module if

fixed and write V E MK(n,r )

f E V ,

To make

V

g E F

(denoted

KF-module

NK(n,r).

V

by a d o t )

F-action

~L.(n,r). K-

KF-module

F

to "reverse" on

V

so defined will not, in

B u t i f we r e p l a c e

We denote by

can be

g ~ g-i of

F = F K.

define

into a left

(transposed matrix) in the above definition, we get a left V

Z ;

is the set of all homogeneous polynomials

K -module in a natural way:

by (g'f)(v) = f(g-lv). general,

Dr, K

which takes

It is clear now that the family of morphisms

(fg)(v) = f(gv)

multiplication;

KF-map:

is the restriction to

the traditional practice in group-theory is to use the map

on

There is

in the sense of the last definition.

The dual space made into right

Dr, Q

@K

K[el,.°.,en]

~]K : Dr,Z ~ K -~ Dr, K

Contravariant duality

fg E V

g E F,

We can show that the family

the relevant

2.7

K[el,...,en];

KF-module, such that

K-algebra automorphism of

-~ ge (~ E n).

Z.

to be

are regarded as commuting indeterminate..

has a unique structure as

of the unique

in

EK

K-map

in fact tha action on

e

of

e K : Eer -~ Dr(EK) taking e i = ell @ ... ~ e i to the K r e(i) = ell "'" eir ' for all i E I(n,r). It is well-known that

a surjectivc

r,K

Dr, K = Dr(EK)

homogeneous subspace of the polynomial

e I = el,K,..., e n = en, K

D

symmetric power

V°

g

-i

by

g

tr

KF-module structure

the space

V ,

equipped

32

(2.7a)

V°

(g-f)(v) = f(gtrv)

all

is called the "contravariant dual" to

g E F, f E V , v E V.

V;

an analogous dual applies to

rational modules over all semisimple algebraic groups a great

deal in recent years by Wong

F,

and has been used

[W], Verma [V] and Jantzen

[J].

It is convenient to express (2.7a) in terms of the action of It is easy to see that the J(
all

K-linear map

i,j E I(n,r)

In fact one has clearly, for any

J(~)(ci,j) = ~(cj,i),

and by taking

~

= e

we find that

is regarded as

the contravariant dual

(2.7c)

MK(n,r) ,

V°( = V )

J(eg) = e t r ' g

(,) : V × W ~ K

V, W

g E F.

gives an exact contravariant function on V ~ (V)

of

K-spaces,

V ~ (V°) °.

be modules in

MK(n,r).

Then a

K-bilinear form

is called contravariant if it has the property

(~v,w) = (v,J(~)w),

So if

~ E SK(n,r) , f E V , v E V.

and that the usual natural isomorphism

Let

all

reads

all

V ~ V°

gives a natural isomorphism Definitio N

i,j E I(n,r),

SK(n,r)-module the action (2.7a) which defines

(~.f)(v) = f(J($)v),

It is clear that

~ E SK(n,r),

all

g V E MK(n,r)

given by

is an involutory anti-automorphism of

(2.7b)

(2.7d)

J : SK(n,r ) -~ SK(n,r)

SK(n,r).

all

~ E SK(n,r) , v E V, w E W.

The proof of the next proposition is standard.

33 (2.7e)

Proposition

correspondence A

: V~

W°

If

V, W 6 ~ ( n , r )

between contravariant

in

~(n,r),

(,)

forms

(,) : V × W + K,

and morphisms

given by

A(v)(w)

The form

are given, there is a bijective

= (v,w),

is non-singular

all

v ~ V, w 6 W.

(= non-degenerate)

if and only if

F

is an

isomorphism. Example 1

We can use the last proposition to show that the module

self-dual,

that is that

~r

~ (~r)o

: E~r × ~ r

<,>

i,j ~ I(n,r).

But a simple calculation,

on

defined by

condition

<ei,ej>

=

6ij,

using (2.6a), shows that

(2.7d). Call <,>

all <,>

the canonical form

E~r .

Example 2

Let

Z-defined by

{VK} VZ

and

be a family of modules {6K}

V Z.

For each

a basis of

K,

write

(V K E ~ ( n , r ) )

as in the last section.

running over some finite index set Va, K =

B) be a basis of 6K(Va, Q ® 1 K)

For each Then a,Q

K,

write

{fa,K}

for the basis of

VzO = {f ~ VQ : f(Vz) --c Z} }.

which is

Let VQ

so that

{Va,Q} which

(~ Z-generates

{Va, K : a 6 B}

is

V K.

We can now show easily that the family

{f

is

For there is clearly a non-singular

bilinear form

satisfies the contravariant

~ K

E~r

is a

It is not hard to see that

6K : VZo ® K ~ VKo

which take

° {V K}

V K* = V oK

Z-form of {VK} o

is

fa,Q ~ IK °+ fa,K'

is defined over

o VZ,

dual to having

Z-defined by all

a 6 B.

Z.

{Va,K}°

Z-basis V oZ

and the maps

34

Example 3

Let

which are

Z-defined by

K

and

{WK}

is defined over

and for each

be families

VZ, {6 K}

we have a bilinear form

{(,)K } K,

{VK}

and

(VK

WZ, {~K }

(')K : VK x WK ~ K,

Z

if

v Z E VZ,

(,)Q

wZ E WZ

maps

and

WK

both in

respectively.

M~(n,r)~

If for every

we say that the family

Vz x Wz

into

z,

and for each

there holds

(SK(Vz ® IK), nK(wZ ® IK)) K = (Vz,Wz).l K. If all the (')K

are contravariant, W Ko

then we can

(given, for each

show

K,

by Proposition

family of morphisms

AK : VK

is defined over

For example, the family of canonical forms

Z.

EK~)r is defined over EK~r ~ (E~r) ° K

2.8

AK(n,r)

Z,

is a

these two

KF

on

derived from these forms.

as

KF-bimodule

KF-bimodule,

i.e.

actions commute.

there holds an equation (see 2.3).

<'>K

(2.7e))

and so therefore is the family of isomorphisms

We saw in the introduction (p.4) that the space f:F ÷ K

that the

KF

it is a left and right If we take an element

&(c) = Z c t ~ c't ' t

of all functions KF-module, and

c E AK(n,r)

for suitable elements

,

then

c t ,c't c AK(n,r)

By formulae (ic), (Id),

(2.8a)

g o c = ~ c~(g)c t , t

for any

g g F .

KF-actions, hence

c o g = Z ct(g)c' t' t

This shows that ~(n,r)

AK(n,r) is closed to both the left and right

is itself a

K -bimodule.

linearity, to give the action of an arbitrary element

Now extend (2°8a) by < c K :

35

c o < = Z ct(K)_- c tv

< o c = Ec'(K)ctt'

By (2.4b), we find (2.4c) shows that MK(n,r)

.

~(n,r)

AK(n,r)

Similarly of right

bimodule,

(2.8b)

K o c = c o ~ = O , ,

regarded

AK(n,r)

belongs

KF-modules

as left

K ~ Y = Ker e . K£-module,

to an analogously

(see 4.4).

AK(n,r)

belongs

defined

becomes

Then

an

to

category

SK(n,r)-

with actions

~ o c = Z$(c~)e t ,

c o % = Z~(ct)c ~ ,

Now let us define a bilinear rule:

for any

if

~ E SK(n,r)

form

, c c AK(n,r)

,

(

,

for

~ E SK(n,r)

•

by the

):SK(n,r)~K(n,r ) ~ K

then

($,c) = J ( O ( c )

This is non-singular, to

(

,

)

(see (2.7b)).

~,q E SK(n,r)

(2.8c)

in fact

,

{cj, i}

are dual bases with respect

The reader may check that for any

c e AK(n,r)

there holds

(~n,c) = (~,J(~) o c) = (~,c o J(q))

In fact, using

(2.3a),

all three expressions shows that left

{$i,j},

(

,

)

(2.7e) we deduce

(2.8b) and the d e f i n i t i o n just given are equal

is eontravariant,

SK(n,r)-modules that

.

to

SK(n,r)

of

( , ) ,

E(~,c~)(q,c t) • and

AK(n,r)

(and even when they are regarded AK(n,r)

~ (SK(n,r)) ° ,

we see that But

(2.8c)

being regarded

as right modules).

an isomorphism of

as

By

KF-bimodules.

2 Polynomial Representations of GLn (K): The Schur algebra

2.1 Notation, etc. Let n be a positive integer, K an inﬁnite ﬁeld, and Γ = GLn (K) the group of all non-singular n × n matrices over K. For each pair µ, ν of elements of n = {1, . . . , n}, let cµν ∈ K Γ be the function which associates to each g ∈ Γ its (µ, ν)-coeﬃcient gµν . Denote by A or AK (n) the K-subalgebra of K Γ generated by the functions cµν (µ, ν ∈ n); the elements of A are, by deﬁnition, the polynomial functions on Γ. Since K is inﬁnite, the cµν are algebraically independent over K, so that A can be regarded as the algebra of all polynomials over K in n2 “indeterminates” cµν (µ, ν ∈ n). For each r ≥ 0 we denote by AK (n, r) the subspace of A consisting of the elements expressible as polynomials which are homogeneous of degree r 2 as K-space; in parin the cµν . Then AK (n, r) has ﬁnite dimension n +r−1 r ticular AK (n, 0) = K · 1A , where 1A denotes the constant function 1A which maps g → 1K for all g ∈ Γ. The K-algebra A has the standard grading AK (n, r). (2.1a) A = AK (n) = r≥0

If integers n, r (both ≥ 1) are given, we write I(n, r) for the set of all functions i : r → n. Such a function is usually written as vector or “multiindex” i = (i1 , . . . , ir ) with values i ∈ n. The symmetric group on the set r = {1, . . . , r} is denoted G(r) or G. It acts naturally on the right on I(n, r) by iπ = (iπ(1) , . . . , iπ(r) ), so that π ∈ G(r) acts as “place-permutation” on each i ∈ I(n, r). We make G(r) act also on the set I(n, r) × I(n, r) by (i, j)π = (iπ, jπ). We write i ∼ j to indicate that the elements i, j of I(n, r) are in the same G(r)-orbit, i.e. that j = iπ for some π ∈ G(r). Similarly (i, j) ∼ (k, l) means that k = iπ and l = jπ for some π ∈ G(r). As an example of the use of this notation, notice that AK (n, r) is spanned, as K-space, by the monomials (2.1b)

ci,j = ci1 j1 ci2 j2 · · · cir jr ,

12

2 Polynomial Representations of GLn (K): The Schur algebra

for all i, j ∈ I(n, r). Of course the pair (i, j) is not uniquely determined by the monomial (2.1b); in fact ci,j = ck,l if and only if (i, j) ∼ (k, l). The space AK (n, r) has as K-basis the set of distinct monomials (2.1b), and these are in bijective correspondence with the G(r)-orbits of I(n, r) × I(n, r). Thus 2 . the number of these orbits is n +r−1 r

2.2 The categories MK (n), MK (n, r) The maps ∆ : K Γ → K Γ×Γ , ε : K Γ → K (see introduction) behave as follows on the functions cµν (µ, ν ∈ n): (2.2a) ∆(cµν ) = cµλ ⊗ cλν , ε(cµν ) = δµν . λ∈n

These follow from the rule for multiplying two matrices, and from the formula for the unit element 1Γ of Γ. Since ∆, ε are both multiplicative we deduce, for any “multi-indices” p, q ∈ I(n, r) of length r ≥ 1, cp,s ⊗ cs,q , ε(cp,q ) = δp,q . (2.2b) ∆(cp,q ) = s∈I

Here δp,q = 1 or 0, according as p = q or p = q. These formulae show that A = AK (n) is a subcoalgebra (hence also a subbialgebra) of F (K Γ ), and that each AK (n, r) is a subcoalgebra of AK (n) (for r = 0 this is because ∆1A = 1A ⊗ 1A ). We shall write MK (n) and MK (n, r) for the categories modAK (n) (KΓ) and modAK (n,r) (KΓ). Thus MK (n) is the category of ﬁnite-dimensional (left) KΓ-modules which aﬀord “polynomial” representations of Γ = GLn (K); and MK (n, r) is the subcategory consisting of those aﬀording representations in which all the coeﬃcients are polynomials homogeneous of degree r in the cµν . By an argument ﬁrst given by Schur [47, p. 5] in case K = C, but valid for any inﬁnite ﬁeld K of any characteristic, we have (2.2c) Theorem. Each KΓ-module V ∈ MK (n) has a direct sum decomposition Vr , V = r≥0

where for each r ≥ 0, Vr is a sub-module of V with cf(Vr ) ≤ AK (n, r), that is Vr ∈ MK (n, r). In other words each polynomial representation of Γ is equivalent to a direct sum of homogeneous ones. Remark. In fact (2.2c) follows from a general theorem onA-comodules, where A is any coalgebra which is a direct sum A = A of subcoalgebras A ( ranging over anindex set P ). This theorem says that if V = (V, τ ) ∈ com(A), then V = V , where for each ∈ P the space V is the unique maximum sub-comodule of V such that cf(V ) ≤ A , i.e. such that V ∈ com(A ) [20, p. 156, (1.6c). The proof given there does not depend on the assumption that the subcoalgebra summands R are minimal].

2.3 The Schur algebra SK (n, r)

13

2.3 The Schur algebra SK (n, r) Theorem (2.2c) shows that each indecomposable module V ∈ MK (n) is homogeneous, i.e. V ∈ MK (n, r) for some r ≥ 0. This means that we may as well conﬁne our attention to homogeneous modules. From now on, let r ≥ 0 be ﬁxed, and deﬁne SK (n, r) to be the dual space of AK (n, r): SK (n, r) = AK (n, r)∗ = HomK (AK (n, r), K). As K-space, SK (n, r) has basis {ξi,j : i, j ∈ I(n, r)} dual to the basis {ci,j : i, j ∈ I(n, r)} of AK (n, r). For i, j ∈ I(n, r), ξi,j is the element of SK (n, r) given by 1 if (i, j) ∼ (p, q) , all p, q ∈ I(n, r). ξi,j (cp,q ) = 0 if (i, j) ∼ (p, q) As with the ci,j we have an equality rule to take into account: ξi,j = ξk,l if and only if (i, j) ∼ (k, l). The dimension of SK (n, r) is of course equal 2 = dim AK (n, r). to n +r−1 r Since AK (n, r) is a coalgebra, its dual SK (n, r) is an associative algebra. We saw in §1 that the product ξη of elements ξ, η of SK (n, r) is deﬁned as follows: if c ∈ AK (n, r) and if ∆(c) = ct ⊗ ct t

where the sum is ﬁnite and the ct , ct ∈ AK (n, r), then ξ(ct )η(ct ). (2.3a) (ξη)(c) = t

The unit element of SK (n, r) will be denoted ε ; it is given by ε(c) = c(1Γ ) for all c ∈ AK (n, r). Applying (2.3a) to a basis element c = cp,q of AK (n, r), we get (see (2.2b)) ξ(cp,s )η(cs,q ). (ξη)(cp,q ) = s∈I(n,r)

Specializing to the case where ξ = ξi,j , η = ξk,l are basis elements of SK (n, r), we deduce a Multiplication Rule for SK (n, r). {Z(i, j, k, l, p, q).1K }ξp,q , (2.3b) ξi,j ξk,l = p,q

where the sum is over a set of representatives (p, q) of the G(r)-orbits of I(n, r) × I(n, r), and Z(i, j, k, l, p, q) = Card{ s ∈ I(n, r) : (i, j) ∼ (p, s) and (k, l) ∼ (s, q) }.

14

2 Polynomial Representations of GLn (K): The Schur algebra

This multiplication rule (rather diﬀerently expressed) is due to Schur (see [47, p. 20]). Some special cases are worth noticing. (2.3c) For any i, j, k, l ∈ I(n, r) there hold (i) ξi,j ξk,l = 0 unless j ∼ k, and (ii) ξi,i ξi,j = ξi,j = ξi,j ξj,j . For example, (i) holds because if ξi,j ξk,l = 0, then by (2.3b) there must exist s, p, q with (i, j) ∼ (p, s) and (k, l) ∼ (s, q). This implies j ∼ s and k ∼ s, hence j ∼ k. 2 = ξi,i , and ξi,i ξj,j = 0 if i ∼ j. Of course From (2.3c) follows that ξi,i if i ∼ j, then (i, i) ∼ (j, j) and hence ξi,i = ξj,j . But the distinct ξi,i form a set of mutually orthogonal idempotents, and their sum is the unit element ε of SK (n, r). This last equation, ξi,i , sum over a set of representatives of the G(r)-orbits (2.3d) ε = i of I(n, r) is proved by evaluating both sides at all the basis elements cp,q of AK (n, r). Of great importance in the modular theory for GLn is the fact that, for ﬁxed n, r, the scheme or family of algebras SK (n, r) is “deﬁned over Z”, in the K following sense. Let us use a superscript K to denote the basis elements ξi,j of SK (n, r). It is clear from (2.3b) that the Z-submodule SZ (n, r) of SQ (n, r), Q (i, j ∈ I(n, r)), is multiplicatively closed—it is which is generated by the ξi,j a Z-order in SQ (n, r). And for any ﬁeld K, there is an isomorphism of K-alQ K gebras SZ (n, r) ⊗ K ∼ . = SK (n, r) which takes each ξi,j ⊗ 1K → ξi,j

2.4 The map e : KΓ → SK (n, r) For each g ∈ Γ we deﬁne the element eg ∈ SK (n, r) by eg (c) = c(g) for all c ∈ AK (n, r). It is clear from (2.3a) and §1 that eg eg = egg for all g, g ∈ Γ; also e1 = ε by the deﬁnition of ε. So if we extend the map g → eg linearly we get a map e : KΓ → SK (n, r) which is a morphism of K-algebras. Any function f ∈ K Γ has a unique extension to a linear map f : KΓ → K. With this convention, the image under e of an element κ = κg g ∈ KΓ, is “evaluation at κ”; i.e. (2.4a)

e(κ) : c → c(κ), all c ∈ AK (n, r).

Propositions (2.4b), (2.4c) give the most important facts about e. (2.4b) Proposition. (i) e is surjective. (ii) Let Y = Ker e, and let f be any element of K Γ . Then f ∈ AK (n, r) if and only if f (Y ) = 0.

2.4 The map e : KΓ → SK (n, r)

15

Proof. (i) If Im e were a proper subspace of SK (n, r) = AK (n, r)∗ , there would exist some 0 = c ∈ AK (n, r) such that eg (c) = c(g) = 0 for all g ∈ Γ, a contradiction. (ii) If f ∈ AK (n, r) and κ ∈ Y , we have e(κ) = 0 and hence f (κ) = 0, by (2.4a). So f (Y ) = 0. Now suppose conversely that f is any element of K Γ such that f (Y ) = 0. By (i) there is an exact sequence e

0 −→ Y −→ KΓ −→ SK (n, r) −→ 0, from which it is clear that there exists an element y ∈ SK (n, r)∗ such that y(e(κ)) = f (κ) for all κ ∈ KΓ. By the natural isomorphism SK (n, r)∗ ∼ = AK (n, r), there exists c ∈ AK (n, r) such that y(ξ) = ξ(c) for all ξ ∈ SK (n, r). Put ξ = e(κ), then we have f (κ) = e(κ)(c) = c(κ), all κ ∈ KΓ. Therefore f = c, and the proof of (2.4b) is complete. (2.4c) Proposition. Let V ∈ mod(KΓ). Then V ∈ MK (n, r) if and only if Y V = 0. Proof. Let {vb } be a basis of V , and (rab ) the invariant matrix aﬀorded by the action of KΓ on this basis (see §1). Clearly Y V = 0 if and only if rab (Y ) = 0 for all a, b. By the last proposition, this is equivalent to saying that all the rab lie in AK (n, r), that is, that cf(V ) ≤ AK (n, r). But of course this is the condition for V to belong to MK (n, r), and so the proof of (2.4c) is complete. These propositions show that the categories MK (n, r) and mod(SK (n, r)) are equivalent, and in a very elementary way; an object V in either category can be transformed into an object of the other, using the rule (2.4d)

κv = e(κ)v, all κ ∈ KΓ, v ∈ V

to relate the action on V of the two algebras KΓ and SK (n, r). Since both actions determine the same algebra of linear transformations on V , the concepts of submodule, module homomorphism, etc. coincide in the two categories. This category equivalence was one of the main techniques used by Schur in his dissertation [47, p. 21]. We might mention that the action of SK (n, r) = AK (n, r)∗ on a module V ∈ MK (n, r), which is given by (2.4d), is the same as that which is obtained by the general procedure outlined in the introduction. If the action of Γ on a basis {vb } of V is given by equations (1e), then the action of SK (n, r) is given by ξ(rab )va , all ξ ∈ SK (n, r), b ∈ B. ξvb = a

For it is clear that (2.4d) holds, whenever κ = g and v = vb . By linearity it holds for all κ ∈ KΓ and v ∈ V .

16

2 Polynomial Representations of GLn (K): The Schur algebra

2.5 Modular theory R. Brauer’s theory of modular representations of ﬁnite groups was extended, by Brauer himself [5] and by Nakayama [42, 43, 44], to ﬁnite dimensional algebras. More recently Serre [49] (see also [20]) extended the theory to aﬃne algebraic groups, or rather to aﬃne algebraic group schemes. (The group scheme GLn is a functor, which associates to each commutative ring K the group ΓK = GLn (K).) We can describe the characteristic modular “reduction” or “decomposition” process as follows. Let AZ (n), AZ (n, r) be the subsets of AQ (n), AQ (n, r) respectively, consisting of those polynomials in the cµν whose coeﬃcients all lie in Z. These are “Z-forms” of AQ (n), AQ (n, r); for example, AZ (n, r) is the Z-span of the Q-basis {cQ i,j } of AQ (n, r), and we have ∆AZ (n, r) ⊆ AZ (n, r) ⊗ AZ (n, r) and ε(AZ (n, r)) ≤ Z (see (2.2b)). For any inﬁnite ﬁeld K, there is a K-coalQ gebra isomorphism AZ (n, r) ⊗ K ∼ = AK (n, r) which takes ci,j ⊗ 1K → cK i,j for all i, j ∈ I(n, r). The Z-order SZ (n, r) which we deﬁned in 2.3, is the set of all elements ξ ∈ SQ (n, r) such that ξ(AZ (n, r)) ≤ Z. Now let VQ be any object in MQ (n, r); we shall regard VQ as a module for SQ (n, r) when this is convenient. By a Z-form of VQ is meant a subset VZ which (i) is the Z-span of some Q-basis {vb } of VQ , and (ii) is closed to the action of SZ (n, r). If RQ = (rab ) is the invariant matrix deﬁned by {vb } (see (1e)), then condition (ii) just says that all the rab lie in AZ (n, r). Still another formulation of (ii) is that τ (VZ ) ⊆ VZ ⊗ AZ (n, r), where (VQ , τ ) is the AZ (n, r)-comodule determined by VQ . That every QΓQ -module VQ in MQ (n, r) contains at least one Z-form, follows from [5, p. 256, §6] or [49, p. 43, lemme 2] or [20, p. 158, (2.2c)]. Now take any inﬁnite ﬁeld K. It is clear that the K-space VK = VZ ⊗ K can be regarded as a left module for SK (n, r) ∼ = SZ (n, r) ⊗ K, hence as a KΓK -module in MK (n, r). The transition from VQ to VK is particularly easy to express in terms of invariant matrices; the invariant matrix RK deﬁned by the K-basis {vb ⊗ 1K } of VK , is (rab ⊗ 1K ), where (rab ) = RQ is the invariant matrix deﬁned by the basis {vb } of VQ . In the case where K has ﬁnite characteristic p, this amounts to “reducing mod p” the coeﬃcients of RQ . Our notation in the preceding discussion conceals a disadvantage: in general there are many diﬀerent Z-forms VZ , VZ , . . . of a given QΓQ -module VQ ∈ MQ (n, r), and the corresponding KΓK -modules VK = VZ ⊗ K, VK = VZ ⊗ K, . . . may be not all isomorphic. However one of the classical results of modular theory, deducible from [5, p. 258, (8)] or [49, p. 44, th´eor`eme 2] or [20, p. 162, (2.5a)], says that, for any type of simple KΓK module Lλ ∈ MK (n, r), the multiplicity mλ (VK ) of Lλ as a composition factor in VK depends only on VQ , i.e. is the same for all Z-forms VZ of VQ .

2.6 The module E ⊗r

17

In the case that VQ = Vµ is a simple QΓQ -module, this multiplicity is often written dµλ , and referred to as a decomposition number for the modular reduction MQ (n, r) → MK (n, r).

2.6 The module E⊗r Fix our inﬁnite ﬁeld K, and write Γ = ΓK = GLn (K). Let E = EK = K · e1 ⊕ · · · ⊕ K · en be an n-dimensional K-space with a basis { eν : ν ∈ n } on which Γ acts “naturally”: geν = gµν eµ = cµν (g)eµ , all g ∈ Γ, ν ∈ n. µ∈n

µ∈n

Since the corresponding invariant matrix is C = (cµν ), we see that the KΓmodule E is an object of AK (n, 1). Now let r ≥ 1, then Γ acts on the r-fold tensor power E ⊗r = E ⊗ · · · ⊗ E in the usual way (⊗ here means ⊗K ). The space E ⊗r has K-basis { ei = ei1 ⊗ · · · ⊗ eir : i ∈ I(n, r) }, and relative to this the action of Γ is given by gej = gej1 ⊗ · · · ⊗ gejr =

gi1 j1 · · · gir jr ei =

i∈I(n,r)

ci,j (g)ei ,

i∈I(n,r)

all g ∈ Γ, j ∈ I(n, r). The corresponding invariant matrix is (ci,j ) = C × · · · × C, and this shows that E ⊗r ∈ MK (n, r). According to what was said in 2.4, E ⊗r can be regarded as an SK (n, r)-module, by the rule (2.6a) ξej = ξ(ci,j )ei , all ξ ∈ SK (n, r), j ∈ I(n, r). i∈I(n,r)

In a very famous paper [48] which appeared in 1927, Schur rederived all the results of his 1901 dissertation [47] by an analysis of this module E ⊗r . Although his method gives a complete answer only when char K = 0, it is still valuable for ﬁelds of ﬁnite characteristic. We make the symmetric group G(r), and hence also its group algebra KG(r), act on the right of E ⊗r by (2.6b)

ei π = eiπ , all i ∈ I(n, r), π ∈ G(r).

It is clear that this action commutes with that of KΓ, or (what is the same) with that of SK (n, r); we can verify from (2.6a) that (ξx)π = ξ(xπ) for all ξ ∈ SK (n, r), x ∈ E ⊗r and π ∈ G(r). We have however a stronger statement.

18

2 Polynomial Representations of GLn (K): The Schur algebra

(2.6c) Theorem (Schur). Let ψ : SK (n, r) → EndK (E ⊗r ) be the representation aﬀorded by the SK (n, r)-module E ⊗r . Then (i) Im ψ = EndKG(r) (E ⊗r ), and (ii) Ker ψ = 0. Hence SK (n, r) ∼ = EndKG(r) (E ⊗r ). Proof. Each element θ ∈ EndK (E ⊗r ) has matrix, say (Ti,j ), relative to the basis {ei } of E ⊗r . Here i, j run independently over the set I = I(n, r), of course, and the Ti,j ∈ K. From (2.6b) follows at once that θ lies in EndKG(r) (E ⊗r ) if and only if (2.6d)

Tiπ,jπ = Ti,j , for all i, j ∈ I and all π ∈ G(r).

Consequently EndKG(r) (E ⊗r ) has a K-basis in one-to-one correspondence with the set Ω of all G(r)-orbits on I ×I, namely if ω is such an orbit, deﬁne the corresponding basis element θω to be that θ ∈ EndK (E ⊗r ) whose matrix (Ti,j ) has Ti,j = 1 or 0 according as (i, j) ∈ ω or not. Now it follows very readily from (2.6a) that, for any (p, q) ∈ I × I, the basis element ξp,q of SK (n, r) is represented on E ⊗r by ψ(ξp,q ) = θω , where ω is the G(r)-orbit containing (p, q). Therefore ψ induces an isomorphism SK (n, r) → EndKG(r) (E ⊗r ), and this proves the theorem. Remark. The proof of (2.6c) shows that SK (n, r) has a faithful matrix representation by the algebra of all nr × nr matrices (Ti,j ) which satisfy condition (2.6d). The basis element ξp,q is represented by the matrix having Ti,j = 1 or 0 according as (i, j) ∼ (p, q) or not. The idempotents ξi,i are represented by diagonal matrices, and the “orthogonal” decomposition (2.3d) is easy to deduce from this. (2.6e) Corollary (Schur [47, 48]). If char K = 0, or if char K = p > r, then SK (n, r) is semisimple. Hence every V ∈ MK (n, r) is completely reducible. Proof. Under the given conditions on char K, the group algebra KG(r) is semisimple (since char K does not divide |G(r)| = r!). Therefore every KG(r)-module, and in particular E ⊗r , is completely reducible. But the endomorphism algebra of a completely reducible module is semisimple, so by (2.6c), SK (n, r) is semisimple. The equivalence of categories MK (n, r) and mod SK (n, r) now completes the proof of (2.6e). ⊗r The family of modules (EK ), with r ﬁxed but with K varying, is clearly “deﬁned over Z” in the sense of the following deﬁnition (which is simply a version of the deﬁnition of GLn -module, GLn being regarded as aﬃne group scheme over Z. See [49, p. 46]).

2.7 Contravariant duality

19

Deﬁnition. Suppose that for each inﬁnite ﬁeld K we have a KΓK -module VK ∈ MK (n, r). We say that the family {VK } is deﬁned over Z if there is a Z-form VZ of VQ , and for each K an isomorphism δK : VZ ⊗ K ∼ = VK in the category MK (n, r). More exactly we say {VK } is Z-deﬁned by VZ and {δK }. ⊗r Example 1. Take VK = EK . The module VZ = i∈I(n,r) Z · ei is a Z-form of VQ (we write eµ,K , ei,K = ei1 ,K ⊗ · · · ⊗ eir ,K for the basis elements ⊗r ), and for each K the K-map δK : VZ ⊗ K → VK taking of EK , EK ⊗r } ei ⊗ 1K → ei,K , for all i ∈ I(n, r), is an isomorphism in MK (n, r). So {EK is deﬁned over Z. Deﬁnition. Suppose {VK }, {WK } are both families of modules in MK (n, r), both deﬁned over Z, by VZ and {δK }, WZ and {ηK }, respectively. Suppose we have for each K a morphism θK : VK → WK in MK (n, r). We say that the family {θK } is deﬁned over Z if θQ maps VZ into WZ , and for each K the diagram shown commutes. VZ ⊗ K

θ Q ⊗ πK

ηK

δK VK

/ WZ ⊗ K

θK

/ WK

Example 2. Deﬁne the rth symmetric power Dr,K = Dr (EK ) of EK to be the rth homogeneous subspace of the polynomial ring K[e1 , . . . , en ]; the elements e1 = e1,K , . . . , en = en,K are regarded as commuting indeterminates. ⊗r → Dr (EK ) taking ei = ei1 ⊗ · · · ⊗ eir to There is a surjective K-map θK : EK the monomial e(i) = ei1 · · · eir , for all i ∈ I(n, r). It is well-known that Dr,K has a unique structure as a KΓ-module, such that θK becomes a KΓ-map; in fact the action on Dr,K of a given g ∈ Γ, is the restriction to Dr,K of the unique K-algebra automorphism of K[e1 , . . . , en ] which takes eµ → geµ for all µ ∈ n. We can show that the family {Dr,K } is deﬁned over Z; the relevant Z-form Dr,Z in Dr,Q is the set of all homogeneous polynomials of degree r in the variables e1 = e1,Q , . . . , en = en,Q , which have coeﬃcients in Z. The isomorphism ηK : Dr,Z ⊗ K → Dr,K takes e(i),Q ⊗ 1K → e(i),K for all i ∈ I(n, r). It is clear now that the family of morphisms {θK } is deﬁned over Z in the sense of the last deﬁnition.

2.7 Contravariant duality In this section we keep K ﬁxed and write Γ = ΓK . The dual space V ∗ = HomK (V, K) of a KΓ-module V ∈ MK (n, r) can be made into a right KΓ-module in a natural way: if f ∈ V ∗ , g ∈ Γ,

20

2 Polynomial Representations of GLn (K): The Schur algebra

we deﬁne f g ∈ V ∗ by (f g)(v) = f (gv) for all v ∈ V . To make V ∗ into a left KΓ-module the traditional practice in group theory is to use the map g → g −1 to “reverse” multiplication; one deﬁnes a left action (denoted by a dot) of Γ on V ∗ by (g · f )(v) = f (g −1 v). However the KΓ-module V ∗ so deﬁned will not, in general, belong to our category MK (n, r). But if we replace g −1 by g tr (transposed matrix) in the above deﬁnition, we get a left KΓ-module structure on V ∗ which is still in MK (n, r). We denote by V ◦ the space V ∗ , equipped with this action (2.7a)

(g · f )(v) = f (g tr v), all g ∈ Γ, f ∈ V ∗ , v ∈ V .

The module V ◦ is called the “contravariant dual” to V ; an analogous dual applies to rational modules over all semisimple algebraic groups Γ, and has been used a great deal in recent years by Wong [56], Verma [53] and Jantzen [29]. It is convenient to express (2.7a) in terms of the action of SK (n, r). It is easy to see that the K-linear map J : SK (n, r) → SK (n, r), deﬁned by J(ξi,j ) = ξj,i for all i, j ∈ I(n, r), is an involutory anti-automorphism of SK (n, r). In fact one has clearly, for any ξ ∈ SK (n, r) (2.7b)

J(ξ)(ci,j ) = ξ(cj,i ), all i, j ∈ I(n, r),

and by taking ξ = eg we ﬁnd that J(eg ) = egtr , all g ∈ Γ. So if V ∈ MK (n, r) is regarded as SK (n, r)-module the action (2.7a) which deﬁnes the contravariant dual V ◦ (= V ∗ ) reads (2.7c)

(ξ · f )(v) = f (J(ξ)v), all ξ ∈ SK (n, r), f ∈ V ∗ , v ∈ V .

It is clear that V → V ◦ gives an exact contravariant functor on MK (n, r), and that the usual isomorphism V → (V ∗ )∗ of K-spaces, gives a natural isomorphism V → (V ◦ )◦ . Deﬁnition. Let V , W be modules in MK (n, r). Then a K-bilinear form (, ):V ×W →K is called contravariant if it has the property. (2.7d)

(ξv, w) = (v, J(ξ)w), all ξ ∈ SK (n, r), v ∈ V , w ∈ W .

The proof of the next proposition is standard. (2.7e) Proposition. If V, W ∈ MK (n, r) are given, there is a bijective correspondence between contravariant forms ( , ) : V × W → K and morphisms Λ : V → W ◦ in MK (n, r), given by Λ(v)(w) = (v, w), all v ∈ V , w ∈ W . The form ( , ) is non-singular (=non-degenerate) if and only if Λ is an isomorphism.

2.8 AK (n, r) as KΓ-bimodule

21

Example 1. We can use the last proposition to show that the module E ⊗r is self-dual, that is that E ⊗r ∼ = (E ⊗r )◦ . For there is clearly a non-singular bilinear form , : E ⊗r × E ⊗r → K, deﬁned by ei , ej = δij , all i, j ∈ I(n, r). But a simple calculation, using (2.6a), shows that , satisﬁes the contravariant condition (2.7d). Call , the canonical form on E ⊗r . Example 2. Let {VK } be a family of modules (VK ∈ MK (n, r)) which is Z-deﬁned by VZ and {δK } as in the last section. Let {va,Q } (a running over some ﬁnite index set B) be a basis of VQ which Z-generates VZ . For each K, write va,K = δK (va,Q ⊗ 1K ) so that { va,K : a ∈ B } is a basis of VK . We can now show easily that the family {VK◦ } is deﬁned over Z. For each K, write {fa,K } for the basis of VK∗ = VK◦ dual to {va,K }. Then VZ◦ = { f ∈ VQ : f (VZ ) ⊆ VZ } is a Z-form of VQ◦ , having Z-basis {fa,Q }. It is not hard to see that {VK◦ } is Z-deﬁned by VZ◦ and the maps δˆK : VZ◦ ⊗K → VK◦ which take fa,Q ⊗ 1K → fa,K , all a ∈ B. Example 3. Let {VK } and {WK } be families (VK and WK both in MK (n, r)) which are Z-deﬁned by VZ , {δK } and WZ , {ηK } respectively. If for every K we have a bilinear form ( , )K : VK × WK → K, we say that the family {( , )K } is deﬁned over Z if ( , )Q maps VZ × WZ to Z, and for each K, and for each vZ ∈ VZ , wZ ∈ WZ there holds δK (vZ ⊗ 1K ), ηK (wZ ⊗ 1K ) = (vZ , wZ )Q · 1K . K

If all the ( , )K are contravariant, then we can show that the family of mor◦ (given, for each K, by Proposition (2.7e)) is deﬁned phisms ΛK : VK → WK ⊗r is deﬁned over Z. For example, the family of canonical forms , K on EK ⊗r ⊗r ◦ ) derived over Z, and so therefore is the family of isomorphisms EK → (EK from these forms.

2.8 AK (n, r) as KΓ-bimodule We saw in the introduction (p. 2) that the space K Γ of all functions f : Γ → K is a KΓ-bimodule, i.e. it is a left and right KΓ-module, and these two actions of KΓ commute. If we take an element c ∈ AK (n, r), then there holds an equation ∆(c) = t ct ⊗ ct , for suitable elements ct , ct ∈ AK (n, r) (see 2.3). By formulae (1c), (1d), (2.8a) g◦c= ct (g)ct , c ◦ g = ct (g)ct , t

t

for any g ∈ Γ. This shows that AK (n, r) is closed to both the left and the right KΓ-actions, hence AK (n, r) is itself a KΓ-bimodule. Now extend (2.8a) by linearity, to give the action of an arbitrary element κ ∈ KΓ:

22

2 Polynomial Representations of GLn (K): The Schur algebra

κ◦c=

ct (κ)ct ,

c◦κ=

t

ct (κ)ct .

t

By (2.4b), we ﬁnd κ ◦ c = c ◦ κ = 0, for any κ ∈ Y = Ker e. Then (2.4c) shows that AK (n, r), regarded as left KΓ-module, belongs to MK (n, r). (n, r) of Similarly AK (n, r) belongs to an analogously deﬁned category MK right KΓ-modules (see 4.4). Thus AK (n, r) becomes an SK (n, r)-bimodule, with actions ξ(ct )ct , c ◦ ξ = ξ(ct )ct , for ξ ∈ SK (n, r). (2.8b) ξ◦c= t

t

Now let us deﬁne a bilinear form ( , ) : SK (n, r) × AK (n, r) → K by the rule: if ξ ∈ SK (n, r), c ∈ AK (n, r), then (ξ, c) = J(ξ)(c). This is non-singular, in fact {ξi,j }, {ci,j } are dual bases with respect to ( , ) (see (2.7b)). The reader may check that for any ξ, η ∈ SK (n, r), c ∈ AK (n, r) there holds (2.8c)

(ξη, c) = (η, J(ξ) ◦ c) = (ξ, c ◦ J(η)).

In fact, using (2.3a), (2.8b) and the deﬁnition of ( , ), we see that all three expressions just given are equal to t (ξ, ct ) (η, ct ). But (2.8c) shows that ( , ) is contravariant, SK (n, r) and AK (n, r) being regarded as left SK (n, r)-modules (and even when they are regarded as right modules). By (2.7e) we deduce that AK (n, r) ∼ = (SK (n, r))◦ , an isomorphism of KΓ-bimodules.

§3

Weights and Characters

3.1

Weishts In this chapter we describe

of modules

in

MK(n,r),

in terms of the Schur algebra

results go back to Schur researches

of Weyl and Chevalley.

ideal domain,

Let

n,r ~ 1

G(r)-orbits

in

weights

(more precisely,

weight

6

of reductive

see [Se,

number of

The elements

as unordered

such that

partitions

of

any

It acts on

w E W,

w

~ = (~w(1)'''''

dominant weight,

W

i.e

into

of

denote the set of all of

A(n,r)

GLn,

i.e.

n

acts on

for each

a weight

to (ordered)

parts.

A+(n,r)

~v

is the

can also be regarded

can be identified with the Weyl group wi = (~(il),... , w(i r)

if

W-orbit of X

A

which gives the

~ f ~ ' ~

r).

parts (zero parts being allowed).

7~(n,r):

Each

will be called

of dimension

These vectors

on the left,

weights correspond Denote by

to the category

This action commutes with that of

~w(n) )"

in classical

split over a

6 = (61,... , 6 n)

~ 6,

W = G(n)

l(n,r)

i ~ I(n,r).

right, and therefore -i

e,~ ....

i p = v. r

The symmetric group GL . n

A(n,r)

they are weights

i = (i I .... ,ir)

p ~ -r-

group schemes

All our

§3].

is specified by the vector

content of any

of

For the generalization

be given~ and let

I(n,r).

SK(n,r) o

[S], and all have been generalized

of rational representations principal

the theory of weights and characters

such that

partitions

of

w ~ W, A(n,r)

~ ~ A(n,r)

on the then

contains exactly one

X1 ~ "'" ~ Xn. r

G(r)

for

Thus dominant

into not more than

the set of all dominant weights.

n

37

3.2

Weight spaces Take a fixed infinite field

weight < .

e E A(n,r)

K.

If

i E l(n,r)

we shall denote the idempotent

This is reasonable, since

<j,j

belongs to the

{.

.

(see

if and only if

2.3) by i ~ j.

The

orthogonal decomposition (2.3d) now reads

s

=

If we apply this to any module

(3.2a)

V =

a decomposition of

V

In fact space

Va

of

[[ <~ ~(A(n,r) V ~ MK(n,r)

we get

~]~> < V, aEi(n,r)

as direct sum of subspaces ([S, pp. 6,7])

V,

< V

Tn(K )

a-weight

which is defined as

all

x(t) E Tn(K)}.

is the diagonal subgroup (a maximal split torus) of

consisting of all diagonal matrices tl,... , tn E K character

~ V.

coincides with the

V a = {v E V : x(t)v = tI ,..tnnv ,

Here

•

X

= K\{0}. : Tn(K ) ~ K

To show that

(3.2b)

ex(t) =

For each ,

by

x(t) = diag(tl,... , tn) c~ E A = A(n,r),

F K = GLn(K)

with

define the multiplicative

51 an X (x(t)) = t I ...t n

~i~V = V~,

first verify the formula

~i en aEAZ tI ... tn ~

,

all

x(t) E Tn(K) ,

by evaluating both sides at each deduce

c. . (i,j E I(n,r)). If v E ~ V , we 1,3 51 ~ V s" x(t)v = ex(t)v = t I ...tnnv; hence ~ V ~ But for distinct

~,~ E A(n,r)

the multiplicative characters

k , X~

are unequal (since

38

K

is infinite),

spaces that

V

and by a familiar argument

, ~ E A(n,r),

< V = V~

is direct.

for each

it

follows

Comparing

that the sum of the

this with

(3.2a), we see

e:

(3.2c)

V =

%~ V ~. ~Efi

Remark.

It follows from (3.2b), and the fact that

image of K'Tn(K) of

SK(n,r )

which has the

commutative, Example V =

under the map

split,

For each

ArE

is any

~

(see 2.4)

(5 E A(n,r))

is infinite,

is the subalgebra as

K-basis.

that the

DK(n,r)

This is a

semisimple algebra. r

satisfying

(= Altr(E)) r-element

e

K

is a

0 ! r ~ n,

KF-module

subset of

~

in

the

r

th

~(n,r).

exterior power

If

(i I < i 2 < ... < ir)

s = {il, .... ir}

write

e s = e i A ... A e i r

Then

V

has a basis consisting

x(t) E Tn(K) = ~(s)

then

x(t)e s = til~.,

is the weight containing

r-element

subsets

s,s'

the weight spaces r-element

3.3

subset

V~

Let

V

n

and

of weight

Let

($)

elements

e s.

Moreover

~I an t.1 es = t I ... tn es' r i = (il,...,ir).

V~ = 0

1

or zero:

for all other

distinct

~(s), ~(s').

So

v~(S)

for any

= K.e

s

~ E A(n,r).

spaces ~(n,r),

w E W = G(n).

and

~

an element of

Then the

if

where

Clearly,

give distinct weights

all have dimension

be a module in

Proposition

are isomorphic.

of

s ~,

Some properties

(3.3a)

of these

K-spaces

A(n,r).

V ~, V w(~)

39

Proof

Let

el,...,

en

of

n~ 1

verify

hence that

(3.3b)

n

be the element of

w E

to

ew(1),... , ew(n)

v ~+ n v w

gives a

Let

0 ~ VI ~

V ~ -~ V ~2

0

Since

Now let to

be an exact sequence in

sequence of

is obtained

is idempotent,

K-spaces

by applying

KF-module

~(n,s)

to every term of

the result follows.

be any non-negative

MK(n,r) ,

regarded as

(3.3c)

~a

r,s

* t I ..... t n E K ,

for all

is exact.

The second sequence

the first.

It is simple to

V ~ ~ V w(~) .

0 ~ V 1 ~ V -~ V 2 ~ 0

Then the naturally induced

elementary

respectively.

K-isomorphism

MK(n,r).

belonging

which maps the basis elements

x(t I .... , tn)n w = X(tw(1),... , tw(n)) ,

Proposition

Proof

F = GL (K) n

integers and

respectively.

V, W

Clearly

in the usual way, belongs

to

be

~-modules

V ~ W = V ®KW ,

~(n,

r+s).

It is

to verify the

Proposition.

Let

where the sum is over all

y E A(n, r+s). e E A(n,r),

Then

(V ® W)~ =

8 (A(n,s)

Z~

V~ ~

such that

(~i ..... an) + (~i ..... Pn ) = (~I ..... ~n )" Next suppose with a subset of i,j E l(n,r).

L

is a field containing

SL(n,r)

Then

~K ~ =

K ~i,j

~L ,

~ E A(n,r).

into an

identify

with the subset

(3.3d) V~=

Pr@position.

$~K V.

for all

SL(n,r)-module

with

~° ,j'

V ® 1L

of

VL,

VL = ~

SK(n,r) for all

So if we make

by "extension of scalars",

The weight-space

In particular,

We identify

by identifying

VL = V ®K L V

K.

and

we have the L

dimK V~ = dim L V L .

VL

is the

L-span of

W ~,

40

Contravariant spaces.

If

duality (see 2.7) behaves well with respect to weight-

V,W E MK(n,r )

and

(,) : V × W ~ K

is a contravariant

form, then

(V~,W ~) = 0

for any distinct weights

[W, p. 42],

[J, p. 6]).

For

(~aV,~W)

= (V, ~ W )

= 0.

only if the restrictions E A(n,r).

(3.3e)

Taking

J(~)

,

(,)~ : V

× W

(see

so by the contravariant

It follows that

W = V°

Proposition.

= ~

~,~ E A(n,r)

(,)

~ K

is non-singular

are non-singular

bilinear

property if and

for all

there follows the

dimK V~ = dimK(V°)~ ,

for all

V E MK(n,r )

and

E A(n,r). Finally let as in 2.6.

{V K}

Because the idempotent$

we have a direct sum These

(3.3f)

in

and

Z-defined by

SQ(n,r)

actually lie in

~ V Z = V Z ~ VQ~ Z-module

VZ, {5 K}

VZ,

for all

Sz(n,r) ~ E A(n,r).

are themselves free

We have the

Proposition

6K : V Z ® K ~ V K

3.4

V Z = Z @ ~ Vz,Q

%aYQVz, being surmnands of the free

Z-modules.

free

be a family of modules,

Z-module

For each ~Q V Z ® 1K

of ~Q~ VZ,

K, •

VK

is the

Hence

K-span of the image under equals the rank of the

dim K V aK

and so is independent of

K.

Characters Let

V

be a

KFK-mOdule in

we assign the monomial

~i ~n X 1 ... Xn

MK(n,r).

To each weight

of degree r

in

n

~ E A(n,r)

indeterminates

XI,...,X n

over the rational field

Q.

character,

cf.

is defined to be the polynomial

[J', p• 274]) of

V

Then the character

(or formal

4]

@v(XI,...,Xn)

~i • X1

dimK v~

Z

=

an ... X n

~EA(n,r) Thus

~V

is an element

homogeneous

of degree

(3.4a)

of the polynomial r.

By (3.3a)

ring

Z[XI,.,.~Xn] ,

it is symmetric,

and is

Jn fact

Z dimK V% • mt(X 1 ..... Xn) ,

+v(XI ..... Xn) =

k the sum being over all dominant monomial

symmetric

of the distinct Xl,... , X n. character

function

monomials

V =

ArE

k E A+(n,r).

(see for example

obtained

For example,

of

weights

let

r

(see 3.2)

from

[M, p. ii]),

kI A n X1 ... X n

be in the range is the

r th

Here

~k

is the

i.e.

the sum

by permuting

0 ! r ! n,

elementar7

then the

symmetric

function

e = ~(i,i,...,I,0 ..... 0) = XIX 2 ... X r + ... --r The propositions If

0 ~ V1 ~ V + V 2 + 0

in 3.3 give rise to propositions is an exact

sequence

in

~(n,r),

about characters. then

~V

=

~V 1

+

2 ,

by (3.3b).

It follows

series for

V,

by induction

on the length

g

say

V : V ° D V1 D %/2 D .. . D Vii = 0,

that

(3.4b)

~V = o=ig %o-I/Vo From

(3.3c) we have

of a composition

42 when

V E MK(n,r),

W E ~(n,s);

this extends of course to a similar formula

for tensor products of any finite number of factors. = (~i''''' ~r ) ![i + "'" + ~ r

is any partition of

= r)

r

....> I~r ~ 0

~I

and

then the s~wmetric function

~ ( X I ..... Xn)

is the character of the module ~V

(i.e.

For example if

lies in the ring

~(n,r)

~i

.o°

e

~Ir

£ ~i E ®... ® A Zr E.

Since every character

of all symmetric functions in

Z[XI,... , Xn],

and since by the fundamental theorem on symmetric functions ([M, (2.4), p. 13]) S(n,r) is Z-spanned by the e (3.4d)

Theorem.

by all characters

above, we have the well-known

The additive subgroup of ~V' V E MK(n,r),

group is independent of the field

is

Z[XI,,.., Xn]

S(n,r).

In particular, this additive

K.

We must next connect our "formal" character character

~V

of

V = (V,p) E ~ ( n , r ) ,

~v(g) - Trace p(g),

~V

is an element of

~(n,r),

(rab)

(3.4e)

Theorem (see [S, p, 17]).

g E rK = GLn(K)

afforded by any basis

there holds

are the eigenvalues of

g.

Let

~V

with the natural

defined hy

all

in fact

matrix

which is generated

g E r

~V

[Va }

= GLn(K ).

is the trace of the invariant of

V.

V E MK(n,r ).

~T(g) = %T(
Then for any where

43

Proof

By (3.3d), the character

module L.

VL = V ® K

L (~(n,r)

@V

obtained by extending

Since this process replaces

coincides with that

K

on

FK'

~V

K.

C

be the

by a function on

by the

to a larger field I = GLn(L)

which

we may legitimately assume, in proving (3.4e),

n x n

Define elements

(I)

matrix

(c v) ,

and let

fl' .... fn ( ~ ( n , r )

u

be an indeterminate

by

det(ul-C) = u n - fl un-I + ... + (-l)nfn

It is clear that

f (g) = ~r(~ , ~ ), r 1 ~''" n %

of

~(n,r)

by

~

% (K1 . . . . . ~n )

-I

g

r,

=

as above.

Then we have, for any

having

dim V ~ diagonal terms

traces

we have

~i ~i

g ( FK

~"(g) "

Relative to a basis of

V = Z~ Vg~ diag(~l,...,~ n)

Now we may write

Define the element

is diagovalizable i.e. that there is some

= diag(~l,...,~n).

decomposition

of

I < r < n. -- -

(bg ( Z ) .

~ = Z (b .iK) f~l ...f~rr "

(2) Now suppose

for

~i ~r = Z b~ -~I "'" ~r

the sum being over the partitions

zgz

K

V

is algebraically closed. Let

over

@V

is unchanged if we replace

V

z ~ FK

such that

which is adapted to the

is represented by a diagonal matrix

an "'" ~n '

for each

~ (A(n,r).

Taking

44

2

We have now two polynomials in

n

and

~(g) = ~v(g)

~V'

and by (2) and (3,),

diagonalizable elements of are distinct belongs to satisfying polynomial

d(g) # O, (i).

Corollary.

F K = f~n(K).

D,

we have

where

~ = ~V'

~i''''' ~t

namely

for all

g

~(g) = ~v(g )

for all

~i''''' ~t

Proof

the natural characters ~i''''' ~t elements of If

~(n,r).

z .

are the characteristics of a set of VI,... , V t ~ ~ ( n , r ) . S(n,r).

(see [CR, p. 184, (27.13)] shows that FI,... , F t

are linearly independent

is a non-trivial relation

p = char K

Then by (3.4e)

g ~ FK

(It is here that we need absolute irreducibility.)

Zl~ 1 + ... + zt~ t = 0

assume, in case

of

of

and this completes the proof of (3.4e).

are linearly independent elements of

A theorem of Frobenius-Schur

D

is the discriminant of the

mutually non-isomorphic, absolutely irreducible modules Then

in the set

Since every matrix whose eigenvalues

d ~ ~(n,r(r-l))

It follows

Suppose that

variables c v ,

is finite, that

p

(z~ E Z),

we may

does not divide all the

(Zl.iK)~l(g) + ... + (zt'iK)~t(g) = 0

for all

g ~ FK-

But this contradicts the Frobenius-Schur theorem.

3.5

Irreducible modules in

~(n,r)

The next theorem, forerunner of far-reaching generalizations by Weyl [We]

and Chevalley(

=ee[S~])

, is due in the case

K = ~

to Schur

Is, p. 37]. The leading term of a polynomial in to the usual lexicographical of monomials

X1

... X

an n

Z[XI,... , Xn]

(Gaussian) ordering of weights

(see [S, p. 17]).

is taken relative a ~ A(n,r),

or

45 (3.5a)

Theorem

Let

any infinite field. (i)

For each

FX, K E ~ ( n , r )

n,r

be given integers,

whose character

These

(iii)

Every irreducible module

Proof to

(i)

k.

(k E A+(n,r))

[S, p. 37]

Let

U

X kl 1 ... X kn n .

a E K

such that

U' = {u 6 U : 0(u) = a.u} equal to the scalar map (ii) for

Z-basis of

V E ~(n,r)

~ = (~l,...,~r) V =

~(n,r). Fk, K

for

We may take

be the partition conjugate A~IE

~V U

of

• k,K(k E A+(n,r))

V

~

and since e(u) = a.u

e

to be

FX, K.

Uk

u E U k.

is a submodule of

kn ... X n .

By

To prove that

U

KFK-endomorphism

Since our assumption on

must map

for

kl

has character

whose character has

it is enough to show that every

U,

into itself, there

But the set

hence

U = U'

and

@

is

a-~. mk

If we express the functions

equations of the form

~... ® A~rE

is therefore

U

The monomial symmetric functions S(n,r).

~I Xn X 1 ... X n

is isomorphic to

is scalar (see [CR, p. 202, (29.13)].

shows that dim U k = i,

is some

form a

The leading term of

is absolutely irreducible,

~U

has leading term

there is some composition factor

leading term

of

be

~ E A+(n,r).

~V = hi~I " " ~ ~r

0

~,K

We saw in 3.4 that the module

(3.4b)

K

these exists an absolutely irreducible module

(ii)

exactly one

Let

Then

k E A+(n,r)

~,K

n ~ I, r t 0.

~X,K = ex +

Z

~

also form a basis for

(k E A+(n,r))

form a basis

@X,K

in terms of these, we get

~

and it follows at

,

S(n,r).

46 (iii) by

Suppose

Fk, K

L

L

is an algebraically closed field containing

the module

Fk, K L .

irreducible, so is ¢~ = ~k,K

as

~ , K ® K L 6 ~(n,r).

6 A+(n,r))

Fk,KL ,

Fk, K

Denote

is absolutely

Fk, K L has the same character

On the other hand

FX, K , by (3.2d).

isomorphic to one of the

Since

K.

Any irreducible

X 6 ~(n,r)

must be

since otherwise the characters

~X' ~k

would be linearly independent by (3.4e) Corollary, and that

would contradict (ii) above. Now let

V

be any irreducible module in

a minimal submodule of that

Fk,K L ~ X "

V L = V ~K L.

hence the space

HomSK(n,r)(Fk,K,. , V) # O.

Therefore

MK(n,r) , and let

There is some

k 6 A+(n,r),

IIomsL(n,r)(Fk,KL , VL) # 0,

X

be

then, such it follows

V ~ F>~,K, and the proof of Theorem

(3.5a) is complete.

Remarks. (i)

The proof above shows that

isomorphism in

~(n,r),

FX, K

by the properties

is defined, uniquely up to

(a)

Fk, K ~ ~(n,r)

is

irreducible, and (b) FX, K has character CX,K whose leading term is kI kn X 1 ... X Moreover if L is any field containing K, then n

Fk,K ® K L 6 ~(n,r)

has the corresponding properties

fact its character is the same as that of ~k,K = ~k,L ~k,K Write

if

K, L

for

~k,K'

{~h,p : k6 A+(n,r)} dk~

are infinite fields with

is the same, for all infinite fields ~k,p

Fk, K.

is a

if

p = char. K. Z-basis of

which appear in the equations

K

In other words, K _c L.

It follows that

of given characteristic.

For each

~(n,r).

(a), (b), since in

If

p,

the set

p > 0,

the coefficients

47

=

Z ~EA+(n,r)

@k,O

k E A+(n,r),

dk~'k, p,

are the decomposition numbers relative to the modular reduction from to

~(n,r),

(ii)

where

K

is any field of characteristic

Suppose that

K

is fixed.

ring for the category corresponding

~(n),

to a module

was said in 3.3, that R(~(n))

R(MK(n ))

and let

[V]

V E ~(n).

[V] -~ ~V

onto the ring

Let

p

MQ(n,r)

(see 2.5).

be the Grothendiock

be the element of

R(MK(n))

Then it as clear from (3.4a) and what

defines an isomorphism of rings from

S(n) =

% _S_(n,r) of all symmetric functions in r>0

Z[X I,..., X n]. (iii)

The symmetric functions

~k,0

were determined by Schur [S,

and are now known as Schur functions or finite characteristic

p,

S-functions

for

and write

X1

... X ~n n ; s(w)

~k

,P

For

have not yet

We sketch here a proof, rather different from

Sehur's, of his theorem for characteristic X~

(see [M, p.24]).

the irreducible characters

been calculated explicitly.

23],

recall that

for the "sign"

(+i)

zero.

W = G(n)

If

e E A(n,r)

acts on

A(n,r)

of a permutation

w E W.

we write (see 3.1), Define

a = ~ ( X l , .... Xn) = Z s(w) X w(~), an alternating function in =~ wEW which in fact is expressible as the n × n determinant Xl,..., 1 Let 6 = (n-l, n-2,...,l,O) E A(n, ~ n(n-1)). Define the S-function iX

= [k(Xl ..... Xn)

for any

(see [M, (3.2). (4.3)]) (in fact

the

k E A+(n,r)

~k (k E A+(n,r))

has leading term X%), n

~l = a~+6/-9~ " form a

-

Then

Z-basis of

S(n,r)

and there holds the formal identity

1

l _ X vV ~ ~,v=l

by

~. "'IXi]I.

Z + kEA (n)

S_.(X)S=x(Y),

48

where and

Y = (YI""' A+(n) =

Yn )

is a set of variables independent of

X = (XI,..., Xn),

~ A+(n,r). r~0

Schur's theorem [S, §23]

is that

¢'k,0 = ~

'

for all

k 6 A+(n,r).

To prove this, it is enough to show that (I) remains true when replaced by

#~0's.

given degree

r

(2)

For then, considering only the part of (I) which is of

in (both)

%

X

and

Y,

S%(X)Sx(Y) =

k~A+(n,r) = If we write

~% =

Z vk~ ~

we shall have

Z +

=

~k o(X)@x,O (Y)"

k(A (n,r) (k 6 A+(n,r))

be a signed permutation matrix, i.e.

the

to order and sign (cf. [M, p. 35]). X k,

we must have

So we must prove (I), with

'

we get an integral matrix

which, on account of (2), must be orthogonal.

leading term

S=>'s are

This matrix must therefore

S=k's coincide with

But since both

S) = ....

~)

k,O

Sk's

g=X

for all

replaced by

and

~k,0's ~'k

coincide on the two sides of the proposed equation.

Z (r)

I] X ~v y ~v ~,v=l

=

~+

X~A (n,r)

r Let

module

have

~k,0's.

It is clearly r ~ O,

So we must prove

~X,o(X)~x,O(7),

where the sum on the left is over all non-negative that

up

X 6 A+(n,r)

sufficient to prove that the sums of terms, of each given degree

(3)

(v~v)

integrsl

matrices such

= r. K

@r EK

irreducible faithfully on

be an infinite field of characteristic is completely reducible Fk, K (k E A+(n,r)) E 6Y,

zero.

Because the

(2.6e), and contains each (absolutely)

with positive multiplicity

see (2.6c)) we have

(SK(n,r)

acts

49

•~i(n,r) =

Now

~(n,r)

is a

~(n,r).

y = diag(y I ..... yn )

cf(~l,K ) .

K~-bii'~odule, using right and left translation

operators g o c,c o g (see sub-bimodule of

~ ~ + h~A (n,r)

~ i. Each coefficient space Take diagonal matrices

(~ ,yy E K ),

ef(F~.,K)

is a

x = diag(xl,...,Xn),

and calculate the trace

f(x,y)

of

the linear transformation

(c ~ y o c o x : c E ~ ( n , r ) ) .

Using the basis of monomials is obtained by substituting

ci, j x

for

for X,

But from (4) we get another basis of A+(n,r)

(r~b)

for

the coefficient functions

F~, K,

relative

~(n,r), y

~(n,r), rab

to some b a s i s of

theorem (see 3.4) shows t h a t t h e s e f u n c t i o n s If we choose a basis for Fk, K

for

we find that Y

f(x,y)

in the left side of (3).

by taking for each

appearing in an invari~nt matrix

F~, K. (The F r o b e n i u s - S c h u r k rab are l i n e a r l y i n d e p e n d e n t . )

which is adapted to the weight-space

decomposition, so that each basis element belongs to some weight-space it is easy to show that the trace of the map is

~(x)~k(y);

hence by (4),

of the variables

(c -~ y o c o x : c E c f ( ~ , K ) )

f(x,y) = Z~ ~ ( x ) ~ ( y ) .

since it holds for arbitrary substitutions XI''''' Xn' YI''''' Yn"

F~, K,

But this proves (3), * Xl''''' Xn' Y I ' ' ' " Yn ~ K

3 Weights and Characters

3.1 Weights In this chapter we describe the theory of weights and characters of modules in MK (n, r), in terms of the Schur algebra SK (n, r). All our results go back to Schur [47], and all have been generalized in classical researches of Weyl and Chevalley. For the generalization to the category of rational representations of reductive group schemes split over a principal ideal domain, see [49, §3]. Let n, r ≥ 1 be given, and let Λ(n, r) denote the set of all G(r)-orbits in I(n, r). The elements α, β, . . . of Λ(n, r) will be called weights (more precisely, they are weights of GLn , of dimension r). A weight α is speciﬁed by the vector α = (α1 , . . . , αn ) which gives the content of any i = (i1 , . . . , ir ) ∈ α, i.e. for each ν ∈ n, αν is the number of ∈ r such that i = ν. These vectors α can also be regarded as unordered partitions of r into n parts (zero parts being allowed). The symmetric group W = G(n) can be identiﬁed with the Weyl group of GLn . It acts on I(n, r) on the left, wi = (w(i1 ), . . . , w(ir )) for any w ∈ W and i ∈ I(n, r). This action commutes with the action of G(r) on the right, and therefore W acts on Λ(n, r): w−1 α = (αw(1) , . . . , αw(n) ) for α ∈ Λ(n, r) and w ∈ W . Each W -orbit of Λ(n, r) contains exactly one dominant weight, i.e. a weight λ such that λ1 ≥ · · · ≥ λn . Thus dominant weights correspond to (ordered) partitions of r into not more than n parts. Denote by Λ+ (n, r) the set of all dominant weights.

3.2 Weight spaces Fix an inﬁnite ﬁeld K. If i ∈ I(n, r) belongs to the weight α ∈ Λ(n, r) we shall denote the idempotent ξi,i (see 2.3) by ξα . This is reasonable, since ξi,i = ξj,j if and only if i ∼ j. The orthogonal decomposition (2.3d) now reads ε= ξα . α∈Λ(n,r)

24

3 Weights and Characters

If we apply this to any module V ∈ MK (n, r) we get (3.2a) V = ξα V , α∈Λ(n,r)

a decomposition of V as a direct sum of subspaces ξα V . In fact [47, pp. 6,7], ξα V coincides with the α-weight-space V α of V , which is deﬁned as αn 1 V α = { v ∈ V : x(t)v = tα 1 · · · tn v, all x(t) ∈ Tn (K) }.

Here Tn (K) is the diagonal subgroup (a maximal split torus) of ΓK = GLn (K) consisting of all diagonal matrices x(t) = diag(t1 , . . . , tn ) with t1 , . . . , tn in K ∗ = K\{0}. For each α ∈ Λ = Λ(n, r), deﬁne the multiplicative charαn 1 acter χα : Tn (K) → K ∗ by χα (x(t)) = tα 1 · · · tn . α To show that ξα V = V , ﬁrst verify the formula αn 1 tα (3.2b) ex(t) = 1 · · · tn ξα , all x(t) ∈ Tn (K), α∈Λ

by evaluating both sides at each ci,j (i, j ∈ I(n, r)). If v ∈ ξα V , we deαn α 1 duce x(t)v = ex(t) v = tα 1 · · · tn v; hence ξα V ⊆ V . But for distinct elements α, β ∈ Λ(n, r) the multiplicative characters χα , χβ are unequal (since K is inﬁnite), and by a familiar argument it follows that the sum of the weight-spaces V α , α ∈ Λ(n, r), is direct. Comparing this with (3.2a), we see that ξα V = V α for each α: V α. (3.2c) V = α∈Λ

Remark. It follows from (3.2b), and the fact that K is inﬁnite, that the image of K · Tn (K) under the map e (see 2.4) is the subalgebra DK (n, r) of SK (n, r) which has the ξα (α ∈ Λ(n, r)) as K-basis. This is a commutative, split, semisimple algebra. Example. For each r satisfying 0 ≤ r ≤ n, the rth exterior power V = Λr E (= Altr (E)) is a KΓ-module in MK (n, r). If s = {i1 , . . . , ir } is any r-element subset of n (i1 < i2 < · · · < ir ) write es = ei1 ∧ . . . ∧ eir . These nr elements es form a K-basis of V . Moreover if x(t) ∈ Tn (K) and α = α(s) is αn 1 the weight containing (i1 , . . . , ir ), then x(t)es = ti1 · · · tir es = tα 1 · · · tn es . Clearly, distinct r-element subsets s, s of n give distinct weights α(s), α(s ). So the weight-spaces V α all have dimension 1 or zero: V α(s) = K · es for any r-element subset s ⊆ n, and V α = 0 for all other α ∈ Λ(n, r).

3.3 Some properties of weight spaces Let V be a module in MK (n, r), and α an element of Λ(n, r).

3.3 Some properties of weight spaces

25

(3.3a) Proposition. Let w ∈ W = G(n). Then the K-spaces V α , V w(α) are isomorphic. Proof. Let nw be the element of Γ = GLn (K) which maps the basis elements e1 , . . . , en of E to ew(1) , . . . , ew(n) respectively. It is simple to verify n−1 w x(t1 , . . . , tn )nw = x(tw(1) , . . . , tw(n) ), for all t1 , . . . , tn ∈ K ∗ , hence that v → nw v gives a K-isomorphism from V α onto V w(α) . (3.3b) Proposition. Let 0 → V1 → V → V2 → 0 be an exact sequence in MK (n, r). Then the naturally induced sequence of K-spaces 0 → V1α → V α → V2α → 0 is exact. Proof. The second sequence is obtained by applying ξα to every term of the ﬁrst. Since ξα is idempotent, the result follows. Now let r, s be any non-negative integers and V , W be KΓ-modules belonging to MK (n, r), MK (n, s) respectively. Clearly V ⊗ W = V ⊗K W , regarded as KΓ-module in the usual way, belongs to MK (n, r + s). It is elementary to verify the (3.3c) Proposition. Let γ ∈ Λ(n, r + s). Then V α ⊗ W β, (V ⊗ W )γ = α,β

where the sum is over all α ∈ Λ(n, r), β ∈ Λ(n, s) such that α + β = γ. Next suppose L is a ﬁeld containing K. We identify SK (n, r) with a subset K L with ξi,j , for all i, j ∈ I(n, r). Then ξαK = ξαL , of SL (n, r) by identifying ξi,j for all α ∈ Λ(n, r). So if we make VL = V ⊗K L into an SL (n, r)-module by “extension of scalars”, and identify V with the subset V ⊗ 1L of VL , we have the α

(3.3d) Proposition. The weight-space VL = ξαL VL is the L-span of the α α α weight space V = ξαK V . In particular, dimK V = dimL VL . Contravariant duality (see 2.7) behaves well with respect to weight-spaces. If V, W ∈ MK (n, r) and ( , ) : V × W → K is a contravariant form, then (V α , W β ) = 0 for any distinct weights α, β ∈ Λ(n, r) (see [56, p. 42], or [29, p. 6]). For we have J(ξα ) = ξα , so by the contravariant property (ξα v, ξβ w) = (v, ξα ξβ w) = 0. It follows that ( , ) is non-singular if and only if the restrictions ( , )α : V α × W α → K are non-singular for all α ∈ Λ(n, r). Taking W = V ◦ , there follows the

26

3 Weights and Characters

(3.3e) Proposition. dimK V α = dimK (V ◦ )α , for all α ∈ Λ(n, r). Finally let {VK } be a family of modules, Z-deﬁned by VZ , {δK } as in 2.6. Because the idempotents ξαQ in SQ (n, r) actually lie in SZ (n, r), we have a di α rect sum VZ = α ξαQ VZ , and ξαQ VZ = VZ ∩VQ , for all α ∈ Λ(n, r). These ξαQ VZ , being summands of the free Z-module VZ , are themselves free Z-modules. We have the α

(3.3f ) Proposition. For each K, VK is the K-span of the image under the α map δK : VZ ⊗ K → VK of ξαQ VZ ⊗ 1K . Hence dimK VK equals the rank of the free Z-module ξαQ VZ , and so is independent of K.

3.4 Characters Let V be a KΓK -module in MK (n, r). To each weight α ∈ Λ(n, r) we assign the monomial X1α1 · · · Xnαn of degree r in n indeterminates X1 , . . . , Xn over the rational ﬁeld Q. Then the character (or formal character, cf. [32, p. 274]) of V is deﬁned to be the polynomial (dimK V α ) · X1α1 · · · Xnαn . ΦV (X1 , . . . , Xn ) = α∈Λ(n,r)

Thus ΦV is an element of the polynomial ring Z[X1 , . . . , Xn ], and is homogeneous of degree r. By (3.3a) it is symmetric, in fact (3.4a) ΦV (X1 , . . . , Xn ) = (dimK V λ ) · mλ (X1 , . . . , Xn ), λ

the sum being over all dominant weights λ ∈ Λ+ (n, r). Here mλ is the monomial symmetric function (see for example [39, p. 11]), i.e. the sum of the distinct monomials obtained from X1λ1 · · · Xnλn by permuting X1 , . . . , Xn . For example, let r be in the range 0 ≤ r ≤ n, then the character of the exterior power V = Λr E (see 3.2) is the rth elementary symmetric function er = m(1,1,...,1,0,...,0) = X1 X2 · · · Xr + · · · . The propositions in section 3.3 give rise to propositions about characters. Suppose 0 → V1 → V → V2 → 0 is an exact sequence in MK (n, r), then ΦV = ΦV1 + ΦV2 , by (3.3b). It follows by induction on the length l of a composition series of V , say V = V0 ⊃ V1 ⊃ V2 ⊃ · · · ⊃ Vl = 0, that (3.4b)

ΦV =

l σ=1

From (3.3c) we have

ΦVσ−1 /Vσ .

3.4 Characters

(3.4c)

27

ΦV ⊗W = ΦV ΦW

when V ∈ MK (n, r), W ∈ MK (n, s); this extends of course to a similar formula for tensor products of any ﬁnite number of factors. For example if µ = (µ1 , . . . , µr ) is any partition of r (i.e. µ1 ≥ · · · ≥ µr ≥ 0 and µ1 + · · · + µr = r) then the symmetric function eµ (X1 , . . . , Xn ) = eµ1 · · · eµr is the character of the module Λµ1 E⊗· · ·⊗Λµr E. Since every character ΦV lies in the ring Sym(n, r) of all symmetric functions in Z[X1 , . . . , Xn ], and since by the fundamental theorem on symmetric functions ([39, (2.4), p. 13]) Sym(n, r) is Z-spanned by the eµ above, we have the well-known (3.4d) Theorem. The additive subgroup of Z[X1 , . . . , Xn ] which is generated by all characters ΦV , V ∈ MK (n, r), is Sym(n, r). In particular, this additive group is independent of the ﬁeld K. We must next connect our “formal” character ΦV with the natural character ϕV of V = (V, ) ∈ MK (n, r), deﬁned by ϕV (g) = Trace (g), all g ∈ Γ = GLn (K). The map ϕV is an element of AK (n, r), in fact ϕV is the trace of the invariant matrix (rab ) aﬀorded by any basis {va } of V . (3.4e) Theorem (see [47, p. 17]). Let V ∈ MK (n, r) and g ∈ GLn (K), then ϕV (g) = ΦV (ζ1 , . . . , ζn ), where ζ1 , . . . , ζn are the eigenvalues of g. Proof. By (3.3d), the character ΦV is unchanged if we replace V by the module VL = V ⊗K L ∈ ML (n, r) obtained by extending K to a larger ﬁeld L. Since this process replaces ϕV by a function on ΓL = GLn (L) which coincides with ϕV on ΓK = GLn (K), we may legitimately assume, in proving (3.4e), that K is algebraically closed. Let C be the n × n matrix (cµν ), and let u be an indeterminate over K. Deﬁne elements f1 , . . . , fn ∈ AK (n, r) by (1) det (uI − C) = un − f1 un−1 + · · · + (−1)n fn . It is clear that fr (g) = er (ζ1 , . . . , ζn ), for 1 ≤ r ≤ n. Now we may write bµ eµ1 1 · · · eµr r (bµ ∈ Z), ΦV = µ

the sum being over the partitions µ of r, as above. Deﬁne the element ψ of AK (n, r) by ψ = µ (bµ · 1K )f1µ1 · · · frµr . Then we have, for any g ∈ ΓK (2) ΦV (ζ1 , . . . , ζn ) = ψ(g).

28

3 Weights and Characters

Now suppose g is diagonalizable, i.e. that there is some z ∈ ΓK such that zgz −1 = diag(ζ1 , . . . , ζn ). Relative to a basis of V which is adapted to the decomposition V = V α , diag(ζ1 , . . . , ζn ) is represented by a diagonal matrix having dim V α diagonal terms ζ1α1 · · · ζnαn , for each α ∈ V (n, r). Taking traces we have (3) ϕV (g) = ϕV (zgz −1 ) = ΦV (ζ1 , . . . , ζn ). We have now two polynomials in n2 variables cµν , namely ψ and ϕV , and by (2) and (3), ψ(g) = ϕV (g) for all g in the set D of diagonalizable elements of ΓK = GLn (K). Since every matrix whose eigenvalues are distinct belongs to D, we have ψ(g) = ϕV (g) for all g ∈ ΓK satisfying d(g) = 0, where d ∈ AK (n, r) is the discriminant of the polynomial (1). It follows that ψ = ϕV , and this completes the proof of (3.4e). Corollary. Suppose that Φ1 , . . . , Φt are the characteristics of a set of mutually non-isomorphic, absolutely irreducible modules V1 , . . . , Vt ∈ MK (n, r). Then Φ1 , . . . , Φt are linearly independent elements of Sym(n, r). Proof. A theorem of Frobenius-Schur (see [11, p. 184,(27.13)]) shows that the natural characters ϕ1 , . . . , ϕt of V1 , . . . , Vt are linearly independent elements of AK (n, r). (It is here that we need absolute irreducibility.) If there is a non-trivial relation z1 Φ1 + · · · + zt Φt = 0 (zτ ∈ Z), we may assume, in case p = char K is ﬁnite, that p does not divide all the zτ . Then by (3.4e) (z1 · 1K )ϕ1 (g) + · · · + (zt · 1K )ϕt (g) = 0 for all g ∈ ΓK . But this contradicts the Frobenius-Schur theorem.

3.5 Irreducible modules in MK (n, r) The next theorem, forerunner of far-reaching generalizations by Weyl [54] and Chevalley [49], is due in the case K = C to Schur [47, p. 37]. The leading term of a polynomial in Z[X1 , . . . , Xn ] is taken relative to the usual lexicographical (Gaussian) ordering of weights α ∈ Λ(n, r), or of monomials X1α1 · · · Xnαn (see [47, p. 17]). (3.5a) Theorem. Let n, r be given integers, n ≥ 1, r ≥ 0. Let K be an inﬁnite ﬁeld. Then (i) For each λ ∈ Λ+ (n, r) there exists an absolutely irreducible module Fλ,K in MK (n, r) whose character Φλ,K has leading term X1λ1 · · · Xnλn . (ii) These Φλ,K (λ ∈ Λ+ (n, r)) form a Z-basis of Sym(n, r). (iii) Every irreducible module V ∈ MK (n, r) is isomorphic to Fλ,K for exactly one λ ∈ Λ+ (n, r).

3.5 Irreducible modules in MK (n, r)

29

Proof. (i) (see [47, p. 37]) Let µ = (µ1 , . . . , µr ) be the partition conjugate to λ. We saw in 3.4 that the module V = Λµ1 E ⊗ · · · ⊗ Λµr E has character ΦV = eµ1 1 · · · eµr r . The leading term of ΦV is therefore X1λ1 · · · Xnλn . By (3.4b) there is some composition factor U of V whose character has leading term X1λ1 · · · Xnλn . We may take U to be Fλ,K . To prove that U is absolutely irreducible, it is enough to show that every KΓK -endomorphism θ of U is scalar (see [11, p. 202, (29.13)]). Since our assumption on ΦU shows that dim U λ = 1, and since θ must map U λ into itself, there is some a ∈ K such that θU (u) = a · u for u ∈ U λ . But the set U = { u ∈ U : θ(u) = a · u } is a submodule of U , hence U = U and θ is equal to scalar map a · 1U . (ii) The monomial symmetric functions mλ (λ ∈ Λ+ (n, r)) form a basis of Sym(n, r). If we express the functions Φλ,K in terms of these, we get equations of the form Φλ,K = mλ + µ<λ zλµ mµ , and it follows that the characters Φλ,K (λ ∈ Λ+ (n, r)) also form a basis for Sym(n, r). (iii) Suppose L is an algebraically closed ﬁeld containing K. Denote L the module Fλ,K ⊗K L ∈ ML (n, r). Since Fλ,K is absolutely irreby Fλ,K L L . On the other hand Fλ,K has the same character Φλ = Φλ,K ducible, so is Fλ,K as Fλ,K , by (3.3d). Any irreducible X ∈ ML (n, r) must be isomorphic to one L , since otherwise the characters ΦX , Φλ (λ ∈ Λ+ (n, r)) would be of the Fλ,K linearly independent by Corollary (3.4e), and that would contradict (ii) above. Now let V be any irreducible module in MK (n, r), and let X be a minimal submodule of VL = V ⊗K L. There is some λ ∈ Λ+ (n, r), then, L L ∼ , VL ) = 0, it folsuch that Fλ,K = X; hence the space HomSL (n,r) (Fλ,K ∼ lows HomSK (n,r) (Fλ,K , V ) = 0. Therefore V = Fλ,K , and the proof of Theorem (3.5a) is complete. Remarks. (i)

The proof above shows that Fλ,K is deﬁned, uniquely up to isomorphism in MK (n, r), by the properties (a) Fλ,K ∈ MK (n, r) is irreducible, and (b) Fλ,K has character Φλ,K whose leading term is X1λ1 · · · Xnλn . Moreover if L is any ﬁeld containing K, then Fλ,K ⊗K L ∈ ML (n, r) has the corresponding properties (a), (b), since in fact its character is the same as that of Fλ,K . In other words, Φλ,K = Φλ,L if K, L are inﬁnite ﬁelds with K ⊆ L. It follows that Φλ,K is the same, for all inﬁnite ﬁelds K of given characteristic. Write Φλ,p for Φλ,K , if p = char K. For each p, the set { Φλ,p : λ ∈ Λ+ (n, r) } is a Z-basis of Sym(n, r). If p > 0, the coeﬃcients dλµ which appear in the equations dλµ Φµ,p , λ ∈ Λ+ (n, r) Φλ,0 = µ∈Λ+ (n,r)

are by deﬁnition the decomposition numbers relative to the modular reduction from MQ (n, r) to MK (n, r), where K is any ﬁeld of characteristic p (see 2.5).

30

3 Weights and Characters

(ii) Suppose that K is ﬁxed. Let R(MK (n)) be the Grothendieck ring for the category MK (n), and let [V ] be the element of R(MK (n)) corresponding to a module V ∈ MK (n). Then it is clear from (3.4a) and an isomorphism of rings what was said in 3.3, that [V ] → ΦV deﬁnes from R(MK (n)) onto the ring Sym(n) = r≥0 Sym(n, r) of all symmetric functions in Z[X1 , . . . , Xn ]. (iii) The symmetric functions Φλ,0 were determined by Schur [47, p. 23], and are now known as Schur functions or S-functions (see [39, p. 24]). For ﬁnite characteristic p, the irreducible characters Φλ,p have not yet been calculated explicitly. We sketch here a proof, rather diﬀerent from Schur’s, of his theorem for characteristic zero. If α ∈ Λ(n, r) we write X α for X1α1 · · · Xnαn ; recall that W = G(n) acts on Λ(n, r) (see 3.1), and write s(w) for the “sign” (∈ {−1, 1}) of a permutation w ∈ W . Deﬁne aα = aα (X1 , . . . , Xn ) = w∈W s(w)X w(α) , an alternating function in X1 , . . . , Xn , which in fact is expressible as the n × n determiα nant |Xi j |. Let δ = (n − 1, n − 2, . . . , 1, 0) ∈ Λ(n, 12 n(n − 1)). Deﬁne the S-function Sλ = Sλ (X1 , . . . , Xn ) for any λ ∈ Λ+ (n, r) by Sλ = aλ+δ /aδ . Then (see [39, (3.2), (4.3)]) the Sλ form a Z-basis of Sym(n, r) (in fact Sλ has leading term X λ ), and there holds the formal identity (1)

n

1 = 1 − X µ Yν µ,ν=1

Sλ (X) Sµ (Y ),

λ∈Λ+ (n)

where Y = (Y1 , . . . , Yn ) is a second set of variables, independent of the set X = (X1 , . . . , Xn ), and Λ+ (n) = r≥0 Λ+ (n, r). Schur’s theorem [47, p. 23] is that Φλ,0 = Sλ , for all λ ∈ Λ+ (n, r). To prove this, it is enough to show that (1) remains true when Sλ ’s are replaced by Φλ,0 ’s. For then, considering only the part of (1) which is of given degree r in (both) X and Y , we shall have Sλ (X)Sλ (Y ) = Φλ,0 (X)Φλ,0 (Y ). (2) λ∈Λ+ (n,r)

λ∈Λ+ (n,r)

vλµ Φµ (λ ∈ Λ+ (n, r)) we get an integral matrix (vµν ) If we write Sλ = which, on account of (2), must be orthogonal. This matrix must therefore be a signed permutation matrix, i.e. the Sλ ’s coincide with Φλ,0 ’s up to order and sign (cf. [39, p. 35]). But since both Sλ and Φλ,0 have leading term X λ , we must have Sλ = Φλ,0 for all λ ∈ Λ+ (n, r). So we must prove (1), with Sλ ’s replaced by Φλ,0 ’s. It is clearly suﬃcient to prove that the sums of terms, of each given degree r ≥ 0, coincide on the two sides of the proposed equation. So we must prove (3)

n

(r)

µ,ν=1

r

r

Xµµν Yν µν

=

λ∈Λ+ (n)

Φλ,0 (X) Φλ,0 (Y ),

3.5 Irreducible modules in MK (n, r)

31

wherethe sum on the left is over all non-negative integral matrices such that µ,ν rµν = r. Let K be an inﬁnite ﬁeld of characteristic zero. Because the module E ⊗r is completely reducible (2.6e), and contains each (absolutely) irreducible Fλ,K (λ ∈ Λ+ (n, r)) with positive multiplicity (SK (n, r) acts faithfully on E ⊗r , see (2.6c)) we have cf(Fλ,K ). (4) AK (n, r) = λ∈Λ+ (n,r)

Now AK (n, r) is a KΓ-bimodule, using right and left translation operators g ◦ c, c ◦ g (see 2.8). Each coeﬃcient space cf(Fλ,K ) is a subbimodule of AK (n, r). Take diagonal matrices x = diag(x1 , . . . , xn ) and y = diag(y1 , . . . , yn ) (xµ , yν ∈ K ∗ ), then calculate the trace f (x, y) of the linear transformation c → y ◦ c ◦ x (c ∈ AK (n, r)). Using the basis of monomials ci,j for AK (n, r), we ﬁnd that f (x, y) is obtained by substituting x for X, y for Y in the left side of (3). But from (4) we get another basis of AK (n, r), by taking for each λ ∈ Λ+ (n, r) λ λ the coeﬃcient function rab appearing in an invariant matrix (rab ) for Fλ,K , relative to some basis of Fλ,K . (The Frobenius-Schur theorem (see 3.4) λ are linearly independent.) If we choose a shows that these functions rab basis for Fλ,K which is adapted to the weight-space decomposition, so α , it is easy to that each basis element belongs to some weight-space Fλ,K show that the trace of the map c → y◦c◦x (c ∈ cf(Fλ,K )) is Φλ (X)Φλ (Y ); hence by (4), f (x, y) = λ Φλ (X)Φλ (Y ). But this proves (3), since it holds for arbitrary substitutions x1 , . . . , xn , y1 , . . . , yn ∈ K ∗ of the variables X1 , . . . , Xn , Y1 , . . . , Yn .

§4.

The modules

4.1

Preamble

Dk.~K In this section and the next we shall define, for each

k = (kl,... , kn) Dk, K

and

in

Vk, K

A+(n,r)

(see Introduction).

We shall define in

~(n,r)

case

(see 4.4).

K = ~,

and for each infinite field

~,K

Both have character

K,

the modules

~k,0 = S~\(XI''"'Xn)"

in terms of certain determinantal functions

These modules have been known, in the "classical"

for a very long time -- they were discovered (under the guise

of "primary convariants") by J. Deruyts in 1892 [D, p. 71] (1) .

More recently

they have been described, for fields of arbitrary characteristic, by M. Clausen [CI, p , i 8 0 ] , and by G. D. James [Ja, p. 129]. refers to the module

DX, K

in fact James

as a "Weyl module", but we prefer to reserve this term for

Vh, K

which was defined by Carter-Lusztig

as we show in 5.2, is the contravariant dual of However James's construction in [Ja ] ours, since it can be used to identify

Dh, K

[CL, p. 211], and which,

DX, K. gives more information than

with the "induced module" of

a one-dimensional character on the lower triangular Borel subgroup

Bn(K) 4.2.

of

F K = GLn(K).

k-tableaux

We mention this identification

For the rest of

The diagram (or "shape") of

§4, k

[k] = {(s,t) : 1 e s,

in 4.8.

k = (k I ..... kn) ~ A+(n,r)

is fixed.

is defined to be the subset i ~ t ! k } S

of

Z x Z

(cf. [DKR, p. 66]).

bijective, of

(I)

[k]

into a set.

A

k-tableau is a map, not necessarily Since

[k]

has

r

elements, there exists

I am indebted to J. Towber for this reference. In his article IT], Towber

gives an account of the history of Dh,K; see particularly IT, po 448].

51

at least one bijection

T : IX] ~ r.

We shall arbitrarily

biJection,

and call it the basic k -tabLeau ~

of

is

(s,t)

(4.2a)

x(s,t),

we may depict

. . . . . .

x(2,1)

x(2,2)

...

x(3,1)

x(3,2)

...

x(l,k I)

x(2,k2)

°i,

p 6 r

appears

we say that

p

R(T)

is the subgroup

preserve

T

as

x(l,2)

Thus every element

T

If the image under

x(l,l)

• .,

of

T

T = Tk .

choose one such

is in row

s

once in (4.2a).

and in column of

the rows of (4.2a),

exactly

G = G(~)

t of

T.

p = x(s,t),

The row stabilizer

consisting

and the column

If

of all

stabilizer

n E G

C(T)

which

is defined

similarly. If k-tableau

i = (il,... , Jr)

is an element

iT : [k] ~ ~

T i.

= x(s,t), (s,t)

we often

place,

of

by

refer

to

of

In general, i

I(n,r), Ti

as the entry

P

we denote

the

is not bijective. in place

P ,

If

or in the

T.. I

Example basic

Suppose k-tableau

i i 3 i5)

r = 5, T = I1 \2

Ti =

i2

4.3°

Bideterminants

of l(n,r). formula

i4

n = 3 3 4

51 /

The entries

We define

Let

and

K

k = (3,2,0).

Then,

of

T.l

for any

belong

be an infinite

an element

(T i : Tj)=

We might

take for our

i 6 I(3,5),

to the set: _3 = {1,2,3}.

field,

and let

(T i : Tj) K

of

i,J

be elements

AK(n,r)

by the

52

(4.3a)

(T i : Tj) =

Here

s(o)

ci,jo

=

c

~ s(o)ci,jo = o~C(T)

is the sign of -i

io

"

o.

for any

Z o~C(T) s(°)ci°'J

The second equality comes from the fact that

~ ~ G(r).

,j

Apart from the interchange of rows and columns, bideterminant

in the sense of Desarmenien,

= (~I''''' ~r )

be the partition of

might not be an element of Then it is easy to see that t

of

[k],

of the

Gt x ~t

x(s,t),

(If

~t = O,

we take this determinant

to be

s(o)(T i : Tj),

Let

for all

(T i : Tj)

22

'

Ti =

Cla

Clb

Cl d

Clelclf|

C2a

e2b

C2d

C2e I

be

n

parts.)

Ti,

= 1,...,~ t

i.)

From this, or directly

is zero if there are equal entries at or of

Tj.

Also

(T i : Tjo) = (Tio : Tj) =

a E C(T).

k = (3,2,0),

T~ =

(notice that

determinants

s,s'

any two places in a column of

k

Let

is the product, over all columns

x(s',t)

from (4.3a), we see that

Example I.

conjugate to

since it might have more than

(T i : Tj)

is a

Kung and Rota [DKR, p. 67].

~

A+(n,r),

(T i : Tj)

T

be as in the example in 4.2. .

Then

(Tg : Ti)

is equal to

Let

53 ,Example 2.

Let

(4.3h)

be the element of

Tg

i

1

2

2

3

3

whose

l(n,r)

•. ,2 , .

k-tableau is

I1 • J

In other words,

gx(s,t) = s,

for all

(s,t) ~ [hi. Then

(Tg : T~) = C(~l)C(~ 2) ... where, for any integer of the

4.4.

n × n

matrix

Definition of The space

m(0 ~ m ~ n),

denotes the

DX, K

AK(n,r )

h,j ~ I(n,r) = I

is a blmodule for

KF,

Z ~(ci,j)Ch, i i~l

(4.4a')

Ch, j o ~ =

Z ~(Ch,i)ci, j, iEI

~(n,r), ~(n,r)

As left module

acting SK(n,r).

~(n,r)

and

belongs to the category

and as right module it belongs to the analogously defined category of all right, finite-dimensional

coefficient space lies in ~(n,r),

F = FK

we have (see

~ o Oh, j =

~ ~ SK(n,r ).

with

Equivalently, it is a bimodule for

(4.4a)

for all

mth leading minor

C = (c v).

by right and left translations. If

c(m)

~(n,r).

KF

(or

SK(n,r))-mo~ules whose

In fact the coefficient space of

whether regarded as left or right

KF-module, is precisely

AK(n,r).

54

Now let that

~

~

be the element of

belongs to the weight

Definition

Dk, K

is the

l(n,r)

defined in (4.3b).

It is clear

k = (kl,... , in).

K-span of the bideterminants

(Tz : T!),

for all

i E l(n,r).

Dk, K

is therefore a subspace of

(4.4a), multiply by

L ~.~.

s(o),

~ o(Tg,j) =

It follows that

Dk, K

~(n,r).

and sum over all

a

% ~(ci,j)(T g : Ti) , i61 is a left

Replace in

all

SK(n,r)-submodule

h

C(T).

by

Zo

in

We get

< E SK(n,r ).

of

AK(n,r).

If we

compare (4.4b) with (2.6a), we see also that the surJective linear map = ~K : EK ®r + Dk,K' an

SK(n,r)-module

Remark basic

k-tableau

i 6 l(n,r),

T

and of

(see 4.2).

T' = ~T

and if

g

for some

Z, = Z -i

In case

~ 6 G(r).

we have

k = (r,0,...,O)

T i = (i I .... , Jr)

map this to the monomial Dr, K

j E l(n,r),

is

is isomorphic to the

and

ell "'" eir rth

depends our choice of

However any other bijective

diagram is a single row of length

i E I(n,r),

for all

(Tg : T i)

Then

T i' = TiT

k-tableau

(% of

and

Therefore

T.

we shall ~ i t e r,

T'

for any

(Ti, : Ti) = (Tz : T i ).

is in fact independent of the choice of

Example i. k

e3 -" (T~:j)

homomorphlsm.

Our definition of

can be written

Dk, K

which takes

Dr, K

for

Tg = (I,I,...,i).

: T i) = Cl,il ... Cl,ir,

DX,K.

For any If we

2.6, Example 2, we see that

symmetric power of

E.

The

55 Example 2.

Assume

we shall write and

r ~ n.

In case

k = (i,i,...,I,0,...,0)

with

r

l's,

for Dk~ K. The k diagram is a single column, (Ir),K has entries 1,2 .... ,r. For any i E I(n,r), ( % : Ti) is the

T

D

r-rowed determinant we see that

D

det(c~, i ).

Mapping this onto

is isomorphic to the

rth

ell A...A e ir

(see 3.2), r

exterior power

A EKO

(ir),K 4.5.

The basis theorem for As usual,

K

is an infinite field.

(4.5a) ...............Basis ...... theorem for such that

Ti

Dk, K

Dk~ K.

DX, K

Our aim is to prove the

has

K-basis consisting of all

is "standard", i.e. the entries in each row of

Ti

(Tg : T i)

are weakly

increasing

(!)

from left to right, and the entries in each column are strictly

increasing

~)

from top to bottom.

This theorem generalizes the theorem of Specht-Garnir for Specht modules (see [P] and [Ja, §§8, 13]).

It may be deduced from a more general basis

theorem of D@sarm~nien, Kung and Rota [DKR, p. 78]. also to prepare for the transition to the module

For completeness, and

Vk, K

in

§5,

we give here

a proof of (4.5a) based on a combinatorial lemma of Carter-Lusztig (see 4.6). We begin by showing that (4.5b) All the

The set

{(T~ : Ti) : T i

(Tg : T i) (i E I(n,r))

k~(n,r) = ~(n,r) o ~k"

in = j.

lie in the "right"

eg, i (i E I(n,r))

if and only if there is some

The condition

stabilizer of the basic

is linearly independent. k-weight-space

For it is clear from (2.3c) that

is spanned by the elements o~, i = cg,j

standard}

~ E G(r)

g~ = g is equivalent to X-tableau (4.2a).

(remember

$o

kAK(n,r) ~k= ~,~).

such that

~ E R(T), c~, i = cg,j

~

Now = g and

the row if and only if

56

T i, Tj

can be obtained from one another by a row permutation (thus the

correspond to James's For each ~s(i) if

(4.5c)

"k-tabloids" [Ja. pp. I0, 127]).

i 6 I(n,r), define

If

i,j 6 I(n,r)

and

~(i)

usual lexicographic way.

If

Ti

s

(~l(i) ..... ~n(i)), of

T i.

Clearly

where

~(i) = ~(j)

by what we have just said; hence

We order these vectors

(4.5d)

~(i) =

is the sum of the entries in row

c4, i = cg,j,,

c~,i

~(i) # ~(j),

then

cg, i # c

(they are in fact elements of

j.

A(n,r))

in the

The reader may verify

is standard and if

1 # ~ ~ C(T),

then

~(i~) > ~(i).

Now suppose we have a non-trivial linear relation

Z fi(Tg : T i) = O, i6H

fi 6 K,

where the sum is over a non-empty subset that

Ti

is standard.

(4.5e)

H

of those

fi # 0

i ~ I(n,r)

such

By (4.3a), this gives

Z i~H

Z fi s(~) cg = 0. ~fC(T) ,in

By (4.5c) we can equate to zero the partial sum of all terms (i,~)

in (4.5e)

such that

the

~(i~)

least of the for

is equal to any given

~(i~)

(i ~ H,

i ~ H, ~ ~ C(T),

~ ~ A(n,r).

~ C(T)). only if

By (4.5d),

~ = i.

Take for

6,

we can have

~ = ~(i~),

So the partial sum has the

form Z f =0, i~H' i c~,i for some non-empty subset

H'

of

H.

But since

Ti

is standard for all

57

i E H',

the

c~, i

appearing in (4.5f) are linearly independent, and this

gives a contradiction.

4.6.

This proves (4.5b).

The Carter-Lusztig lemma To complete the proof of the basis theorem (4.5a) we must show that

every bideterminsnt combination of

(Tg : Ti)

(Tg : T i)

(i E I(n,r))

for which

Ti

is expressible as a linear

is standard.

This is the harder

part of the proof of (4.5a), and we deduce it from a combinatorial lemma (4.6a) of Carter-Lusztig.

The proof of (4.6a) is given in [CL, pp. 214, 215],

and does not depend essentially on other results in

[CL].

The reader may

note some variations of expmession between Carter-Lusztig's paper and this one: our

their "semi-standard" is our "standard"; their Ti ;

their

oT

to the "weight" of

is our

i.

the "type"

k'

corresponds to

of

T( = T i)

corresponds

Finally, in our (4.6a) we have no > restricted

to tableaux of a given type

(4.6a)

Tio ;

T

f

k v.

Carter-Lusztig lemma Let

f : I(n,r) ~ F

be any map with values in an abelian group

F,

satisfying the following three conditions: (i)

f(i) = O,

if

Ti

has equal entries at two distinct places in the

same column. (ii)

f(io) = s(q)f(i),

(iii) (Garnir relations) for any ~+i and

i E l(n,r)

of the basic G(J)

for any

i E l(n,r)

and

~ E C(T).

Z s(v)f(iv) = O, vEG(J)

and any non-empty subset k-tableau

T.

Here

h

J

of the

(h+l)th column

is any element of

is a transversal of the set of cosets

{vX : v E Y},

{1,2 .... ,r-l}, where

Y

58

is the subgroup of

G(r)

consisting of all

element outside

U J,

and

Then {f(i) : T. i Remarks

~

Im f

~ G(r)

which fix every

X = C(T) @ Y.

lies in the subgroup of

F

generated by the set

standard}.

Condition (ii) shows that the sum in (iii) is independent of the

choice of transversal

G(J).

Condition (iii) is equivalent, by an argument

given in [CL, p. 212], to condition (37) of

[CL, p, 214].

A slightly weaker

form of (4.6a), which uses a larger set of "Garnir relations" (but is still adequate for our purposes) can be proved by an adaptation of the proof of Garnir [Ga] for Specht modules; see [P, pp. 93, 94] or [Ja, pp. 29, 30]. Lemma (4.6a) completes the proof of the basis theorem (4.5a), as soon as we observe (4.6b)

The function

f(i) = (Tg : Ti)

satisfies conditions (i), (ii) and

(iii) of (4.6a). Proof

That

f(i)

satisfies (i) and (ii) has already been said in 4.3.

The proof of (iii) is like that for Specht polynomial~. ~ Every element of the set with (with

v ~ G(J)

and

B = Y • C(T)

~ ~ C(T).

f(i) = (T~ : Ti))

Z v~G(J)

has unique expression

transposition in

becomes

Z ~C(T)

{~,~}, R(T).

, = vo,

So the left side of the Garnir relation (iii)

s(~)s(~)cg,iv °

An argument given in [P, pp. 92, 93] shows that union of subsets

Noos as follows.

in which

K

For such a pair

=

Z ~EB B

s(~)c~ i~' '

can be written as disjoint

(which depends on

~)

is some

5g

s(.)c~,i~

because

c%,i~ K

+ s(~)c~,i~ K

= O,

~K = % for all

= c~K,i ~ = c%,i~ (notice

K ~ R(T)).

Hence the sum above is zero, as required.

4.7.

Some consequences of the basis theore______mm Let

~. ~

~ { A(n,r)

Putting

~(=

be a given weight, and

~a,a )

for

o (Tg : Tj) = (Tg : Tj)

each bideterminant i.e.

~

or zero, according as

(Tg : Tj)

(4.7a) with

j.

(j E I(n,r)) Dk,K,

j E e

or not.

Consequently

is a "weight element" of where

~

Dk, K,

is the weight

Then (4.5a) gives the first statement below:

For each i E ~

an element of

in formula (4.4b) we see that

it belongs to the weight-space

containing

a 6 I(n,r)

and

~ ~ A(n,r), Ti

Dk, Kq

standard.

has

K-basis consisting of all

Hence the character

~D

(T~ : T i)

is equal to X,K

~ ( x I .... , Xn). The second statement in (4.7a) follows fact that the coefficient of number of

h-tableaux

Ti

X~

in

from the first, and from the

S=%(XI,... , X n)

which have "content"

can be proved by a direct combinatorial

(i.e.

is precisely the weight)

~

-- this

argument from the definition of

(see [M, p. 42]). Since when

(4.7b)

~

char K = O,

If

is the character of an irreducible module in we deduce

char K = 0,

then

DX, K

is irreducible.

~(n,r)

~A

60 We show next that the family infinite field

K,

we may write

is defined over

{~,K }

(~

Z.

For each

to denote the element

: Ti) K

(~

: T i)

defined in 4.3.

(4.7c)

The

Z-span

(i E l(n,r)) is a D~, Z to

~,Z

of the elements

Z-form of

~,Q.

6K : ~ , Z

K~

and the maps

(~

: Ti) Q

The family ~,K

{~,K }

is

Z-defined by

which take each

(T~ : Ti) Q ® 1K

D~,K~+ k~(n,r)

is also defined

(Te : Ti) K. Moreover the family of inclusions

over

Z.

Proof

The Carter-Lusztig !emma (4.6a), applied to the function

f(i) = (T~ : Ti)Q,

shows that

(Tg : Ti) Q

{(Tg : Ti) Q : T i But by (4.5a), this set is a basis of to the left action of Dk, Q.

The maps

5K

Sz(n,r ).

is in the

standard}. Dk, Q.

By (4.4b),

It follows that

described above are clearly

are isomorphisms by (4.5a) applied to

Z-span of

DX, K.

DX, Z

D~, Z is a

is invariant Z-form of

SK(n,r)-morphisms , and

The last statement of (4.7c)

is immediate from the definitions. Remark

Dk, Z

is a direct summand, as

Z-module, of

inc Az(n,r ) the exact sequence 0 ~ DX, Z -----> from (4.7c) and the next lemma.

is

Az(n,r);

Z-split

equivalently,

This follows

61 (4.7d)

Lemma.

Z-defined by

Suppose VZ

{VK} , {WK}

and {6K} , W Z

family of morphisms in (i)

If all the 0 + VZ

(ii)

If

}~(n,r),

@Q----> W Z

is

also defined over

eK : V K ~ W K

is a

(see 2.6).

Then

Z

Z-split.

9K

are surJective, then the exact sequence

VZ ~ W Z ~ 0

is

Z-split.

Let of

M = (m~)

be the matrix of

VQ, WQ J which {9K}

and that for each

K,

appropriate bases of has rank

d

Z-generate

are equal to

M' = (m~)

9Q,

VZ, W Z

is defined over ~

= (mB~ • IK)

VK, W K.

= dimQVQ,

relative to bases respectively.

Z

implies

M

9K

The assumption

9K

relative to

are injective, this means

and that it still has rank p.

{v },

is an integral matrix,

is the matrix of

If all the

reduction modulo any rational prime M

Suppose

~(n,r),

are injective, then the exact sequence

that the family

M

{~K }.

all the

Proof {w~}

9K

and

are families of modules in

d,

even after

Hence all elementary divisors of

i, and therefore there exists an integral matrix

such that

M'M = identity.

This proves (i). The proof of (ii)

is similar.

4.8

James's construction of

DX, K

G. D. James has given [Ja, p. 129] a construction of a W X, D~ K"

KFK-mOdule

which he calls a "Weyl module", but which is in fact isomorphic to If

e,~ E A(n,r),

then the elements

ci,j(i ~ ~, J E ~)

may be identified with James's "s-tabloids of type not necessary that either

~ = (~i ..... an)

dominant, i.e. be proper partitions of X~(n,r)

in our language (see 4.5).

r.)

or

of

~(n,r)

~" [Ja, p. 127].

~ = (~i .... , ~n )

James's space

S°'%

(It is

be becomes

He defines, for all pairs of integers

62

s E {i .... ,n} and

v E {0,..., ks+ 1 }

a linear map

k ~s,v :

~(n,r)

~ ~(n,r)

by the rule:

(4.8a)

Ss,v(C~,i) = Zh

sunmed over all ks+ I - v

h 6 l(n,r)

of the entries

Definition such that

%s,v(C) = O

obtained by replacing,

s + I by

[Ja, p. 129]

Ch'i'

Wk

in

e

Tg),

s.

is the set of all elements

for all

(or in

s E {i .... ,n}

c E k~(n,r)

v E {0, .... ks+ I - i}.

By a rather delicate combinatorial argument, James proves a theorem [Ja, 26.3, p. 128] from which it follows readily that the set

{(T$ : T i) : T i

standard}; hence

~

= Dk, K.

definition has a very important group-theoretical

Wk

has as basis

However James's

interpretation,

which we

shall now give. Let

U

= U (K)

be the subgroup of

lower triangular unipotent matrices in generated by the elements

Us(t)

F.

F = GLn(K)

consisting of all

It is well-known that

U-

is

(s 6 {I .... ,n-l}, t E K) where

I 0

"°° i

us(t)

t 1

=

0

(row

S)

(row

s+l)

.

'. i

Now write F

on

~(n)

g = Us(t)

and

(see Introduction)

i E l(n,r).

By definition of the action

63

(4.8b)

cg i o g = '

it is clear that

(4.8c)

For all

such that

h E l(n,r) the entries

then

Cg,h(g ) = tw,

s + i

by

s.

c £ k~(n,r),

ks+l ks+l-V Z t v=0

s E {!,...,n-l}

w

is the number of

Ce,h(g ) = tw,

in

Z

(or in

it is clear that

Te) ,

w

t 6 K.

of

(i E l(n,r)).

%s,v(C) = c, K

Naturally we prove

for all

c E X~(n,r).

is infinite, we see that for any

the conditions

~s,v(C) = O,

all

s E {i, .... n-l},

v E {0,..., ks+l-l}

are equivalent to the conditions c o Us(t) = c,

all

s E {i .... ,n-l},

which in turn are equivalent to the conditions

(4.8e)

for any

~s,v(C),

and

c = ce, i

So by (4.8d), and using the fact that c E kAK(n,r) ,

where

Thus (4.8b) gives a formula

(4.8d), by taking first the case v = ks+l,

(s+l,s)},

In other words,

which can be obtained by replacing,

c o Us(t) =

If

is zero unless

p E r, (gp,h@) E {(i,i) .... ,(n,n),

(gp,hp) = (s+l, s).

(4.8d)

for all

C~,h(g)ch, i •

Cg'h(g) = gglhl "'" gP~rrh

and if (4.8c) is satisfied, p

Z hEI(n,r)

c o u = c,

all

u E Un(K).

t E K,

64 k

~(n,r)

is the right

Extend the character Bn(K) = Tn(K)Un(K) ,

X k of

k-weight-space of Tn(K )

by defining

allows us to reformulate James's

(4.8e)

Theorem

c £ ~(n,r)

The module

Xk

~(n,r)

FK = GL (K)), n of

for all

u ~ Un(K).

This

theorem as follows.

Dk, K = W

k

is the set of all elements

which satisfy the conditions

This shows that

for

(see 4.5).

(see 3.2) to the Borel subgroup

Xk(u) = 1

c 0 b = Xk(b)c,

category

~(n,r)

Bn(K).

Dk, K

all

b 6 B-(K). n

is the induced module

Ind~-(Kk)

in the

(or, in fact, in the category of all rational modules of a

K-B-(K)-module n

Kk

which affords the character

For a discussion of a theorem equivalent to (4.8e), for

semisimple algebraic groups, see [J"~ §i.].

4 The modules Dλ,K

4.1 Preamble In this section and the next we shall deﬁne, for each λ = (λ1 , . . . , λn ) in Λ+ (n, r) and for each inﬁnite ﬁeld K, the modules Dλ,K and Vλ,K (see introduction). Both have character Φλ,0 = Sλ (X1 , . . . , Xn ). In 4.4, we shall deﬁne Dλ,K in terms of certain determinantal functions in AK (n, r). These modules have been known, in the “classical” case K = C, for a very long time—they were discovered (under the guise of “primary covariants”) by J. Deruyts in 1892 [13, p. 72]1 . More recently they have been described, for ﬁelds of arbitrary characteristic, by M. Clausen [8, p. 180], and by G. D. James [27, p. 129]. In fact James refers to Dλ,K as a “Weyl module”, but we prefer to reserve this term for the module Vλ,K which was deﬁned by Carter-Lusztig [6, p. 211], and which, as we show in 5.2, is the contravariant dual of Dλ,K . However James’ construction in [27] gives more information than ours, since it can be used to identify Dλ,K with the “induced module” of a one-dimensional character on the lower triangular Borel subgroup Bn− (K) of ΓK = GLn (K). We mention this identiﬁcation in 4.8.

4.2 λ-tableaux For the rest of §4, λ = (λ1 , . . . , λn ) ∈ Λ+ (n, r) is ﬁxed. The diagram (or “shape”) of λ is deﬁned to be the subset [λ] = { (s, t) : 1 ≤ s, 1 ≤ t ≤ λs } of Z × Z (cf. [15, p. 66]). A λ-tableau is a map, not necessarily bijective, of [λ] into a set. Since [λ] has r elements, there exists at least one bijection T : [λ] → r. We shall arbitrarily choose one such bijection, and call it 1 I am indebted to J. Towber for this reference. In his article [52], Towber gives an account of the history of Dλ,K : see particularly [52, p. 448].

34

4 The modules Dλ,K

the basic λ-tableau T = T λ . If the image under T of (s, t) is x(s, t), we may depict T as x(1, 1) x(1, 2) · · · (4.2a)

···

x(1, λ1 )

x(2, 1) x(2, 2) · · · x(2, λ2 ) x(3, 1) x(3, 2) · · · ···

···

Thus every element ∈ r appears exactly once in (4.2a). If = x(s, t), we say that is in row s and column t of T . The row stabilizer R(T ) of T is the subgroup of G = G(r) consisting of all π ∈ G which preserve the rows of (4.2a), and the column stabilizer C(T ) is deﬁned similarly. If i : r → n is an element of I(n, r), we denote the λ-tableau i ◦ T : [λ] → n by Ti . In general, Ti is not bijective. If = x(s, t), we often refer to i as the entry in place , or in the (s, t) place, of Ti . Example. Suppose r = 5, n = 3 and λ = (3, 2, 0). We might take for our

1 3 5 i1 i3 i5 basic λ-tableau T = . Then, for any i ∈ I(n, r), Ti = . 2 4 i2 i4 The entries of Ti belong to the set 3 = {1, 2, 3}.

4.3 Bideterminants Let K be an inﬁnite ﬁeld, and let i, j be elements of I(n, r). We deﬁne an element (Ti : Tj ) = (Ti : Tj )K of AK (n, r) by the formula (4.3a) (Ti : Tj ) = s(σ)ci,jσ = s(σ)ciσ,j . σ∈C(T )

σ∈C(T )

Here s(σ) denotes the sign of σ. The second equality comes from the fact that ci,jσ = ciσ−1 ,j , for any σ ∈ G(r). Apart from the interchange of rows and columns, the element (Ti : Tj ) is a bideterminant in the sense of D´esarm´enien, Kung and Rota [15, p. 67]. Let µ = (µ1 , . . . , µr ) be the partition of r conjugate to λ (notice that µ might not be an element of Λ+ (n, r), since it might have more than n parts). Then it is easy to see that (Ti : Tj ) is the product, over all columns t of [λ], of the µt × µt determinants . det cix(s,t) ,jx(s ,t) s,s =1,...,µt

(If µt = 0, we take this determinant to be 1.) From this, or directly from (4.3a), we see that (Ti : Tj ) is zero if there are equal entries at any two places in a column of Ti , or of Tj . Also (Ti : Tjσ ) = (Tiσ : Tj ) = s(σ)(Ti : Tj ), for all σ ∈ C(T ).

4.4 Deﬁnition of Dλ,K

35

Example 1. Let λ = (3, T be as in the example in 4.2. Consi 2, 0), 1 1 1 a d f der Tl = , Ti = . Then 2 2 b e c1a c1b c1d c1e c ∈ AK (3, 5). (Tl : Ti ) = c2a c2b c2d c2e 1f Example 2. Let l be the element of I(n, r) whose λ-tableau is ⎞ ⎛ 1 1 ··· ··· 1 ⎟ ⎜ ⎜ 2 2 ··· 2 ⎟ ⎟. (4.3b) Tl = ⎜ ⎟ ⎜ ⎟ ⎜ 3 3 ··· ⎠ ⎝ ··· In other words, lx(s,t) = s, for all (s, t) ∈ [λ]. Then (Tl : Tl ) = c(µ1 )c(µ2 ) · · · , where, for any integer m (0 ≤ m ≤ n), c(m) denotes the mth leading minor of the n × n matrix C = (cµν ).

4.4 Deﬁnition of Dλ,K The space AK (n, r) is a bimodule for KΓ, with Γ = ΓK acting by right and left translations. Equivalently, it is a bimodule for SK (n, r). If h, j ∈ I(n, r) = I, we have (see 2.8) (4.4a) ξ ◦ ch,j = ξ(ci,j )ch,i i∈I

and (4.4a )

ch,j ◦ ξ =

ξ(ch,i )ci,j ,

i∈I

for all ξ ∈ SK (n, r). As left module AK (n, r) belongs to the category MK (n, r), (n, r) of and as right module it belongs to the analogously deﬁned category MK all right, ﬁnite-dimensional KΓ- (or SK (n, r)-) modules whose coeﬃcient space lies in AK (n, r). In fact the coeﬃcient space of AK (n, r), whether regarded as left or right KΓ-module, is precisely AK (n, r). Now let l be the element of I(n, r) deﬁned in (4.3b). It is clear that l belongs to the weight λ = (λ1 , . . . , λn ). Deﬁnition. Dλ,K is the K-span of all bideterminants (Tl : Ti ), i ∈ I(n, r). Hence Dλ,K is a subspace of AK (n, r). Replace h by lσ in (4.4a), multiply by s(σ), and sum over all σ in C(T ). We get

36

(4.4b)

4 The modules Dλ,K

ξ ◦ (Tl : Tj ) =

ξ(ci,j )(Tl : Ti ), all ξ ∈ SK (n, r).

i∈I

It follows that Dλ,K is a left SK (n, r)-submodule of AK (n, r). If we compare (4.4b) with (2.6a), we see also that the surjective linear map ⊗r ϕ = ϕK : EK → Dλ,K ,

which takes ej → (Tl : Tj ) for all j ∈ I(n, r), is an SK (n, r)-module homomorphism. Remark. Our deﬁnition of l and (Tl : Ti ) depends on our choice of basic λ tableau T (see 4.2). However any other bijective λ-tableau T can be written T = πT for some π ∈ G(r). Then Ti = Tiπ for any i ∈ I(n, r), and if l = lπ −1 we have (Tl : Ti ) = (Tl : Ti ). Therefore Dλ,K is in fact independent of the choice of T . Example 1. In case λ = (r, 0, . . . , 0) we shall write Dr,K for Dλ,K . The λ diagram is a single row of length r, and Tl = (1 1 . . . 1). For any i ∈ I(n, r), we have Ti = (i1 i2 . . . ir ) and (Tl : Ti ) = c1,i1 · · · c1,ir . If we map this to the monomial ei1 · · · eir of 2.6, example 2, we see that Dr,K is isomorphic to the rth symmetric power of E. Example 2. Assume r ≤ n. In case λ = (1, . . . , 1, 0, . . . , 0) with r 1’s, we shall write D(1r ),K for Dλ,K . The λ diagram is a single column, and Tl has entries 1, 2, . . . , r. For any i ∈ I(n, r), (Tl : Ti ) is the r-rowed determinant det (cl,1 ). Mapping this onto ei1 ∧ . . . ∧ eir (see 3.2), we see that D(1)r ,K is isomorphic to the rth exterior power of Λr EK .

4.5 The basis theorem for Dλ,K As usual, K is an inﬁnite ﬁeld. Our aim is to prove the (4.5a) Basis theorem. Dλ,K has K-basis consisting of all (Tl : Ti ) such that Ti is “standard”, i.e. the entries in each row of Ti are weakly increasing (≤) from left to right, and the entries in each column are strictly increasing (<) from top to bottom. This theorem generalizes the theorem of Specht-Garnir for Specht modules (see [45] and [27, §8, 13]). It may be deduced from a more general basis theorem of D´esarm´enien, Kung and Rota [15, p. 78]. For completeness, and also to prepare for the transition to the module Vλ,K in §5, we give here a proof of (4.5a) based on a combinatorial lemma of Carter-Lusztig (see 4.6). We begin by showing that (4.5b) The set { (Tl : Ti ) : Ti standard } is linearly independent.

4.6 The Carter-Lusztig lemma

37

Proof. All the (Tl : Ti ) (i ∈ I(n, r)) lie in the “right” λ-weight-space AK (n, r) = AK (n, r) ◦ ξλ .

λ

For it is clear from (2.3c) that λAK (n, r) is spanned by the elements cl,i , where i ∈ I(n, r) (remember ξλ = ξl,l ). Now cl,i = cl,j if and only if there is some π ∈ G(r) such that lπ = l and iπ = j. The condition lπ = l is equivalent to π ∈ R(T ), the row stabilizer of the basic λ-tableau ((4.2a)). So cl,i = cl,j if and only if Ti , Tj can be obtained from one another by a row permutation (thus the cl,i correspond to James’ “λ-tabloids” [27, pp. 10, 127]). For each i ∈ I(n, r), deﬁne β(i) = (β1 (i), . . . , βn (i)), where βs (i) is the sum of the entries in row s of Ti . Clearly β(i) = β(j) if cl,i = cl,j , by what we have just said; hence (4.5c) If i, j ∈ I(n, r) and β(i) = β(j), then cl,i = cl,j . We order these vectors β(i) (they are in fact elements of Λ(n, r)) in the usual lexicographic way. The reader may verify (4.5d) If Ti is standard and if 1 = π ∈ C(T ), then β(iπ) > β(i). Now suppose we have a non-trivial linear relation fi (Tl : Ti ) = 0, fi ∈ K, fi = 0, i∈H

where the sum is over a non-empty subset H of those i ∈ I(n, r) such that Ti is standard. By (4.3a), this gives (4.5e) fi s(π) cl,iπ = 0. i∈H π∈C(T )

By (4.5c) we can equate to zero the partial sum of all terms (i, π) in (4.5e) such that β(iπ) is equal to any given β ∈ Λ(n, r). Take for β, the least of the β(iπ) (i ∈ H, π ∈ C(T )). By (4.5d), we can have β = β(iπ), for i ∈ H, π ∈ C(T ), only if π = 1. So the partial sum has the form (4.5f ) fi cl,i = 0, i∈H

for some non-empty subset H of H. But since Ti is standard for all i ∈ H , the cl,i appearing in (4.5f) are linearly independent, and this gives a contradiction. This proves (4.5b).

4.6 The Carter-Lusztig lemma To complete the proof of the basis theorem (4.5a) we must show that every bideterminant (Tl : Ti ) (i ∈ I(n, r)) is expressible as a linear combination

38

4 The modules Dλ,K

of (Tl : Ti ) for which Ti is standard. This is the harder part of the proof of (4.5a), and we deduce it from a combinatorial lemma (4.6a) of CarterLusztig. The proof of (4.6a) is given in [6, p. 214, 215], and does not depend essentially on other results in [6]. The reader may note some variations of expression between Carter-Lusztig’s paper and this one: their “semi-standard” is our “standard”; their T corresponds to our Ti ; their σT is our Tiσ ; the “type” λ of T (= Ti ) corresponds to the “weight” of i. Finally, in our (4.6a) we have not restricted f to tableaux of a given type λ . (4.6a) Carter-Lusztig lemma. Let f : I(n, r) → F be any map with values in an abelian group F , satisfying the following three conditions: (i) f (i) = 0, if Ti has equal entries at two distinct places in the same column. (ii) f (iσ) = s(σ) f (i), for any i ∈ I(n, r) and σ ∈ C(T ). (iii) (Garnir relations) ν∈G(J) s(ν) f (iν) = 0, for any i ∈ I(n, r) and any non-empty subset J of the (h + 1)th column Ch+1 of the basic λ-tableau T . Here h is any element of {1, 2 . . . , r − 1}, and G(J) is a transversal of the set of cosets { νX : ν ∈ Y }, where Y is the subgroup of G(r) consisting of all π ∈ G(r) which ﬁx every element outside Ch ∪J, and X = C(T )∩Y . Then Im f lies in the subgroup of F generated by the set { f (i) : Ti standard }. Remarks. Condition (ii) shows that the sum in (iii) is independent of the choice of transversal G(J). Condition (iii) is equivalent, by an argument given in [6, p. 212], to condition (37) of [6, p. 214]. A slightly weaker form of (4.6a), which uses a larger set of “Garnir relations” (but is still adequate for our purposes) can be proved by an adaptation of the proof of Garnir [19] for Specht modules; see [45, pp. 93, 94] or [27, pp. 29, 30]. Lemma (4.6a) completes the proof of the basis theorem (4.5a), as soon as we observe (4.6b) The function f (i) = (Tl : Ti ) satisﬁes conditions (i), (ii) and (iii) of (4.6a). Proof. That f (i) satisﬁes (i) and (ii) has already been said in 4.3. The proof of (iii) is like that for Specht polynomials, and goes as follows. Every element of the set B = Y · C(T ) has unique expression π = νσ, with ν ∈ G(J) and σ ∈ C(T ). So the left side of the Garnir relation (iii) (with f (i) = (Tl : Ti )) becomes s(ν) s(σ) cl,iνσ = s(π) cl,iπ . ν∈G(J) σ∈C(T )

π∈B

An argument given in [45, pp. 92, 93] shows that B can be written as disjoint union of subsets {π, πκ}, in which κ (which depends on π) is some transposition in R(T ). For such a pair s(π) cl,iπ + s(πκ) cl,iπκ = 0, because cl,iπκ = clκ,iπ = cl,iπ (notice lκ = l for all κ ∈ R(T )). Hence the sum above is zero, as required.

4.7 Some consequences of the basis theorem

39

4.7 Some consequences of the basis theorem Let α ∈ Λ(n, r) be a given weight, and suppose a ∈ I(n, r) is an element of α. Putting ξα (= ξa,a ) for ξ in formula (4.4b) we see that ξα ◦ (Tl : Tj ) = (Tl : Tj ) or zero, according as j ∈ α or not. Consequently each bideterminant (Tl : Tj ) (where j ∈ I(n, r)) is a “weight element” of Dλ,K , i.e. it belongs to the weightα , where α is the weight containing j. Then (4.5a) gives the ﬁrst space Dλ,K statement below: α (4.7a) For each α ∈ Λ(n, r), Dλ,K has K-basis consisting of all (Tl : Ti ) with i ∈ α and Ti standard. Hence the character of Dλ,K is equal to the Schur function Sλ (X1 , . . . , Xn ).

The second statement in (4.7a) follows from the ﬁrst, and from the fact that the coeﬃcient of X α in Sλ = Sλ (X1 , . . . , Xn ) is precisely the number of λ-tableaux Ti which have “content” (i.e. weight) α—this can be proved by a direct combinatorial argument from the deﬁnition of Sλ (see [39, p. 42]). Since Sλ is the character of an irreducible module in MK (n, r) when char K is zero, we deduce (4.7b) If char K = 0, then Dλ,K is irreducible. We show next that the family Dλ,K is deﬁned over Z. For each inﬁnite ﬁeld K, we may write (Tl : Ti )K to denote the element (Tl : Ti ) deﬁned in 4.3. (4.7c) The Z-span Dλ,Z of the elements (Tl : Ti )Q (i ∈ I(n, r)) is a Z-form of Dλ,Q . The family {Dλ,K } is Z-deﬁned by Dλ,Z and the maps δK : Dλ,Z ⊗ K → Dλ,K , (Tl : Ti )Q ⊗ 1K → (Tl : Ti )K Moreover the family of inclusions Dλ,K → λAK (n, r) is also deﬁned over Z. Proof. When applied to the function f (i) = (Tl : Ti )Q , the Carter-Lusztig lemma (4.6a) shows that (Tl : Ti )Q is in the Z-span of { (Tl : Ti )Q : Ti standard }. But by (4.5a), this set is a basis of Dλ,Q . By (4.4b), Dλ,Z is invariant to the left action of SZ (n, r). It follows that Dλ,Z is a Z-form of Dλ,Q . The maps δK described above are clearly SK (n, r)-morphisms, and are isomorphisms by (4.5a) applied to Dλ,K . The last statement of (4.7c) is immediate from the deﬁnitions. Remark. Dλ,Z is a direct summand, as Z-module, of AZ (n, r); equivalently, inc

the exact sequence 0 → Dλ,Z → AZ (n, r) is Z-split. This follows from (4.7c) and the next lemma.

40

4 The modules Dλ,K

(4.7d) Lemma. Suppose {VK }, {WK } are families of modules in MK (n, r), and are Z-deﬁned by VZ and {δK }, WZ and {ηK }. Suppose θK : VK → WK is a family of morphisms in MK (n, r), also deﬁned over Z (see 2.6). Then θQ

(i) If all the θK are injective, then the exact sequence 0 → VZ → WZ is Z-split. θQ

(ii) If all the θK are surjective, then the exact sequence VZ → WZ → 0 is Z-split. Proof. Let M = (mβα ) be the matrix of θQ , relative to bases {vα }, {wβ } of VQ , WQ which Z-generate VZ , WZ respectively. The assumption that the family {θK } is deﬁned over Z implies M is an integral matrix, and that for each K, MK = (mβα ·1K ) is the matrix of θK relative to the appropriate bases of VK , WK . If all the θK are injective, this means that M has rank d = dimQ VQ , and that it still has rank d, even after reduction modulo any rational prime p. Hence all elementary divisors of M are equal to 1, and therefore there exists an integral matrix M = (mαβ ) such that M M = identity. This proves (i). The proof of (ii) is similar.

4.8 James’s construction of Dλ,K G. D. James has given [27, p. 129] a construction of a KΓK -module W λ , which he calls a “Weyl module”, but which is in fact isomorphic to Dλ,K . If α, β ∈ Λ(n, r), then the elements ci,j (i ∈ α, j ∈ β) of AK (n, r) may be identiﬁed with James’s “α-tabloids of type β” [27, p. 127]. (It is not necessary that either α = (α1 , . . . , αn ) or β = (β1 , . . . , βn ) be dominant, i.e. be proper partitions of r.) James’s space S ◦,λ becomes λAK (n, r) in our language (see 4.5). He deﬁnes, for all pairs of integers s ∈ {1, . . . , n} and v ∈ {0, . . . , λs+1 } a linear map ψs,v : λAK (n, r) → AK (n, r) by the rule: (4.8a)

ψs,v (cl,i ) =

ch,i ,

h

summed over all h ∈ I(n, r) obtained by replacing, in l (or in Tl ) λs+1 − v of the entries s + 1 by s. Deﬁnition (see [27, p. 129]). W λ is the set of all elements c ∈ λAK (n, r) such that ψs,v (c) = 0 for all s ∈ {1, . . . , n} and v ∈ {0, . . . , λs+1 − 1}. By a rather delicate combinatorial argument, James proves a theorem (see [27, 26.3, p. 128]) from which it follows readily that W λ has as basis the set { (Tl : Ti ) : Ti standard }; hence W λ = Dλ,K . However James’s deﬁnition

4.8 James’s construction of Dλ,K

41

has a very important group-theoretical interpretation, which we shall now give. Let U − = Un− (K) be the subgroup of Γ = GLn (K) consisting of all lower triangular unipotent matrices in Γ. It is well-known that U − is generated by the elements us (t) (s ∈ {1, . . . , n − 1}, t ∈ K) where ⎞ ⎛ 1 ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ (row s) ⎜ 1 ⎟ ⎜ us (t) = ⎜ ⎟ (row s + 1) t 1 ⎟ ⎜ ⎟ ⎜ . .. ⎠ ⎝ 1 with t in position (s + 1, s). Now write g = us (t) and i ∈ I(n, r). By deﬁnition of the action of Γ on AK (n) (see introduction) (4.8b) cl,i ◦ g = cl,h (g) ch,i . h∈I(n,r)

It is clear that cl,h (g) = gl1 h1 · · · glr hr is zero unless (4.8c) For all ∈ r, (l , h ) ∈ {(1, 1), . . . , (n, n), (s + 1, s)}, and if (4.8c) is satisﬁed, then cl,h (g) = tw , where w is the number of such that (l , h ) = (s + 1, s). In other words, cl,h (g) = tw , for any h ∈ I(n, r) which can be obtained by replacing, in l (or Tl ), w of the entries s + 1 by s. Thus (4.8b) gives a formula

λs+1

(4.8d)

c ◦ us (t) =

tλs+1 −v ψs,v (c),

v=0

for all c ∈ λAK (n, r), s ∈ {1, . . . , n − 1} and t ∈ K. Naturally we prove (4.8d) by taking ﬁrst the case c = cl,i (i ∈ I(n, r)). If v = λs+1 , it is clear that ψs,v (c) = c, for all c ∈ λAK (n, r). So by (4.8d), and using the fact that K is inﬁnite, we see that for any c ∈ λAK (n, r), the conditions ψs,v (c) = 0, all s ∈ {1, . . . , n − 1}, v ∈ {0, . . . , λs+1 − 1} are equivalent to the conditions c ◦ us (t) = c, all s ∈ {1, . . . , n − 1}, t ∈ K, which in turn are equivalent to the conditions (4.8e)

c ◦ u = c, all u ∈ Un− (K).

42

4 The modules Dλ,K

Recall from 4.5 that λAK (n, r) is the right λ-weight-space of AK (n, r). Extend the character χλ of Tn (K) (see 3.2) to the Borel subgroup Bn− (K) = Tn (K) Un− (K), by deﬁning χλ (u) = 1 for all u ∈ Un− (K). This allows us to reformulate James’s theorem as follows. (4.8f ) Theorem. The module Dλ,K = W λ is the set of all c ∈ AK (n, r) which satisfy the conditions c ◦ b = χλ (b)c, all b ∈ Bn− (K). This shows that Dλ,K is the induced module IndΓB − (Kλ ) in the category MK (n, r) (or, in fact, in the category of all rational modules for GLn (K)), of a K · Bn− (K)-module Kλ which aﬀords the character χλ of Bn− (K). For a discussion of a theorem equivalent to (4.8f), for semisimple algebraic groups, see [30, §1].

§5

The Carter-Lusztig modules

5.1

Definition of Let

%~,K

~ ~ #~+(n,r)

Denote by

NK

~,K

be given, and let

the kernel of the

defined in 4.4.

Let

VX, K

the canonical form

on

E~rK ' Vk,K

{x (

is also a submodule of

o Vk, K ~ (Dk, K) ,

contravariant

to

DX, K

If

K

is an

NK,

relative to

SK(n,r)-submodule

of

Since

<, > is non-singular,

we

form

(,) : Vk, K × D k K -~ K

by

@r x ~ Vk,K, y ~ E K .

all

dual modules in

V%,K

~ = Sk(X 1 ..... Xn) o ' Vk, K is irreducible, and is isomorphic

has finite characteristic,

V~~,K

and

5.2

Vh~,K is Carter-Lusztig's

then in general

are not isomorphic.

We shall identify

Vk, K

Lusztig in [CL, pp. 211, 222]. more exactly.

"

(by 3.3e))

if char K = 0, then

(see 4.7). Dk,K

,

NK

and because (contravariant)

have the same character

In particular,

and E~.

(x, ~K(y)) = <x,y~

l~(n,r)

--> 0

.~r : <x, Nk;~ = 0}

is contravariant

may define a non-singular,

Hence

,K

(see 2.7, example I):

Vk, K =

<,>

D~

~(n,r)

be the orthogonsl complement to

<,>

(5.1b)

~r ~K : EK ~ ~ , K

SK(n,r)-epJmorphism

®r ~ K --> EK

0 --~ N K

Since

be any infinite field.

We have then an exact sequence in

(5.1a) Delinition.

K

"Weyl module" with the module

~

defined by Carter-

First we must describe

N K = Ker ~K

@6

(5.2a)

NK

(i)

is the

R1

K-span of the subset

consists of all

ei

R = R1 U R2 U R 3

such that

i E I(n,r)

and

of

E~,

where

Ti

has equal

entries in two distinct places in some column. (ii)

R2

cons~ists of all

e i - s(o)elo ,

(iii)

R3

consists of all elements

where

Y

i ~ I(n,r)

s(w)eiv,

where

and

o ~ C(T).

i ~ l(n,r) _

o~G(J) and

J

is a non-empty subset of

Proof the

All the elements of K-span

N'

F = EK®r'-'/m onto function

of

R.

~,K'

R

Ch+I(T)

lie in

for some

N = NK,

h E {1,2 ..... r-l}.

by (4.6b), and so

There is therefore a well-defined given by

~(x+N') = ~K(X),

all

K-map

x ~ E~ r.

N ~

contains from

Now the

f : I(n,r) ~ F given by f(i) = ei+N' clearly satisfies the hypotheses

of (4.6a), hence

F

is K-spanned by the set {e i + N' : T i standard}.

this set onto a basis of implies

NC

N';

hence

~,K'

by (4.5a).

N = N',

Therefore

and (5.2a) is proved.

Ker ~ = O,

But ~ maps

which

As a corollary

we have

(5.2b)

~,K

x E ~EKr which satisfy the

is the set of all elements

following conditions: (i)

<x,

ei>=

0

for all

i ~ I(n,r)

such that

Ti

has equal entries

Jn two distinct places in the same c61umn. (ii)

~

(iii)

= s(o)x %

for all

s(~)x~ -I = 0

o E C(T). any non-empty subset

J

of

~h+l(T), h ~ {l,2,...,r-l}

~ G(J) Proof (5.2a) shows that <x, R

>= 0

for

VX, K

s = 1,2,3.

consists of all

x E E~ r

such that

(5.2b) is an almost immediate consequence

S

of this, together with the fact that an "invariance" condition

<,>

is non-singular, and satisfies

67

(5.2c)

<x~, y> = <x, y~

for all

x,y 6 E K ,

n ~ G(r).

form the deflnitien of

<,>

>

This last condition is verified trivially

(see 2.7, example I).

It is now easy to see that "Weyl module"

-I

Vk, K

~oincides with Carter-Lusztig's

~± ", conditions (28), (29) of [CL, p. 211] are essent~ally

(i), (ii), (iii) of (5.2b).

Example..... I.

If

k = (r,0, "'" ,0)

(i), (il) of (5.25) are vacuous. transpositions all s ~ e t r i c

x

in

contravariant sense to the

over all

~1

e~ = e I

Pn

...e n

for

•

So

Assume

{v

Vk, K

Dr, K

: a ~ A(n,r)},

(v ,e~) = 6

r ~ n

x ~ = x,

for all

is the space of

EK~r. Therefore this module is dual in the

where

The non-singular contravariant form

so that

Conditions

Vk,K"

Condition (iii) says

rth symmetric power

has basis

is given by

[x ample 2. We write

Vr, K

I ~ ~.

(see 5.])

V r,K

v = (h, h +I) (h = i,..., r-l). tensors•

Example i).

we write

for all

~'~

{e ~ : ~ ( A(n,r)}

and that

of

EK va

(see 4.4, =

~ei,

sum

(,) : Vr, K × Dr, K ~ K

a,~ 6 A(n,r).

is a basis of

k = (I,...,I,0 .... ,0)

D

Here r,K"

with

r

l's.

V

for Vk, K. Condition (iii) of (5.2b) is vacuous, and (ir),K conditions (i), (ii) show that V is the space of all antisymmetric (ir),K whatever tensors in E,er K . V (ir),K is an irreducible module in ~ ( n , r ) , the characteristic of

K,

since its character

er(Xl,...,X n)

non-trivial expression as a sum of symmetric functions in Therefore

V

(Ir),K

Is isomorphic to the

rth

has no

Z(XI,...,Xn].

exterior power

68

D(ir), K =

~&rEK

ArEK ~ V

(see 4.4, example 2). which takes each

e

(ir),K

'fhere is an isomorphism

= e s

A ...4

ei

iI

(see 3.3) to r

(eil ® "'" ~ elr )( ~6G Z s(~)~). 5.3

The Carter-Lusztig Carter-Lusztig

is a cyclic module

basis for

Vk, K

have given a basis for [CL, pp. 216-219],

different proof of these results.

VX,K,

V~, K

It Is interesting that the Carter-Lusztig DX, K

If

X

given in (4.5a);

are connected by a certain unimodular matrix

(see (5.3)) which has appeared in work of D~sarmenien

Notation.

VX, K

We shall give here a slightly

basis (see (5.3b)) is not the dual of the basis of these bases of

and shown that

[De, ~.7~ ].

is any subset of the syum~etric group

Ix]

=

z

~,

{x}

G(r),

we shall write

=

n~X These are elements of the group-rlng even belong to

ZG(r).

(5.3a)

be the element of

Let

fg = eg{C(T)}

Proof For

lies in

Verify that y ~ RI U

as in (5.2a) y = e~{G(J)}. B = G(J)C(T) up

g

B

R2 (iii).

KG(r),

l(n,r)

of course.

K = Q,

given in (4.3b).

they

Then

Vk, K .

= O,

for all

there is no problem.

y E ~i U R 2 U R 3

Suppose then that

In the notation just introduced,

Using (5.2c),

<e~{C(T)},

{~, ~K}, each

(see (5.2a)).

y =

% s(v)eiv, v~G(J)

this reads

ei{G(J)} > = <e~, el{B}>,

= YC(T) -- see the proof of (4.6b).

as union of pairs

If

K

where

As in that proof, we break

being a transposition

6g

in

R(T).

Using (5.2c) again, and the fact that

<e%, ei(s(~)~

+ s(~K)~<)> = 0

for each pair.

e%<

= e% ,

Therefore

we have

= 0,

and this completes the proof of (5.3a).

Remark.

fg

is the element denoted by

Now let by

fg.

Since

As

V'

be the

K-space,

V'

V'

is

in [CL, p. 216].

SK(n,r)-submodule of

V~, K

is spanned by the elements

~i,jfg = (~i,jeg){C(T)}, Hence

~

~i,jfg(i,j

we see by (2.6a) that

K-spanned by the elements

which is generated E I(n,r)).

~i,jfg = 0

unless

b i = ~i,gfg(i ~ I(n,r)).

In fact we have the following much more precise statement.

(5.3b)

Theorem

[CL, Theorem 3.5, p. 218].

{b i = ~i,~fg

is a

K-basis for

generated as

Proof

Let

Vk, K.

SK(n,r)

In particular

(or

i,j ~ I(n,r).

that there is some ~ ~ R(T)

iR(T) of

h ~ l(n,r)

~ ~ G(r)

(see (4.3b)).

standard}

i.e.

VX ,K

is

f~.

From (2.6a) we have

~i,ge$ =

where the sum is over all

Ti

V' = Vk, K,

KFK)-module by

(5.3c)

if

: i ~ I(n,r),

The set

with

~i: e h , h such that

h = i~, f = ~ .

(i,~) ~ (h,g), But

g = ~

So the sum in (5.3c) is over the

i.e. such if and only

R(T)-orbit

i.

Now we bring in the form and calculate

(,) : VX, K x D~, K ~ K

introduced in 5.1,

70

(bi, (Tg : Tj)) = =

< ~i,ge~ ,

=

<

and

~(i,j)

i

=

~ eh , h~iR(T)

% s(o),

belong to the same Now suppose

% ~C(T)

o E C(T)

If

this implies

o = i,

(see 4.5), and if the matrix I

(~(i,j)),

= {k E I(n,r)

that

either

i > J

singular.

Note.

5.4

i

and

for all

and

J

i

that

jo

There

R(T)-orbit.

~(i) = ~(jo)

~(jo) > ~(j).

Let

~

denote

running over the set I

i,j ~ I ,

any total order

then

~(i,i) = i.

~(i,j) = (bi, (Tg : Tj)),

standard}

~(i,j) # O.

are in the same

If we give

and clearly

{(Tg : Tj) : Tj

A proof that

such

In any case we have

standard}.

{b i : T i

standard}

~

such

is a unimodular ~(i,j) ~ 0 Hence

and since

is a basis of

is a basis of

>

VX, K.

~ (,)

implies is nonis

Dk, K,

it

This proves (5.3b).

is unimodular is given by D~sarm~nien in [De, ~ . ~ ] .

Some consequences of the basis theorem For each

b i = ~i,~f~ of

o ~ C(T)

then (4.5d) implies

i = j ,

But since

follows that

is equal to

For by what we have shown above,

or

non-singular and

jo

i = j.

~(i) > ~(j) = i ~ J

triangular matrix.

~(i,j),

using (5.3c).

are both standard, and that

with

: Tk

s(o)ejo > ,

sum over all

such that

¢ ~ 1

using (5.2c)

R(T)-orbit.

Ti, Tj

must exist

< ~i,~eg{C(T)}, ej>

ej{C(T)} > ,

This last expression, which we denote (5.3d)

=

i.

From

i ~ I(n,r),

satisfies (5.3b)

it is clear by (2.3c) that the element

~i,i bi = b i , we deduce

thus

~ K, b i ~ V~,

where

is the weight

71

(5.4a)

Let

a (£(n,r).

{b i : i ~ ~,

In particular,

Then

Ti

Vk, K

has

K-basis

standard}

V k,K ~ = K.fg

(since

b~ = ~g,g fg

= f~).

A well-

The element

fg

known argument (see e.g. [J", p. 2]) shows (5.4b)

VX, K

has a unique maximal submodule

does not lie in character

Proof Vk,K,

~ax ,K '

~k,p,

By (5.3b)

V' =

FX,K = Vk,K/~ax ,K

has

Any proper submodule

M

The irreducible module

where

VX, K

p = char. K.

is generated by

since it does not contain

lie in

~ax ,K "

fg,

a proper

Z Vk,K, ask

sum of all proper submodules

M

f&.

has

M k= M N VX, K = 0,

K-subspace of of

and is the unique maximal submodule

VX, K

V~, K.

lies in

V',

hence is proper,

and does not contain

Since

DX, K

is dual to

VX, K

VX, K

M

Therefore the

third statement in (5.4b) now follows from the definition of (see 3.5, Remark (i)) and the fact that kI kn ~k = XI ... Xn + . . . .

and so

of

f~.

The

~k,p

has character

we have the following corollary to

(5.4b).

(5.4c) Hence

Dk, K

has a unique minimal submodule

minK , FX, K D~,

_min uk, K

and

min Dk, K

~

(Fk,K)°.

are isomorphic modules.

Proof.

The second statement follows from the first, and the fact that any

module

V

V°

in

~k(n,r)

(see (3.3e)).

If

has the same character as its (contravariant) dual V

is irreducible

this implies

V ~ V °.

72

Remark

Since

Fk, K

same is true of

has

minK . Dk,

k-weight space of dimension minK Dk,

Therefore

SK(n,r)-module

by, the element

k-weight-space

of

DX, K

is

1

(by (5.4c))

the

contains, hence is generated as

(Tg : Tg).

For (4.7a) shows that the

K (T~ : T~).

This proves (5.4d) below.

A

quite different proof comes from a standard argument for semisimple algebraic groups, using the fact (cf. 4.8) that the action of the upper and lower unipotent F = GL (K). n (5.4d)

(T~ : Tg)

of

@r V~,Q = {x ( EQ : < x, NQ > 0},

Vk, Z = E~Z

Lemma.

(i)

bi,Q

~r EZ ,

The sets

=

Vk, Z.

Vk, Z.

It

Vk, Q

is defined over

NO = Ker

is a

that,

lies in

Since (i) is a Q-basis of

Z-form of

Since

j ~ I(n,r).

Take any

t h e p r o o f of

(5.3b)

j gives

Ti .

(see 5.1).

it has

Vk, Q.

<x,

Sz(n,r) ,

i 6 I(n,r),

(by (5.3b)), the proof of (5.4e)

we have T. J

Z-basis

hence that (i) is a subset of

x = ~k.b. ll

such that

Recall that

standard}

every element

x E E~r Z ,

~

the standard

~r,

We certainly have

k.1 ~ Q"

Ti

for each

Vk, Q

be a c h i e v e d i f we show t h a t

some

~Q

Z.

are both closed to the action of

is clear

~i,~ e~,Q{C(T)}

Z-span of (i). for

Vk, K

where

N Vk, Q

generates the irreducible module DX, K .

{bi, Q : i E l(n,r),

hence so is

will

triangular subgroups of

min

Dk, K

We show next that the family

Proof.

is stable under

See [St. p. 214].

The element

(5.4e)

(Tg : Tg)

is

x ( Vk, Z

is

(sum is over <x,

ej>

standard.

e.> = ~k. !,~ ( i , j ) , j 1

6

in the

Ti Z

standard)

for all

The c a l c u l a t i o n t h e sum b e i n g o v e r

s

But t h e " D e s a r m e n i e n m a t r i x "

~ = ~(i,j)),

in case

in

73

K = Q,

is integral and unimodular.

standard

Remark

E~;

Tj) follows

(5.4e) shows that

hence the inclusion

0 ~ l~, Z ~ EZ ~

in the category EZ~ ® K.

8K

(5.4f)

described.

as submodule of {EK ~}

is

Z-defined

From (5.4e) and (5.3b) it is

induces an isomorphism

bi, Q ® IK~

The family

If we tensor with

0 ~ V~, Z ~ K ~ E~r ~ K

VX, z ~ K

6 K : el, Q ® 1 K ~ el, K.

Z-submodule of

Z-split.

we showed that the family

8~ : Vk,Z®

which maps

is

We shall regard

In 2.6, Example 1

immediate that

Uence (5.4e) is proved.

is a pure

we get an exact sequence

MK(n,r ).

E~ ~r and maps

Ti).

(all

VX, Z = Sz(n,r)f ~.

Vk, Z = E~ r n VX,Q

K

~k i ~ (i,j) ~ Z

(all standard

It is clear that

any infinite field

by

ki ~ Z

So from

bi, K

{Vk, K } is

for all

K -+ VX, K.

i E I(n,r).

Z-defined by

The family of inclusions

So is the family of contravariant

Vk, Z

VX, K ~+ E ~r K

forms

From all this we deduce:

and the maps

6'K just

is also defined over

(')K : VX,K x Dk, K -+ K

Z.

defined in

5.1.

5.5

Contravariant

forms on

J.C. Jantzen (see

V%, K

~], CJ'] ....

)

has studied contravariant forms

on the Weyl modules for a simply-connected,

semisimple algebraic group;

particular,

SLn(K)

his results apply to the group

alteration to our case

F = GL (K) . n

,

in

and extend with little

In this section we shall give an

independent description of the contravariant

forms on the modules

V%, K .

74

We saw that element

is generated

f~ = e~{C(T)}

in the submodule form

V%, K

< , >

on

(5.5a)

of

E mr ,

Emr{C(T)} E mr ,

of

as

S(= SK(n,r))-module

and it follows

E mr

that

by the

VI, K

is contained

We have the canonical

and from (5.2c) we deduce

<x{C(T) }, y> = <x, y{C(T)}>

This allows us to define a "contracted"

contravariant

that

for all

v e r s i o n of

x, y g E mr

< , >

on

Emr{C(T)}

by

the rule

(5.5b)

If

x, y e E ~r ,

define

<<x{C(T)},

y{C(T)} >> = <x, y{C(T)} >

Any ambiguity

arising

from the fact that an element

expressed

x{C(T)}

= x'{C(T)}

as

is eliminated

by (5.5a).

contravariant

form on

symmetric, rule

(5.5b) gives

We might

also m e n t i o n

form on

If we restrict

V%, K ,

=

of

E mr ,

is a symmetric, it to

V%, K

which is moreover

= <e~,e~{C(T)}>

fields

x, x'

may be

~ s(o) oEC(T) << ' >>K '

K , is defined over

Z

we get a

non-zero,

<e~,e~o>

since

= 1

constructed

in

in the sense of

3.

Any contravariant factor, with

.

<< , >>

that the family of forms

this way for all infinite 2.7, Example

It is clear that

<>

Emr{C(T)}

for distinct elements

Emr{C(T)}

contravariant

of

<< , >> .

form

(

,

)

on

V%, K

For the contravariant

coincides, property

up to a scalar

(2.7d),

together with

75

the fact that values

VX, K = Sf~ ,

shows that

(f~,v) , v E VX, K .

with each

v

If

(

,

v s V%, K

)

is

determined by the

is decomposed as sum

belonging to the weight space

V%, K~

v = Ev~ ,

(~ c A(n,r)) ,

then

since weight spaces for distinct weights are orthogonal with respect to (

,

)

(see 3.4),

by the values space is

(f%,v)

K.f%

for

(f~,v) = (f~,vx) , v

(see 5.4a).

Therefore (since for all

we have

in the So

(

<> = i)

i.e.

%-weight space

,

if

)

(

,

)

x VX, K .

is determined

But this weight

is completely determined by (f~,f%) •

(f~,f%) = k ,

then

(v,w) = k<> ,

v,w e VX, K o

In the work of Jantzen which we have mentioned, and also in the earlier work of W.J~ Wong ([W],[W']) , a Weyl module submodule

(5.5c) space

(~W',

of

V .

In our case the result reads as follows.

Theorem 3B, p.362~).

M = {v g VX, K : <
M

M # V%, K , of

is that it provides a method of calculating the maximal

Vmax

submodule

Proof.

V

V%, K (~ ~ X)

of

I,K

since

The radical of

>>= O}

<< , >> ,

i,e.

coincides with the unique maximal

f% ~ax ,K

Since

M .

V~, K

by the contravariant property.

Therefore

M £~%ax -

lies in the sum V'

V'

is orthogonal to

.

v~%ax lies in ,K

M

Also

But we saw in the proof

,K

of all the weight-spaces X VX, K = K.f~ ,

we have

(~ax VX,K ) = <
the

V%, K .

is a submodule of

(5.4b) that

the importance of the contravariant form on

and the proof of (5.5c) is complete.

76

Example.

We shall calculate

of 5.2, Example

i).

one row, and so

Since

the form

I = (r,O,.,.,O),

C(T) = {i} .

to

Vr, K

of the canonical

{v

: ~ e A(n,r)}

and

< , >

E ~r .

form

Since

(r,~)

maxK , M = Vr,

1

has only

is just the restriction to the basis

I, the form is given by where

r~ f

C~n -

of this form is spanned by those

the integer

(notation

Relative

= (r,~).l K ,

C~I . . . .

divides

on

Vr, K

the diagram of

<< , >>

(r, ~)

M

on

Therefore,

given in 5.2, Example

= O (e # B)

So the radical

<< , >>

v

p = char K

for which

.

the irreducible

module

F r , K = V r,~~/vma~ r,~

(see

(5.4b)) has basis

(v

The

a-weight

the character with

(r,e)

+ M : ~ s A(n,r),

space of

Fr, K

of

is

Fr, K

~ O mod p .

~r,p

expansion

(5.5d)

(XI+°..+Xn)r

is

~

r,p

~ 0 m o d p} .

K . ( v ~ + M) =

IX ~ 1

°'°

Since the integers

the multinomial

we have the result:

is

(r,~)

=

~ esA(n,r)

,

for all

x~n n

(r,~)

~i

(r,~)X 1

(which is a polynomial

t h e sum o f t h o s e m o n o m i a l s

X l l ' " .xan n

'

e e A(n,r)

sum over all

.

Hence

~ e A(n,r)

are the coefficients

in

~n

... X n

over

Z ,

which have non-zero

by definition)

coefficients

77

when

(5.5d) is reduced mod

p .

The reader may deduce

case of Steinberg's

"Tensor Product T h e o r e m " ( ~ t ,

r = r0 + r l P

(O < ro,rl,... , ! p - l )

+ ...

i ~r,p(Xl,...,Xn) M = 0

=

symmetric

5.6

function"

Z-forms

of

Fr, 0 = Vr, K .

h =r

and

~ ~eA(n,r)

Of course in case

The character

X ~ (~i, p.14])

p = 0

is the "complete

.

V%,Q

D%,Q

are irreducible, mr

~Q : EQ

In fact the map

isomorphism.

: if

then

In this section we work over the rational V%,Q

p.218])

i

E ~ri,p(X ~ ...,X p ) . i>O ' n

and we can take

from this a special

For

~

÷ DX,Q

is certainly

field

Q .

The modules

and isomorphic

to each other

i n d u c e s a map

~:Vx,Q ~ DX, Q

a homomorphism, and i t

(see 5.1).

w h i c h i s an

is non-zero

because

(5.6a)

~(f ) =

Therefore

by Schur's

We would Z-form

Z ocC(T)

L ,

a

the argument

L % = Z. Yfz

Z-form of

essential

lemma

V%,Q

for some ,

and

the Z-forms

(since

lying in

V%,Q = Q'fz

O # y ~ Q .

(y-IL)% = Z . f

if we confine our attention

"normalized"

s(o)(T~:Tzo)

=

V%,Q

at the end of 3.3 shows that

of rank 1 '

Z oEC(T)

IC(T) I(T%:T%)

•

~ is an isomorphism.

like to describe

a free Z-submodule that

s(o) %(e%o ) =

by the condition

to

.

For any such

L % = L ~V~,Q

has dimension

It is clear that .

Therefore

Z-forms

L

i),

y-iL

is so is also

we shall lose nothing of

V%,Q

which are

78

(5.6b)

L % = Z.f

We already know two such normalized Sz(n,r).f %

Z-forms, namely

~r V%, Z = E Z {'~ V I , Q

=

(see (5.4e)), and

(5.6c)

XI, z = g-l(ic(T ) IDI,z) .

XI, Z

is a Z-form of

is a

Z-form of

VI,Q , because

DI,Q •

DI, Z

-

hence also

IC(T) IDI,Z

-

It is normalized, because

XI,ZI = ~-I(Ic(T) ID%I,Z) = %-I(z. IC(T) l.(r :T~)) = Z.f% , by (4.7c) and (5.6a).

Our aim is the following theorem.

(5.6d) ~

Verma IV, p.681]). Then

Proof.

contain

write

SZ

L

for

f% ,

Sz(n,r) ).

spaces

La

for

Z-form of

V%,Q

which

it contains

Sz.f ~ = V%, Z

(we shall

<> = <>

using the contravariant property of

SzL ~ L . ~ ~ I .

be any

On the other hand

= <<SzL,f~>> ~ <> , and the fact that

L

Vk, Z C_ L c_ X%, Z .

satisfies (5.6b).

Since

Let

So

But

f%

<< , >>

is orthogonal to all the weight

<> = <> = <>

= Z<> = Z . We have hereby proved that

L

Y%,Z = {y ~ V%,Q : <> ~C Z}

lies in the set

79

So it will'be

enough

to prove

follows

from the argument

let

be any element

z

DX, Q

~(z)

.~ kj 3

=

the sum being over

just given,

of

Y%,Z

"

taking

Using

That

X%,Z ~ Y % , z

L = X%, Z .

the basis

Conversely

theorem

(4.5a) for

Q .

Because

IC(T)I(T%:Tj) ,

j e I(n,r) z e YI,Z

such that

Our definition the case

is standard;

the

k. J

lie

(5.5b) gives

e Z ,

for all

a particularly

simple

i e I(n,r)

formula

.

for

<<

,

>>

in

char K = O , namely 1 <> =IC(T-~

(5.6f)

For if

T. J

we have

<> = <> i i

by

Y%,Z = X%,Z

we may write

(5.6e)

in

that

u = x{C(T)}

(5.2c),

following

(5.6g)

{C(T)} 2 =

for all

u, v E EmriC(T) j "- -~

as in (5.5b), we have

IC(T) I{C(T)}

.

We use

= <x,y{C(T)} 2>

(5.6f)

to

make

the

calculation:

<>

= I kj = E k. ~(i,j) J J J comes

of the D~sarm~nien T. , l

for all standard therefore

, v = y{C(T)}

and also

The last equality

standard

,

from the calculation

coefficient

~(i,j)

the unimodularity T. . J

the proof

Referring

of (5.6d)

.

preceding

Since

(5.6g)

of the Desarmenien to (5.6e),

is complete.

the definition lies in

matrix

this proves

Z

for all

shows that

that

(5.3d)

k° ~ Z J

z s X%, Z , and

5 The Carter-Lusztig modules Vλ,K

5.1 Deﬁnition of Vλ,K Let λ ∈ Λ+ (n, r) be given, and let K be any inﬁnite ﬁeld. Denote by NK ⊗r the kernel of the SK (n, r)-epimorphism ϕK : EK → Dλ,K deﬁned in 4.4. We have then an exact sequence in MK (n, r) (5.1a)

ϕK

⊗r 0 −→ NK −→ EK −→ Dλ,K −→ 0.

Deﬁnition. Let Vλ,K be the orthogonal complement to NK , relative to the ⊗r (see 2.7, example 1): canonical form , on EK (5.1b)

⊗r Vλ,K = { x ∈ EK : x, NK = 0 }.

⊗r Since , is contravariant and NK is an SK (n, r)-submodule of EK , Vλ,K ⊗r is also a submodule of EK . Since , is non-singular, we may deﬁne a nonsingular, contravariant form ( , ) : Vλ,K × Dλ,K → K by ⊗r (x, ϕK (y)) = x, y , all x ∈ Vλ,K , y ∈ EK . ◦ Hence Vλ,K ∼ , and because (contravariant) dual modules in MK (n, r) = Dλ,K have the same character (by (3.3e)), ΦVλ,K = Sλ (X1 , . . . , Xn ). In particular, if char K = 0, then Vλ,K is irreducible, and is isomorphic to Dλ,K (see 4.7). If K has ﬁnite characteristic, then in general Vλ,K and Dλ,K are not isomorphic.

5.2 Vλ,K is Carter-Lusztig’s “Weyl module” λ

We shall identify Vλ,K with the module V deﬁned by Carter and Lusztig in [6, pp. 211, 222]. First we must describe NK = Ker ϕK more exactly. ⊗r (5.2a) NK is the K-span of the subset R = R1 ∪ R2 ∪ R3 of EK , where

44

5 The Carter-Lusztig modules Vλ,K

(i) R1 consists of all ei such that i ∈ I(n, r) and Ti has equal entries in two diﬀerent places in some column. (ii) R2 consists of all ei − s(σ)eiσ , where i ∈ I(n, r) and σ ∈ C(T ). (iii) R3 consists of all elements ν∈G(J) s(ν) eiν , where i ∈ I(n, r) and J is a non-empty subset of Ch+1 (T ) for some h ∈ {1, 2 . . . , r − 1}. Proof. All the elements of R lie in N = NK by (4.6b), and so N contains the K-span N of R. There is therefore a well-deﬁned K-map ψ ⊗r ⊗r /N onto Dλ,K , given by ψ(x + N ) = ϕK (x), all x ∈ EK . Now from F = EK the function f : I(n, r) → F given by f (i) = ei + N clearly satisﬁes the hypotheses of (4.6a), hence F is K-spanned by the set { ei + N : Ti standard }. But ψ maps this set onto a basis of Dλ,K , by (4.5a). Therefore Ker ψ = 0, which implies N ⊆ N ; hence N = N , and (5.2a) is proved. As a corollary, we have ⊗r which satisfy the following (5.2b) Vλ,K is the set of all elements x ∈ EK conditions:

(i) x, ei = 0 for all i ∈ I(n, r) such that Ti has equal entries in two distinct places in the same column. (ii) xσ = s(σ) x for all σ ∈ C(T ). (iii) ν∈G(J) s(ν) xν −1 = 0 for any h ∈ {1, 2 . . . , r − 1} and any non-empty subset J of Ch+1 (T ). ⊗r such that x, Rs = 0 Proof. (5.2a) shows that Vλ,K consists of all x ∈ EK for s = 1, 2, 3. (5.2b) is an almost immediate consequence of this, together with the fact that , is non-singular, and satisﬁes an “invariance” condition

(5.2c)

xπ, y = x, yπ −1

⊗r , π ∈ G(r). This last condition is veriﬁed trivially from the for all x, y ∈ EK deﬁnition of , (see 2.7, example 1).

It is now easy to see that Vλ,K coincides with Carter-Lusztig’s “Weyl λ

module” V ; conditions (28), (29) of [6, p. 211] are essentially (i), (ii), (iii) of (5.2b). Example 1. If λ = (r, 0, . . . , 0) we write Vr,K for Vλ,K . Conditions (i) and (ii) of (5.2b) are vacuous. Condition (iii) says xν = x, for all transpositions ν = (h, h + 1) (h = 1, . . . , r − 1). So Vλ,K is the space of all symmetric ⊗r . Therefore this module is dual in the contravariant sense tensors x in EK th to the r symmetric power Dr,K of EK (see 4.4, example 1). Vr,K has basis { vα : α ∈ Λ(n, r) }, where vα = ei , sum over all i ∈ α. The non-singular contravariant form ( , ) : Vr,K ×Dr,K → K (see 5.1) is given by (vα , eβ ) = δα,β for all α, β ∈ Λ(n, r). Here eβ = eβ1 1 · · · eβnn , so that { eβ : β ∈ Λ(n, r) } is a basis of Dr,K .

5.3 The Carter-Lusztig basis for Vλ,K

45

Example 2. Assume r ≤ n and that λ = (1, . . . , 1, 0, . . . , 0) with r 1’s. We write V(1r ),K for Vλ,K . Condition (iii) of (5.2b) is vacuous, and conditions (i), (ii) show that V(1r ),K is the space of all antisymmetric tensors ⊗r in EK . V(1r ),K is an irreducible module in MK (n, r), whatever the characteristic of K, since its character er (X1 , . . . , Xn ) has no non-trivial expression as a sum of symmetric functions in Z[X1 , . . . , Xn ]. Therefore V(1r ),K is isomorphic to the rth exterior power D(1r ),K = Λr EK (see 4.4, example 2). There is an isomorphism Λr EK → V(1r ),K which takes (see 3.3) ei1 ∧ . . . ∧ eir −→ (ei1 ⊗ · · · ⊗ eir ) s(π) π . π∈G

5.3 The Carter-Lusztig basis for Vλ,K Carter-Lusztig have given a basis for Vλ,K , and shown that Vλ,K is a cyclic module [6, pp. 216–219]. We shall give here a slightly diﬀerent proof of these results. It is interesting that the Carter-Lusztig basis (see (5.3b)) is not the dual of the basis of Dλ,K given in (4.5a); these bases of Vλ,K are connected by a certain unimodular matrix Ω (see (5.3d)) which has appeared in work of D´esarm´enien [14, p. 74]. Notation. If X is any subset of the symmetric group G(r), we shall write π, {X} = s(π) π. [X] = π∈X

π∈X

These are elements of the group ring KG(r), of course. If K = Q, they even belong to ZG(r). (5.3a) Let l be the element of I(n, r) given in (4.3b). Then fl = el {C(T )} lies in Vλ,K . Proof. Verify that fl , y = 0, for all y ∈ R1 ∪ R2 ∪ R3 (see (5.2a)). For y ∈ R1 ∪ R2 there is no problem. Suppose then that y = ν∈G(J) s(ν) eiν , as in (5.2a)(iii). In the notation just introduced, this reads y = el {G(J)}. Using (5.2c),

el {C(T )}, ei {G(J)} = el , ei {B} , where B = G(J)C(T ) = Y · C(T )—see the proof of (4.6b). As in that proof, we break up B as a union of pairs {π, πκ}, each κ being a transposition in R(T ). Using (5.2c) again, and the fact that el κ = el , we have for each such pair el , ei (s(π)π + s(πκ)πκ) = 0. Therefore fl , y = 0, and this property completes the proof of (5.3a). Remark. The element fl is denoted by Φµ in [6, p. 216].

46

5 The Carter-Lusztig modules Vλ,K

Now let V be the SK (n, r)-submodule of Vλ,K which is generated by fl . As K-space, V is spanned by the elements ξi,j fl (i, j ∈ I(n, r)). However, since ξi,j fl = (ξi,j el ){C(T )}, we see by (2.6a) that ξi,j fl = 0 unless j ∼ l. Hence V is K-spanned by the elements bi = ξi,l fl (i ∈ I(n, r)). In fact we have the following much more precise statement. (5.3b) Theorem (see [6, Theorem 3.5, p. 218]). The set { bi = ξi,l fl : i ∈ I(n, r), Ti standard } is a K-basis of Vλ,K . In particular, V = Vλ,K , i.e. Vλ,K is generated by fl as SK (n, r)- (or KΓK -)module. Proof. Let i, j ∈ I(n, r). From (2.6a) we have (5.3c) ξi,l el = eh , h

where the sum is over all h ∈ I(n, r) such that (i, l) ∼ (h, l), i.e. such that there is some π ∈ G(r) with h = iπ, l = lπ. But l = lπ if and only if π ∈ R(T ) (see (4.3b)). So the sum in (5.3c) is over the R(T )-orbit iR(T ) of i. Now we bring in the form ( , ) : Vλ,K × Dλ,K → K introduced in 5.1, and calculate (bi , (Tl : Tj )) = bi , ej = ξi,l el {C(T )}, ej = ξi,l el , ej {C(T )} , using (5.2c) eh , s(σ)ejσ , using (5.3c). = h∈iR(T )

σ∈C(T )

This last expression, which we denote Ω(i, j), is equal to (5.3d) Ω(i, j) = s(σ), sum over all σ ∈ C(T ) such that jσ and i σ belong to the same R(T )-orbit. Now suppose Ti , Tj are both standard, and that Ω(i, j) = 0. There must exist σ ∈ C(T ) such that jσ and i are in the same R(T )-orbit. If σ = 1, this implies i = j. In any case we have β(i) = β(jσ) (see 4.5), and if σ = 1 then (4.5d) implies β(jσ) > β(j). Let Ω denote the matrix (Ω(i, j)), with i and j running over the set I ∗ = { k ∈ I(n, r) : Tk standard }. If we give I ∗ any total order > such that β(i) > β(j) implies i > j for all i, j ∈ I ∗ , then Ω is a unimodular triangular matrix. For by what we have shown above, Ω(i, j) = 0 implies either i > j or i = j, and clearly Ω(i, i) = 1. Hence Ω is non-singular. But since Ω(i, j) = (bi , (Tl : Tj )), and since ( , ) is non-singular and { (Tl : Tj ) : Tj standard } is a basis of Dλ,K , it follows that { bi : Ti standard } is a basis of Vλ,K . This proves (5.3b). Note. A proof that Ω is unimodular is given by D´esarm´enien in [14, p. 74].

5.4 Some consequences of the basis theorem

47

5.4 Some consequences of the basis theorem For each i ∈ I(n, r), it is clear by (2.3c) that the element bi = ξi,l fl satisα , where α is the weight of i. From (5.3b) we ﬁes ξi,i bi = bi ; thus bi ∈ Vλ,K deduce α (5.4a) Let α ∈ Λ(n, r). Then Vλ,K has K-basis { bi : i ∈ α, Ti standard }. λ = K ·fl (since bl = ξl,l fl = fl ). A well-known argument In particular, Vλ,K (see e.g. [31, p. 2]) shows max

(5.4b) Vλ,K has a unique maximal submodule Vλ,K . The element fl does not max max lie in Vλ,K . The irreducible module Fλ,K = Vλ,K /Vλ,K has character Φλ,p , where p = char K. Proof. By (5.3b) Vλ,K is generated by fl . Any proper submodule M of Vλ,K , λ = 0, and so M lies in since it does not contain fl , has M λ = M ∩ Vλ,K V =

α Vλ,K ,

α=λ

a proper K-subspace of Vλ,K . Therefore the sum of all proper submodules M max of Vλ,K lies in V , hence is proper, and is the unique maximal submodule Vλ,K , and does not contain fl . The third statement in (5.4b) now follows from the deﬁnition of Φλ,p (see 3.5, Remark (i)) and the fact that Vλ,K has character Sλ = X1λ1 · · · Xnλn + · · · . Since Dλ,K is dual to Vλ,K , we have the following corollary to (5.4b). min min (5.4c) Dλ,K has a unique minimal submodule Dλ,K , and Dλ,K ∼ = (Fλ,K )◦ . min Hence Dλ,K , Fλ,K are isomorphic modules.

Proof. The second statement follows from the ﬁrst, and the fact that any module V in MK (n, r) has the same character as its (contravariant) dual V ◦ (see (3.3e)). If V is irreducible this implies V ∼ = V ◦. Remark. Since Fλ,K has λ-weight-space of dimension 1 (by (5.4b)) the same min min is true of Dλ,K . Therefore Dλ,K contains, hence is generated as SK (n, r)module by, the element (Tl : Tl ). For (4.7a) shows that the λ-weight-space of Dλ,K is K · (Tl : Tl ). This proves (5.4d) below. A quite diﬀerent proof comes from a standard argument for semisimple algebraic groups, using the fact (cf. 4.8) that (Tl : Tl ) is stable under the action of upper and lower unipotent triangular subgroups of Γ = GLn (K). See [50, p. 214]. min

(5.4d) The element (Tl : Tl ) of Dλ,K generates the irreducible module Dλ,K . We show next that the family Vλ,K is deﬁned over Z. Recall from 5.1 that Vλ,Q = { x ∈ EQ⊗r : x, NQ = 0 }, where NQ = Ker ϕQ .

48

5 The Carter-Lusztig modules Vλ,K

(5.4e) Lemma. Vλ,Z = EZ⊗r ∩ Vλ,Q is a Z-form of Vλ,Q . It has Z-basis B = { bi,Q : i ∈ I(n, r), Ti standard }. Proof. The sets EZ⊗r , Vλ,Q are both closed to the action of SZ (n, r), hence so is Vλ,Z . It is clear that, for each i ∈ I(n, r), bi,Q = ξi,l el,Q {C(T )} lies in EZ⊗r , hence that B is a subset of Vλ,Z . Since B is a Q-basis of Vλ,Q (by (5.3b)), the proof of (5.4e) will be achieved if we show that every element x ∈ Vλ,Z is in the Z-span of B. We certainly have x = ki bi (sum is over Ti standard) for some ki ∈ Q. Since x ∈ EZ⊗r , we have x, ej ∈ Z for all j ∈ I(n, r). Take any j such that Tj is standard. The calculation in the proof of (5.3b) ki Ω(i, j), the sum being over the standard Ti . But the gives x, ej = “D´esarm´enienmatrix” Ω = (Ω(i, j)), in case K = Q, is integral and unimodular. So from ki Ω(i, j) ∈ Z (all standard Tj ) follows ki ∈ Z (all standard Ti ). Hence (5.4e) is proved. Remark. (5.4e) shows that Vλ,Z = SZ (n, r)fl . It is clear that Vλ,Z = EZ⊗r ∩ Vλ,Q is a pure Z-submodule of EZ⊗r ; hence that the inclusion 0 → Vλ,Z → EZ⊗r is Z-split. If we tensor with any inﬁnite ﬁeld K we get an exact sequence 0 → Vλ,Z ⊗ K → EZ⊗r ⊗ K in the category MK (n, r). We shall regard Vλ,Z ⊗ K as submodule of EZ⊗r ⊗ K. ⊗r } is Z-deﬁned by EZ⊗r and In 2.6, example 1 we showed that the family {EK maps δK : ei,Q ⊗ 1K → ei,K . From (5.4e) and (5.3b) it is immediate that δK induces an isomorphism δK : Vλ,Z ⊗ K → Vλ,K ,

which maps bi,Q ⊗ 1K → bi,K for all i ∈ I(n, r). From all this we deduce: (5.4f ) The family {Vλ,K } is Z-deﬁned by Vλ,Z and the maps δK just de⊗r scribed. The family of inclusions Vλ,K → EK is also deﬁned over Z. So is the family of contravariant forms ( , )K : Vλ,K × Dλ,K → K deﬁned in 5.1.

5.5 Contravariant forms on Vλ,K J.C. Jantzen (see [29], [32], ...) has studied contravariant forms on the Weyl modules for a simply-connected, semisimple algebraic group; in particular, his results apply to the group SLn (K), and extend with little alteration to our case Γ = GLn (K). In this section we shall give an independent description of the contravariant forms on the modules Vλ,K . We saw that Vλ,K is generated as S (= SK (n, r))-module by the element fl = el {C(T )} of E ⊗r , and it follows that Vλ,K is contained in the submodule E ⊗r {C(T )} of E ⊗r . We have the canonical contravariant form , on E ⊗r , and from (5.2c) we deduce that

5.5 Contravariant forms on Vλ,K

(5.5a)

49

x{C(T )}, y = x, y{C(T )} for all x, y ∈ E ⊗r .

This allows us to deﬁne a “contracted” version of , on E ⊗r {C(T )} by the rule (5.5b) If x, y ∈ E ⊗r , deﬁne

x{C(T )}, y{C(T )} = x, y{C(T )} . Any ambiguity arising from the fact that an element of E ⊗r {C(T )} may be expressed as x{C(T )} = x {C(T )} for distinct elements x, x of E ⊗r , is eliminated by (5.5a). It is clear that

, is a symmetric, contravariant form on E ⊗r {C(T )}. If we restrict it to Vλ,K , we get a symmetric, contravariant form on Vλ,K , which is moreover non-zero, since rule (5.5b) gives

fl , fl = el , el {C(T )} = s(σ) el , elσ = 1. σ∈C(T )

We might also mention that the family of forms

, K , constructed in this way for all inﬁnite ﬁelds K, is deﬁned over Z in the sense of 2.7, example 3. Any contravariant form ( , ) on Vλ,K coincides, up to a scalar factor, with

, . For the contravariant property (2.7d), together with the fact by the values (fl , v), v ∈ Vλ,K . that Vλ,K = Sfl , shows that ( , ) is determined vα , with each vα belonging to the If v ∈ Vλ,K is decomposed as a sum v = α (α ∈ Λ(n, r)), then since weight-spaces for distinct weights weight-space Vλ,K are orthogonal with respect to ( , ) (see 3.3, p. 25), we have (fl , v) = (fl , vλ ), i.e. ( , ) is determined by the values (fl , v) for all v in the λ-weightλ . But this weight-space is K · fl (see (5.4a)). So ( , ) is comspace Vλ,K pletely determined by (fl , fl ). Therefore (since

fl , fl = 1) if (fl , fl ) = k, then (v, w) = k

v, w , for all v, w ∈ Vλ,K . In the work of Jantzen which we have mentioned, and also in the earlier work of W.J. Wong [56, 57], the importance of the contravariant form on a Weyl module V is that it provides a method of calculating the maximal max of V . In our case the result reads as follows. submodule V (5.5c) [57, Theorem 3B, p. 362]. The radical of

, , that is, the space M = { v ∈ Vλ,K :

v, Vλ,K = 0 }, coincides with the unique maximal max submodule Vλ,K of Vλ,K . Proof. Notice that M is a submodule of Vλ,K , by the contravariant property. max / M . Therefore M ⊆ Vλ,K . But we saw in the proof Also M = Vλ,K , since fl ∈ max α (α = λ). of (5.4b) that Vλ,K lies in the sum V of all weight-spaces Vλ,K λ Since V is orthogonal to Vλ,K = K · fl , we have (Vλ,K , Vλ,K ) =

Vλ,K , Sfl ⊆

SVλ,K , fl ⊆

V , fl = 0. max

max

max

max

This shows that Vλ,K lies in M , and the proof of (5.5c) is complete.

50

5 The Carter-Lusztig modules Vλ,K

Example. We shall calculate the form

, on Vr,K (notation of 5.2, example 1). Since λ = (r, 0, . . . , 0), the diagram of λ has only one row, and so C(T ) = {1}. Therefore,

, is just the restriction to Vr,K of the canonical form , on E ⊗r . Relative to the basis { vα : α ∈ Λ(n, r) } given in 5.2, example 1, the form is given by vα , vβ = 0 (α = β) and vα , vα = (r, α)·1K , where r! . (r, α) = α1 ! · · · αn ! So the radical M of this form is spanned by those vα for which p = char K divides the integer (r, α). max max Since M = Vr,K , the irreducible module Fr,K = Vr,K /Vr,K (see (5.4b)) has basis { vα + M : α ∈ Λ(n, r), (r, α) ≡ 0 modulo p }. The α-weight-space of Fr,K is K ·(vα +M ), for all α ∈ Λ(n, r). Hence the charX1α1 · · · Xnαn , sum over all α ∈ Λ(n, r) with (r, α) ≡ 0 acter of Fr,K is Φr,p = modulo p. Since the integers (r, α) are the coeﬃcients in the multinomial expansion (5.5d) (X1 + · · · + Xn )r = (r, α) X1α1 · · · Xnαn , α∈Λ(n,r)

we have the result: Φr,p (which is a polynomial over Z, by deﬁnition) is the sum of those monomials X1α1 · · · Xnαn which have non-zero coeﬃcients when (5.5d) is reduced modulo p. The reader may deduce from this a special case of Steinberg’s “Tensor Product Theorem” [50, p. 218]: If 0 ≤ r0 , r1 , . . . ≤ p − 1 such that r = r0 + r1 p + · · · , then

i i Φr,p (X1 , . . . , Xn ) = Φri ,p (X1p , . . . , Xnp ). i≥0

Of course in case p = 0, M = 0 and we can take Fr,0 = Vr,K . The character is the “complete symmetric function” hr = α∈Λ(n,r) X α (see [39, p. 14]).

5.6 Z-forms of Vλ,K In this section we work over the rational ﬁeld Q. We have seen in 5.1 that the modules Vλ,Q and Dλ,Q are irreducible, and isomorphic to each other. In fact the map ϕQ : EQ⊗r → Dλ,Q induces a map ϕ : Vλ,Q → Dλ,Q which is an isomorphism. For ϕ is certainly a homomorphism, and it is non-zero because (5.6a) ϕ(fl ) = s(σ) ϕ(elσ ) = s(σ) (Tl : Tlσ ) = |C(T )| (Tl : Tl ). σ∈C(T )

σ∈C(T )

5.6 Z-forms of Vλ,K

51

Therefore by Schur’s lemma ϕ is an isomorphism. We would like to describe all the Z-forms lying in Vλ,Q . If L is any λ such Z-form, then the argument at the end of 3.3 shows that Lλ = L ∩ Vλ,Q λ is a free Z-submodule of rank 1 (since Vλ,Q = Q · fl has dimension 1), so that Lλ = Z · yfl , for some 0 = y ∈ Q. It is clear that y −1 L is also a Z-form of Vλ,Q , and (y −1 L)λ = Z · fl . Therefore we shall lose nothing essential if we conﬁne our attention to Z-forms L of Vλ,Q which are “normalized” by the condition (5.6b)

Lλ = Z · fl .

We already know two such normalized Z-forms, namely Vλ,Z = E ⊗r ∩ Vλ,Q = SZ (n, r) · fl (see (5.4e)), and (5.6c)

Xλ,Z = ϕ−1 ( |C(T )| Dλ,Z ).

Xλ,Z is a Z-form of Vλ,Q , because Dλ,Z —hence also |C(T )| Dλ,Z —is a Z-form of Dλ,Q . It is normalized, because λ λ = ϕ−1 ( |C(T )| Dλ,Z ) = ϕ−1 (Z · |C(T )| · (Tl : Tl )) = Z · fl , Xλ,Z

by (4.7c) and (5.6a). Our aim is the following theorem. (5.6d) (cf. [53, p. 681]). Let L be any Z-form of Vλ,Q which satisﬁes (5.6b). Then Vλ,Z ⊆ L ⊆ Xλ,Z . Proof. We write SZ = SZ (n, r). Since L contains fl , it contains SZ · fl = Vλ,Z . On the other hand

L, Vλ,Z =

L, SZ · fl =

SZ L, fl ⊆

L, fl , using the contravariant property of

, and the fact that SZ L ⊆ L. But we know that fl is orthogonal to all the weight spaces Lα for α = λ. It follows that

L, fl =

Lλ , fl =

Z·fl , fl = Z·

fl , fl = Z. We have hereby proved that L lies in the set Yλ,Z = { y ∈ Vλ,Q :

y, Vλ,Z ⊆ Z }. So it will be enough to prove that Yλ,Z = Xλ,Z . That Xλ,Z ⊆ Yλ,Z follows from the argument just given, taking L = Xλ,Z . Conversely let z be any element of Yλ,Z . Using the basis theorem (4.5a) for Dλ,Q we may write kj |C(T )| (Tl : Tj ), (5.6e) ϕ(z) = j

the sum being over j ∈ I(n, r) such that Tj is standard; the kj lie in Q. Because z ∈ Yλ,Z we have

bi , z =

z, bi ∈ Z, for all i ∈ I(n, r). Our deﬁnition (5.5b) gives a particularly simple formula for

, in the case char K = 0, namely

52

(5.6f )

5 The Carter-Lusztig modules Vλ,K

u, v =

1

u, v , for all u, v ∈ E ⊗r {C(T )}. |C(T )|

For if u = x{C(T )}, v = y{C(T )} as in (5.5b), then u, v = x, y {C(T )}2 by (5.2c), and also {C(T )}2 = |C(T )| {C(T )}. We use (5.6f) to make the following calculation: (5.6g)

bi , z = kj bi , ϕ−1 (Tl : Tj ) = kj Ω(i, j). j

j

The last equality comes from the calculation preceding the deﬁnition (5.3d) of the D´esarm´enien coeﬃcient Ω(i, j). Since (5.6g) lies in Z for all standard Ti , the unimodularity of the D´esarm´enien matrix shows that kj ∈ Z for all standard Tj . Referring to (5.6e), this proves that z ∈ Xλ,Z , and therefore the proof of (5.6d) is complete.

§6.

Representation

6.1

The functor

theory of the s y ~ e t r i c

f:~(n,r)

÷ mod KG(r)

group

(r i n )

In this chapter we shall apply our results on the representations FK = GLn(K) , to the representation G(r)

.

IS] . in

The method is to use a process Suppose first that

A(n,r)

;

u-weight

r

space

l's). V~

determines

ES, sections

in

a functor

III, IVy)

Then there exists a weight

~

(notice that

f:MK(n,r)

that in case MK(n,r)

K = ~

are completely

which uses another functor,

in

representation of

MK(n,r)

profitable

MK(n,r)

theory of

G(r)

the

The correspondence

Sehur proved

(see

n < r

by this means he showed that

The proof which we have given later proof in

by an argument

~(r,r)

to

~',

p.77])°

(Is, pp.61-63])

~(n,r)

.

This

6.5.

Of course Schur used his functor about

.

,

hence are determined up to

Schur's

this time from

second functor will be described

;

~ , p.35].

handle the case

V s MK(n,r)

n ,

this funetor gives an equivalence

reducible,

(see

(I,I ..... 1,0 ..... O)

is a vector of length

KG(r)-module.

KG(r)

of this fact, see (2.6e), is essentially

make deductions

~

÷ mod KG(r)

and mod

isomorphism by their characters

ThenSchur was able to

of the symmetric group

invented by Schur in his dissertation

can be regarded as a left

Mc(n,r)

K

We shall see that for any module

between the categories modules

r ! n .

we denote this by

and contains

V ÷ V~

theory over

of

f ,

and its "inverse"

(see 6.2), to

his starting point was the known .

But since we have already got some knowledge

by the "combinatorial"

methods of §§4,5, it is also sometimes

to work in the other direction.

81

Let us keep Any m o d u l e

K, n, r

V ~ MK(n,r)

for any w e i g h t

fixed for the moment, can be r e g a r d e d as left

e ~ A(n,r)

regarded as left

and w r i t e

,

the w e i g h t - s p a c e

S(~)-module, w h e r e

S(e)

S = SK(n,r)

.

S-module, and therefore V~ = ~ V

(see 3.2) can be

denotes the algebra

~ S~

.

We

get then a functor

(6.1a)

f :M~(n,r) ÷ rood S(~)

w h i c h takes each 0:V + V'

in

S(~)

V s MK(n,r)

~(n,r)

is a

to

,

V ~ c mod S(~)

to its r e s t r i c t i o n

K-algebra with

$

,

and each m o r p h i s m

@~ :V ~ ÷ V ,~

as identity element.

If we choose some

G element

i e I(n,r)

(6. Ib)

w h i c h belongs

i = (i 1 ... 1 ~---~w- - ~

to

~ ,

2 2 ... 2 ~

for example

...

n n ... n)

w e m a y use the m u l t i p l i c a t i o n rules in 2.3 to show that as

K-space, b y the elements

~iz,i

follows that, for any elements if

~, ~'

' ~ c G .

~, 7'

of

is spanned,

F r o m the e q u a l i t y rule in 3.2

G ,

~iz,i = $i~',i

of

G .

So

S(~)

has

K-basis

if and only

{~i~,i } ,

over a set of r e p r e s e n t a t i v e s of the d o u b l e - c o s e t space

N o w suppose that

(6.1c)

S(~)

b e l o n g to the same double coset w i t h respect to the subgroup

G~ = {~ E G:i~ = i}

The e l e m e n t

,

r < n ,

and that

(6.1b) c o r r e s p o n d i n g to

u = (l,2,...,n)

Since the stabilizer in

G

e

e = ~

e

G~G

~

running

/G

is the w e i g h t d e s c r i b e d above. is w r i t t e n

l(n,r)

of this element is

.

G

= {I} ,

the algebra

82

S(~) has

K-basis

multiplication ~,~' in

{~u~,u : ~ c G} .

rule

G .

An elementary

(2.3b) shows that

We have therefore

application

~u~,u~u~,, u = g u ~ ' , u

an isomorphism

of

for all

of K-algebras

~

(6. Id)

S(~)

which takes isomorphism

~u~,u ÷ ~

KG(r)

for all

to be the functor

Hecke ring follows. set

K .

~(G,H) H(G,H)

H\G/H

By means of this

and mod KG(r)

can be identified.

the Schur functor

~, p.22~,

+ mod KG(r)

For the general case, where

over

.

f = f

with no restriction HK(G,G ~)

S(~)

we define

f:MK(n,r)

Remark.

~ ~ G = G(r)

the categories mod

With this identification

,

on

n, r)

S(~)

~

is any weight in

is isomorphic

We may follow Iwahori for any subgroup

has a free

Z-basis

of all double-cosets

of H

H

~,

A(n,r)

(and

to the Hecke ring

p.218~

and define the

of any finite group

as

{XA} ,

where

in

the product of elements

G;

A

G ,

runs over the in

this basis is given by

(6.1e)

where if Hv

XA XB =

¥

is any fixed element of

in the set

rule, see

~,

C g H\G/H

A-Iy ~ B . §~].

ZA'B'C XC '

C, ZA,B, C

(For an explanation

Alternatively

we may define

is the number of H-cosets of this artificial-looking ~(G,H)

to be the

83

e n d o m o r p h i s m ring of the subset r e g a r d e d as right

ZG-module.

Z G - e n d o m o r p h i s m of

EH]ZG

[H]ZG

HK(G,H)

R e t u r n i n g n o w to our case

6.2

~H]

(7 ~ G),

to

G = G(r)

becomes If

K

K-algebra

, H = G

,

the

is any

~(G,H)@ Z K .

we leave it as an

S(~) ÷ ~ K ( G , G )

is an i s o m o r p h i s m of

General theory of the functor

XA

~A] .)

to be the

prove that the K - l i n e a r map

~i~,i ÷ XG ~G

ZG , this subset b e i n g

In this i n t e r p r e t a t i o n

w h i c h takes

c o m m u t a t i v e ring, we define

exercise to

of

g i v e n by

K-algebras.

f:mod S + mod eSe

It soon becomes clear that m a n y p r o p e r t i e s of Schur's functor b e l o n g to a m u c h m o r e general context. need to be f i n i t e - d i m e n s i o n a l ) We define a functor the subspace

eV

E mod eSe .

If

f(0):eV ÷ eV' morphism.

of

Let

S

and let

be any e ¢ 0

K-algebra

be any idempotent in

f:mod S ÷ m o d eSe as follows. V

@:V ÷ V'

is an

eSe-module,

to be the r e s t r i c t i o n of

mod

@ ;

It is important to observe that

If

f

S ,

clearly

S .

V s m o d S , clearly

so we define

is a m o r p h i s m in

(it does not

f(V) = eV then w e define

f(@)

is an

eSe-

is an exact functor,

in

other w o r d s

If

(6.2a)

0 + V' ÷ V ÷ V'' * 0

0 ÷ eV' ÷ eV + eV'' ÷ 0

This is quite elementary. well known,

(I)

is an exact sequence in is an exact sequence in

The next proposition,

mod mod

S , eSe

then .

though easy and u n d o u b t e d l y

does not seem to appear in the literature

(I)

(a special case is

Our functor is a special case of functor described by M. A u s l a n d e r Communications indebted

in A l g e b r a

to J. A l p e r i n

for

I(1974),177-268; this

reference.

see p . 2 4 3 .

I am

84

g i v e n by Curtis and F o s s u m 6.2. appears, of

T.

p.402].

M u c h of the p r e s e n t section

sometimes w i t h different proofs,

Martins

(6.2b)

~F,

If

in the Ph.D. d i s s e r t a t i o n

~ ~a]) .

V g mod

S

is irreducible,

then

eV

is either zero or

is an irreducible module in mod eSe.

Proof.

Let

and also

W

be any n o n - z e r o

SW = SeW ,

to

V .

Hence

if

eV # 0 ,

w h i c h is a n o n - z e r o

eV = e(SeW) = ( e S e ) W ~ then

N o w suppose S-submodules

e S e - s u b m o d u l e of

Vo

IV

is an i r r e d u c i b l e

V ~ mod S , of

W .

V

and define

such that

the largest

S-submodule of

V

also define

a(V) = V/V(e ) .

eV.

Then

S - s u b m o d u l e of This proves eSe-module.

V(e )

eV o = O

-

W = eW ,

V ,

is equal

W = eV .

Therefore

This proves

(6.2b).

to be the sum of all the in other words,

w h i c h is contained in

is

V(e)

(I - e)V .

We

Then we can make a functor

a:mod S ÷ m o d S ;

notice that if into

V'

(e)

0:V ÷ V' hence

'

@

is a m o r p h i s m in

mod S ,

induces a w e l l - d e f i n e d map

then

@

maps

V

(e)

a(@):a(V) ÷ a(V')

.

The virtue of this functor,

is that it gets rid of the part of each m o d u l e

V

f ,

in

w h i c h is annihilated b y f(V)

(6.2c)

.

and does this w i t h o u t d e s t r o y i n g a n y t h i n g

E x p r e s s e d precisely, we have

Let

V a mod S .

induces an i s o m o r p h i s m

T h e n the n a t u r a l map

f(cq.):f(V) ÷ f(a(V)). ,s

~v:V ÷ a(V) = ~/V(e )

85

Proof.

Clearly

f(~v)

f(V)

= eV,

is o n t o

zero

since

V(e)~

Our next which we

can

the

objective

Se

also

If

~:W ÷ W'

a right

Ker

of

~V

to is

f ( ~ v ) = eV f ~ V ( e )

is an i s o m o r p h i s m .

functors

partially,

= Se

meS e W

in

Let

W

,

S-module

from

as i n v e r s e s

mod

S .

h(W) in m o d

We g e t

proposition

c mod

for

eSe

.

W

e mod

(it is a left

eSe-module,

the n e x t

(6.2d)

And

f(~v )

is to d e f i n e

is a m o r p h i s m

is a m o r p h i s m Moreover

Thus

.

the r e s t r i c t i o n

mod

to

eSe

f .

to

As

mod

first

S , attempt

definition

is a l e f t

and

is j u s t

= e.a(V)

.

at l e a s t

h(W)

Since

which

f(a(V)) (l-e)V

serve,

employ

,

eSe

.

ideal

of

is w e l l - d e f i n e d eSe

,

then

in this w a y

shows

that

h

Then

e.h(W)

S ,

and

h(~)

is a left

= ~Se

a functor

,

* h(W')

eSe ÷ m o d

inverse"

and

S-module.

H ~:h(W)

h:mod

is a " r i g h t

= e m W

of c o u r s e )

to

f

S. .

the m a p

~ w + e m w(w

Proof. Thus

that there

for all 0 = q(e and

gives

e.h(W)

= e(Se

the m a p

check that

E W)

defined

= w

, w

E W

.

This

is p r o v e d .

.

= eSe

takes

eSe-map.

is a w e l l - d e f i n e d

m w)

eSe-isomorphism

m eSe W)

above

it is an

s E Se

(6.2d)

an

W

onto

Then

q:Se if

establishes

w

e.h(W)

m eSe W = e ~ W

To p r o v e map

W

e.h(W)

that

;

the

,

is s u c h injectivity

.

as stated.

it is e l e m e n t a r y

it is i n j e c t i v e ,

~eSe W + W g W

= f(h(W))

such that of

that

first q(s

notice

m w)

e m w = 0 the m a p

to

,

= esw we

get

w c e ~ w

,

,

88

h

The trouble w i t h the functor i r r e d u c i b l e module

W

is that it u s u a l l y takes an h(W)

to a m o d u l e

w h i c h is not irreducible.

H o w e v e r we have

(6.2e)

If

W ~ m o d eSe

is irreducible,

m a x i m a l proper submodule of

Proof. Thus

Write

an

V = h(W)

a(V) # 0 ,

N o w let

V'

h(W)

.

.

T h e n by

,

is contained in

Definition.

by

Let

V(e ) .

h*

for all

By f ,

W c mod eSe

irreducibles

(6.2f)

Proof.

If

V .

If

eV' # 0

then

eV = e.h(W)

e.h(W)

. So

V .

eV'

, being

(recall

Then eV' = 0 ,

i.e.

(6o2e).

denote the functor

ah: m o d eSe ÷ mod S ,

so that

= h (W) /h (W) (e)

.

f(h*(W)) ~ W

for all

to irreducibles.

V c mod

There is an

f(a(V)) ~ f(V) ~ W°

is a proper submodule of

eSe-module

This proves

is the unique

is irreducible.

a contradiction.

(6.2c) and (6.2d) this functor i.e.

a(h(W))

(6.2d)) is equal to

h*(W)

to

V(e )

e S e - s u b m o d u l e of the i r r e d u c i b l e

h(W)(e )

(6.2d) and (6.2c),

be any p r o p e r submodule of

V' ~_. SeV' = S(e m W) = h(W) = V , V'

Hence

w h i c h shows that

e.h(W) = e m W ~ W

then

S

is

h*

,

like

W ~ m o d eSe

h , .

By

is a right inverse (6.2e)

h*

takes

We have finally

irreducible and if

eV # 0 ,

S-map B:h(eV) = Se ~eSe eV ÷ V ,

then

w h i c h takes

h*(eV) ~ V

s m ev

87

to

sev ,

equals

V

for all

s e Se ,

because

V

proper submodule hence

of

v ~ V .

is irreducible. h(eV)

.

But

the only maximal proper

Therefore onto

B

The image of

induces

is

So the kernel of

eVc

mod eSe

submodule

an isomorphism

B

of

of

SeV , B

is a maximal

is irreducible

h(eV)

is

which

by (6.2b),

h(eV)(e)

,

by (6o2e).

h(eV)/h(eV)(e ) = a(h(eV))

= h*(eV)

V .

Taking

(6.2g)

all these facts, we arrive at our main theorem.

Theorem.

modules Then

together

Let

be a full set of irreducible

in

mod S ,

indexed by a set

{eV%:

% E A' }

is a full set of irreducible

Moreover

Remarks

if

V%, K ,

then

for any

V ~ mod S

eV t O

that

Let

it will be useful

eSe-module

= 0 )

Vmax ,

modules

Hom S (Se, V) ~ eV

(see

to notice

A' = {% g A : eV%# O} .

DR, V

p.375]).

eV m a x

if

V

image of

V ~ mod S

proper

of

is Se .

modules

has a unique

is either equal

to

eV

submodule

(i.e°

of the

The proof is easy.

In the same context we shall use the following:

symmetric bilinear

.

(isomorphism

to the Carter-Lusztig

or else it is the unique maximal

eV .

mod eSe

Therefore

is a homorphic

that if any

then

in

.

When we come to apply the Schur functor

e(V/V max)

denotes

A •

h*(eV%)

if and only if

maximal proper submodule

3.

V%~

It is well known

irreducible,

2.

% s A' ,

I.

K-spaces),

Again,

{V : % s A}

form on

the restriction

V

such that

of this form to

the proof is an easy exercise.

(eV,

If

(l-e)V) = 0 ,

eV , then

rad(

(

,

)

is a

and if (

, ~ = e.rad(

, , )

)e

88

6.3

Application

I. Specht modules

and their duals

In this section we shall apply the general special

case of the Schur functor

any infinite We take

field,

S = SK(n,r)

identify

eSe

~u~,u ÷ w , for any

with

and

n , r

,

e = Sw

KG(r)

for all

V e MK(n,r)

Notice

correspondence We shall write partitions

Recall

of

are fixed integers = ~u,u

w c G = G(r)

A = A+(n,r)

the effect of X

,

that

DX, K

of

that

%

X

f

is

and

takes

f(V) = eV = V ~

on the modules

of

r A

Z s(o) ~EC(T) c£'i~'

all

%-weight

space

of

f(Dx,K)

is the

= ~l(n,r)

w-weight space

space

o ~%

D%,KW

.

as the set of all A

K-span of the elements

(T£:Ti)

%AK(n,r)

r ~ n)

is a fixed element of

is the

D%, K ,

are in one-to-one

(because

We saw (p°55) that these bideterminants

w-weight

K

r ~ n .

(6.1d), which

A+(n,r)

and think of

From now on,

(T£:Ti) =

the (left)

such that

(see 6.1 for notation)

Notice

with the partitions

(p.54)

Here

.

that the elements

r .

÷ mod KG(r).

by the isomorphism

Our aim is to calculate VX,K

f:~(n,r)

theory of 6.2 to the

of

i e l(n,r)

all lie in the right

~(n,r) D%,K

.

,

.

But by definition

and therefore

lies in

89

(6.3a)

of

%AK(n,r) ~ = E o AK(n,r) o E%

%AK(n,r)

.

Elementary calculations based on formulae (4.4a,a')

and (6.3a) show that

if and only if

%~(n,r) ~ =

v' s vR(T)

I K.c ~sG %,u~

Since

C~U~ = C~u~V

(see 4.5 ), there is an isomorphism of

K-spaces

(6.3b)

%AK(n,r) ~ ÷ KG[R(T)]

which takes

c

~(T)]

u~ +

,

left ideal of the group algebra the other hand

%AK(n,r)m ,

SK(n,r)-module

%AK(n,r) ,

(T%:Ti)

such that

~ G

i c w

and

KG[R(T~

hence is a left

KG-module.

f(D~,K)

On

KG-module by means of (6.1d). c

%,uv

to give

C~UT~

It follows

KG-isomorphism.

T.~

has

K-basis consisting of all

is standard.

The elements

i = u~ (~ c G) .

to

to the left KG-submodule

i

in

The isomorphism

o~C(T) and so it takes

is a

~-weight space of the left

which by (4.4a) is equal to

f(D%, K) = D%, K

(T%:Tu)

NOW

acts on the element

can be written, uniquely, in the form (6.3b) takes

~ c G .

becomes a left

at once that (6.3b) is a left

By (4.7a), p.59,

KG ,

being the

To be explicit, the element ~C%,u~ = SuT,u o e%,u~ ,

for all

(left ideal)

90

ST,K = KG (C(T) } ~R(T)]

of

KG .

We shall define

ST, K

to be the Sg e c h t m o d u l e

corresponding to the bijective

X-tableau

T .

(over

K)

(This is a little

different from the original definition of Specht;

for an explanation

of the latter, and of the equivalence of the two definitions, see ~, p.9d.)

(6.3c)

We have now the

Theorem.

~{C(T)}~R(T) i

ST, K

has

K-basis consisting of the elements

such that

Tu~

is standard.

is an irreducible Tx ,

X-tableau full

set

DX, K

Dm x,K is

(by (4.7a)), NK(n,r)

If we choose for each

SX, K = STX ,K ,

.

in

given by (4.7a)

X s A

ST, K

a bijective

{Sx, K : X s A}

Then the this

last

present

look of

is

f(Dx,K)

the module

E~ r

and s i n c e

a full

statement

case

at

already quoted.

by ( 4 . 7 b )

{DX, K : X ~ A}

a subspaee

in

set

is a

g SX, K

VX, K .

and t h e r e f o r e

= (E~r) m .

S = SK(n,r) no r e s t r i c t i o n

on

From the E Hr ,

on

formula

we s e e

n , r)

that

for

DX, K

follows

has

at

is non-zero

By d e f i n i t i o n

f(Vx,K)

(2.6a)

If char K = 0

= Vm X,K

which gives

any weight

then

character

of irreducible

(6.3c)

•

f(E ~r)

then

then

KG-modules.

irreducible

Now l e t ' s is

char K = O

The first statement comes by applying the isomorphism (6.3b) to

the basis of

since

and w r i t e

of irreducible

Proof.

each

KG-module.

If

modules

S in

once from (6.2g), for

all

(see is

X ~ A .

(5.1b))

a sub-space

the action

~ ¢ A(n,r)

this

of (and with

of

91

(Emr) e =

~ E mr

=

E

K.e i

i ~

So in particular of

(Emr) ~

for all

has

{e

: ~ g G} .

u~

KG-module

is given by

T , ~ ~ G .

Therefore

there

The structure

Teu~ = Su~,ueu~

is a left

= eu~

'

KG-isomorphism

(Emr) ~ ÷ KG ,

takes

e u~

By (5.3b), standard}

.

(see 5.3.).

+ ~ ,

for all

(5.4a) weJknow

Recall

that,

If we put

i = u~

bu~ = eu~[R(T) ~

{C(T)}

(6.3d)

to the element

v rR(T)]

to the left

KG-submodule

We have

that

V%, K

has K-basis

i c l(n,r)

in formula

H~

U~

b i = $i,%f% = ~i,%e~{C(T)}

is carried by the isomorphism

KG . Therefore

of

: ~ s G , T

(5.3e) we get ~u~,~e% = eu~[R(T) ]

This element {C(T)}

,

{b

~ K V%,

is carried

(left ideal)

K

KG .

~ ~ G .

for any

Hence

of

K-basis

as left

(6.3d)

which

(Emr) ~

= KG[R T)]

the following

theorem,

whose proof

is entirely

analogous

to that of (6.3c).

(6.3e) ~(T~

Theorem. {C(T)}

is an irreducible

ST, K

such that

has Tu~

KG-module.

K-basis

consisting

is standard. If we choose

If

of the elements char K = O

for each

X e A

then

S--T,K

a bijective

92

X-tableau

T%

and write

~

= S %,K

full set of irreducible

The

modules

module

ST, K

is in fact dual

the contravariant

of this form.

~ ~ G ,

when we regard

form

, D%, K~

that these KG-modules form, by means

the calculation

(V,~u,u~

=

described

as

=

property

in 5.3

in 5.1.

,

But this becomes

KG-modules

to exhibit

the

(2.7d) gives

, ~u~_l,u d)

(v

(~v, d) = (v,~-id)

by means of (6.1d),

are dual to each other.

given

are dual to each

and this shows

Naturally we can transfer

(6.3d),

to give an invariant

to do this,

the following

this

form

and also to apply explicit

version of

form in question.

The

~ , ~' e G , vT

The m a t r i x relative

d)

We lave it to the reader

Theorem.

the tableaux

)

to the Specht

are dual to each other under

KG-modules

There is an invariant bilinear

for all

,

of the isomorphism~(6.3b),

ST, K x ST, K ÷ K

the invariant

(

v ~ V%, K , d e D w ,K

V%, Km

a

=~ a " , p-460 ].

V%, K , D ,K

The contravariant

(~u~,u v , d)

(6.3f)

(in the usual sense)

the modules

w (see 3.3) V%, K, D%, K

restriction

is

• %,K

ST, K - this was first proved by G.D. James

other under

for all

: ~ ~ A}

'

KG-modules.

We can give another proof:

Therefore

then T%,K

where

and

form

~

ST, K

, ST, K

( , ):ST,KXST,K

~, = Es(o)

,

are row-equivalent.

(~ ,~,

: ~T , ~'T standard)

ordering

of the standard

÷ K

such that

sum over all

~'dT

to a suitable

are dual to each other.

is unipotent ~T .

s C(T) such that

triangular,

93

Remarks.

I.

Since

identified with the basic tableau 2. of

The m a t r i x the

e,

3.

(~ ,~,)

Desarmenlen

=

matrix

~(u~,u~')

For if we take

with

~

.

in the last theorem is just that part

(~(i,j))

Therefore

to

corresponding

to

ST, Z = ZG{C(T) }[R(T)]

these last are

and have

Z-bases

{~R(T)]{C(T) } : T u~

standard}.

6.4

Irreducible

Application

Throughout r ,

can be

u

1,j

e w ;

in

II.

fact

is replaced by

are integers

, -ST, Z = ZG~R(T~{ C ( T ) }

{~{C(T) }[R(T)]

that

satisfying

K

: Tu~

ST, Q ,

standard}

,

char K = p .

has finite

r ~n

Z .

(6.3b),

Z-forms of the QG-modules

K G(r)-modules,

this section we assume n

K

then we may check that the isomorphisms

D ~X,Z ' VX,Z ~

respectively,

and that

Tug

in this section remain true when

K = Q ,

respectively. ST, Q

appearing

and

T ,

T

.

All the results

(6 • 3d) take

the tableau

u = (1,2, .... r) ,

.

characteristic

p ,

In 5.4 we constructed

a

+ full set

{F%, K : X s A (n,r) = A}

Apply the Schur functor

(6.4a) subset of

Let A

A

Of course

and we have by (6.2g)

be the set of all partitions

consisting

{F ~~,K : ~ E A' }

f ,

of irreducible modules

of those

%

of

such that

is a full set of irreducible

in the next theorem.

~(n,r)

.

the theorem

r ,

and let

F~ # 0 ~,K

A'

be the

Then

KG(r)-modules.

this still leaves open the crucial question:

The answer is contained

in

what is the set

A' ?

94

(6.4b)

Theorem

3.2]).

(Clausen

The set

A' of (6.4a)

= (Xl,X2,...,Xr,O, for which all O

and

p-i

Proof.

~C£,

... )

We must

of

the integers

module

min D%,K

show that

Therefore for all

X~ # 0

X

which is zero unless

X

is

~i,~ o (T~:T~)

on

i.e. lie between

in (6.4c)

iG = w

the elements

i.e. iT

this module

X is column

K-space

by the elements

but with by

the X •

p-regular.

by

(T%:T%)

.

~i,j o (T :T~)

,

By (4.4b)

=

~ .

E hcl

Ei,j(Ch,~)(T~:Th)

,

Therefore

by the elements

The element

is the weight

We denote

SK(n,r)-module

.

j~

FX, K ,

as

Z ~i,~(Ch,~)(T~:Th) heI

i g I .

elements

p-regular"

not with

if and only if

as

K-spanned =

to work,

is generated

it is spanned

~

which are "column

hi - %2 ' X2 - X3 ' "'" ' %r

~i,j o (T~:T~)

for all

EJa', Theorem

of those partitions

(see (5.4c)).

i,j c I = I(n,r)

(6.4c)

r

James

.

By (5.4d),

where

consists

It will be convenient

isomorphic

Lemma 6.4, p.184],

(6.4c)

containing

such that

=

(T%:Th)

i .

i e ~ .

So

If

X~

is

i e ~ ,

for all

are all distinct.

,

~-weight

lies in the

iT = iT' => ~ = ~' , (~ e R(T))

Z h~iR(T)

K-spanned then

~,~' So

space

G

s G .

X~ ,

by those

acts regularly In particular

95

(6.4d)

If

i E ~ ,

then

that

is the group of all elements

Suppose preserve

the set of columns

permutation each

e

maps Since

t

of the basic

0q of

is a p e r m u t a t i o n T

x(s,t)

to

IWql = %

q

- %

has length x(S,eq(t)) q+l

,

the order of

'

(see 4.3) that

(T%:Ti) = (T%:Ti0)

So by breaking

up the sum in (6.4d) IHI .

If

%

that

%

enough to show that

(6.4e)

p-regular,

into

~u,~ o (T%:T%)

T .

which

Such a

of all

of (4.2a) and all

for such

, t ~ Wq

(%1 - %2)~(%2 - %3 )' .... as product i s I

p-singular, i.e.

R(T)

t ->- 1

H-orbits,

of determinants

and all

~ c H .

we see that it is IHI

is divisible

by

X~ = 0 .

To prove the other half, we assume

and show that # 0 .

X~ # 0 .

By (6.4d)

and

For this it is (4.3a),

is equal to

E E ocC (T) TeR(T)

s (o) c%o,uT

There is a unique element entries

for all

is zero,

one half of (6.4b).

is column

~u,~ o (T :T%)

,

of

Wq

is

(T%:T i)

is column

hence every term (6.4d)

This proves

of

e

.

where

s -> 1 , H

(T%:Ti)

el,e2,... ,

In the notation for all

from the expression

p ,

%-tableau

of the set

q .

Now it follows

divisible by

~ ~eR(T)

can be specified by a sequence

q -> 1 , col

that

H

~i,~ o (T~:T%) =

in each column of

T~ ,

~ E C(T) namely

which reverses

the order of the

9G

~:x(s,t)

For

example

if

TZ =

We

shall

If

~

y E G = ~o

M = 4)

.

columns are

]

2

2

2

2

2

3 4

same

elements

2

2

3

3

1

1

1

3

2

2

4

1

1

the R(T)

that

TZ

row.

places

of

is p r i m e

c£~,u

in

to

in

(6.4e)

entry

also

in

,

hence

T%~

that

, M-2, TZ~

,

~T

= ~

satisfying

is

The

in

T~

,

Tzo

,

all e n t r i e s

also

H

q>l

in

TZ~

hence

iK # 0

just

is hence

are

in the

M

in

T~,q

consider

arguments The

M are

the p l a c e s

similar group

to of

order

,

glven

, and

there

all e n t r i e s

z = J . has

zero.

tile e x a m p l e , M

Next we

and by

1

T = T ,

(in

entries

that

)q

M

TZ~ T ,

.

clearly

then

implies

say

all

((%q-%q+l) ~

argument

s(~)w,

,

is n o t

= c%o,uT, This

in turn,

= TZo

(6.4e)

.

in

...

in

c~r, u

TZ~ T = T~o M

=

cz~,u

that

the e n t r i e s

.

T%~

£~y = Zo , uT = UT

M-I

p

of

such

Since

as

conclude T e R(T)

are

and h e n c e But

1 ,

the m a x i m u m

, .

1

coefficient

w =

which

t g Wq

2

by e n t r i e s

given

and

4

, T e

first

s -> 1 ,

for

4

that

t e WM

occupies that

1

Consider

in the

in the

1

In

,

we h a v e

1

such

.

(7,5,2,2)

1

prove

~ s C(T)

some

% =

÷ x(q+l-s,t)

shows

that

so the p r o o f

of

the

coefficient

(6.4b)

is c o m p l e t e .

97

This theorem has some interesting lemma c o n c e r n i n g

the

left

module for the algebra

ideal

S{

of

w

S(~) = ~ S ~

consequences.

,

S ,

First we need a

which

is

also

a right

and hence, by (6.1d), a right

KG-~aodule.

(6.4f)

S~ w E mr

S~

has

K-basis

given by

and a right

{~i,u : i e I(n,r)}

~i,u + ei

(i g l(n,r))

By 6.2, Remark i, w

,

2.7, Example

V

is

Vw # 0

and hence of

(6.4g)

If

E mr .

V ~ M~(n,r) to

(James

module of

E mr

But both

E mr

S = SK(n,r)-map

is irreducible,

a submodule

~a',

of

1

ST,K = KG[R(T)~{C(T)}

of 6.3.

<< , >>

Emr{C(T)}

on the space

(E~r) w {C(T)} ,

E mr

be irreducible.

is a homomorphic

and

V

image of

are self-dual

Vw # 0

(Notice,

F%, K

is column

(by

if and only if

we a s s u m e

is isomorphic

r

< n

to a sub-

p-regular.

In 5.5 was defined a contravariant .

.)

the "dual" Specht module

Restrict

and then transfer it to

(6.3d).

V

then

Theorem 3.2])

if and only if

V e MK(n,r)

We have therefore

Next we have a theorem concerning

isomorphism

K-isomorphism

is a left

Now let

if and only if

I, and (5.4c), proof).

isomorphic

Corollary

The

KG-map.

The proof of (6.4f) is routine.

S~

.

this to the

KG{C(T)}

The result is a symmetric,

w-weight

form space

by means of the

invariant form on

KG{C(T)}

98

which we denote by if

~, ~' s G ,

(

,

)

and which is specified

,

then

I s(~-l~ ') (6.4h)

(~{C(T)}

In 5.5 we considered and showed

unique maximal

if

-I

, ~ C(T)

,

, 7r'{C(T)}) = 0

V%, K ,

by the formula:

if

-i , ~ C(T)

the form obtained by restricting ((5.5c))

submodule

that the radical

~ax,K

now to apply the Schur functor,

of

V~, K .

.

<< , >>

to

of this form is the

It is a routine matter

and use Remarks

2,3 of 6.2 to prove

the following.

(6.4i)

Let

restricting ST, K

be the invariant

of

below),

%

is column

then the radical of ST, K ,

From

form on

the form given by (6.4h).

if and only if

p-regular, _-~nax ST, K

( , )

Then

( , )

p-regular.

( , )

K)

automorphism

of

KG

elementary

given by

is non-zero %

device.

B(~) = s(~)~

on

is column sub-module

.

(6.4i) we may deduce a w e l l - k n o w n

by the following

If

obtained by

is the unique maximal

_-max ~ T , K / S T , K ~ f(F

and

S--T,K

theorem of James Let $ ,

denote

for all

(see (6.4k),

the

~ c G •

K-algebra Let

K S

denote

the field

~k = s(~)k ~(M)

,

for

K ,

regarded

~ E G , k s K .

is also a left ideal of

6(M) ~ MmKK s

as one-dimensional

which takes

KG ,

Then if

M

,

by the action

is any left ideal of

and there is a

m m 1K ÷ 6(m)

KG-module

for all

KG ,

KG-isomorphism m s M .

It is trivial

99

to check that

B

maps

{C(T)} ,

respectively, where

T'

of

% )

T .

r

conjugate to

The bilinear form

[R(T)]

is the

%'-tableau

if

[R(T')] (%'

,

{C(T')}

is the partition

obtained by "transposing" the (6.4h) on

KG{C(T)}

syrmnetric, invariant bilinear form the formula:

to

~, 7' g G

( , )

%-tableau

is translated by on

KG~R(T')]

: {1

if

~-l~'e

0

if

B

to a

specified by

then

,

R(T') ,

(6.4j)

Moreover

B(~T,K) = ST,,K

= ST, K mK K s

(6.4k)

(James ~a, Theorems ii.I, 11.5]).

~-tableau, where

the invariant form on (6.4j).

Then

( , )

~

is

submodule

Remark.

~

ST,,K

is a partition of

~

ST,,K

is p-regular if

p-regular, then the radical of max

ST,,K of

ST,,K

'

the module

D~

(IJa, p.39])

module

in

[Ja', §i]

Let

r .

Let

if and only if

and

~'

( , )

T'

be a ( , )

be

~

is

is column p-regular). is the unique maximal

max

ST,,K/ST,,K = f(F ,,K) ~ KS

Comparison with the notation of James in

Dx

ST',K

obtained by restricting the form given by

is non-zero on

p-regular (by definition, If

which shows incidentally that

.

and so (6.4i) translates as follows:

Theorem

bijective

-

-I , ~ R(T')

is isomorphic to

is isomorphic to

between James's two families of irreducible

~a]~

shows that

f(F~',K) ~K Ks

f(F ,K) .

The

So the connection

KG-modules is

100

(6.4~)

D ~'' ~ D% H K K

,

for all column

p-regular

% .

s

The importance (6.4i),

of James's

or of the equivalent

is that it gives a satisfactory

(isomorphism p-regular

classes

~ ,

D%

of) irreducible is isomorphic

of a dual Specht module f

theorem,

~I,K

gives an independent

has a one-sided interesting

h

using the isomorphism ~ S~ W ~ KG(r)

namely

h,h g{w

of (6.1d).

of the

for each column

to the unique irreducible

quotient

We have also seen that Schur's DIm

f(Vl, K)

the functor

is related

First we re-define

labelling

KG-modules:

connection,

inverse,

that

"

"natural"

theorem

h

.

as functors

~ Emr

of

in 6.2.

used by James

of from

for any

V

V e ~(n,r)

such that (6.2d)

(6.4m) Define w e W .

that

Let the

W

V(~)

that for each

,

takes

{~

,

S-submodules

to

eu ,

U

it follows

h(W) ~ = e u ~ W .

be any left ideal of

Ker y = h(W)(~) .

(6.4f)

[Ja'].

W E mod KG we define

is the sum of all

Since

S-map 7:h(W) ÷ EmrW

Then

h*(W) & EmrW

U~ = O .

,

in

and the isomorphism

h(~'$) = E mr mKG W , h (W) = h(W)/h(W)(~)

where

It is

from mod KG(r) ÷ MK(n,r)

(6.4f),

This means

The Schur functor

defined

to a construction

functor

by .

KG ,

regarded

as left

KG-module.

y(x ~ w) = xw ,

for all

x s E ~r ,

Therefore

y

induces

an

S-isomorphism

101

Proof.

Suppose that

written

v = eu m w

But we have e ~ = e u

V ~h(W)(~)

If M .

for some

0 = y(v) = e w u

(~ ~ G)

u~

V = Ker y . w

.

form a

s W

since

This implies

K - b a s i s of

e KG u

of

V~

can be

V ~ ~~ h ( W ) m = e u m W w = 0 ,

.

.

since the elements

Hence

i.e.

V = 0 ,

is not zero, it contains some irreducible submodule

Since the ~ -weight space of M .

M~ ~ 0

,

v

.

y(h(W)(~))

true of

Any element

by

But

M ,

(6.4g)

.

h(W)(w )

is zero,

as i r r e d u c i b l e submodule of

This c o n t r a d i c t i o n implies

the same m u s t be E ~r

,

satisfies

h(W)(~)C

V ,

and the

rest of the proof of (6.4m) is immediate.

Since left

KG

is a Frobenius a l g e b r a

KG-module is i s o m o r p h i c to some left ideal

(61.6)]).

If we combine this w i t h

c o n s t r u c t i n g some irreducible James

~a']).

F ,K

for

Example.

%

S-module where

that

v~

EmrW p

of

e v e r y irreducible KG (~R,

column

W = K~G]

is an ideal of G .

If

is a submodule of does not divide

terms

r = q(p-l) + s .

KG,

and as left

By (6.4m),

i E ~ g A(n,r)

is the basis element of

q

isomorphic to

p-regular.

E ~ r w = Emr[G].

w h e r e there are

p.417,

(this m e t h o d is due to

Of course we can get in this way only modules

V

Vr, K

r,K '

,

h (W) then

and has weights So

It is easy to see that p-I

,

KG-module

W

affords

is isomorphic to the ei~G ] =

IG Iv

,

described in 5.2, Example I.

IG [ = ~l!...~n ~

is the highest such weight.

given by

W

,

(6.4m) and (6.2g), we have a way of

SK(n,r)-modules

the trivial r e p r e s e n t a t i o n of

So

([CR, p.4203)

~ ,

for all

h (W) ~ F%, K ,

~

such

where

%

% = (p-l,...,p-l,s,O,...,O),

and the n o n - n e g a t i v e integers

q,s

are

102

6.5

Application

III.

The functor

MK(N,r)

In this section we fix our infinite integer MK(n,r)

r > 0 . ,

as

such that

n

We consider varies.

N > n .

functor

of 6.2.

the irreducible in a very is based

and

modules

satisfactory

in

N > n ,

I(N,r)

I(n,r)

is the subset

With this convention, SK(N,r ) .

For

(6.5a)

SK(N,r )

for which

two elements

of all

are positive

integers

,

to those in

to characters

(see

,

whose

(6.5b)).

as a subset of

~,

It p.61].

I(N,r)

with components

components

can be regarded

from

and behaves

in his dissertation

I(n,r)

i

"mod S + mod eSe"

MK(n,r)

i = (il,...,ir)

of those

, i

all lie in

as a subalgebra

of

has basis

SK(n,r)

Sk,%

the categories

a very easy way of passing

given by Schur

i,j e I(n,r)

~i,j'

gives

EK(N,r)

j , 1,j s I(N,r)

and we can identify $i,j

d

SK(n,r )

and also the

case of our general

we may regard

consists

n

+ MK(n,r)

way with regard

on a construction

Since namely

The functor

N ,

> n)

a functor

d = %,n:MK(N,r)

as a special

K ,

between

that

We shall produce

which can be viewed

field

connections

Suppose

÷MK(n,r)(N

,

with .

of type

the

K-subspace

For the rule (6.5a),

spanned by those

(2.3b)

for multiplying

has the consequence

that

E N n .

103

if

i,j,k,Z

product p,q

e I(n,r)

~i,j~k,£

do belong

to

it would be if

,

then the coefficient

is zero unless both I(n,r)

,

$i,jSk, ~

as by

follows. a

=

If

then this coefficient

map

~ = (al,...,an)

(al,...,an,O,...,O)

~p,q

p,q e I(n,r)

were computed

We define an i n j e c t i v e

of any

~ ÷ a

,

while

if

is the same as

in

SK(n,r)

.

of

A(n,r)

into

we d e f i n e

A(N,r)

e A(n,r)

,

Then

image of

A(n,r)

: Bn+ 1 = ... = ~

= O} .

the

in the

e A(N,r)

a

under this

map is the set

A(n,r)

Notice

that if

i

= {B e A(N,r)

belong

So another description G(r)-orbits

of

of

I(N,r)

to a weight A(n,r)

(6.5b)

e = E~B ,

This is an idempotent e ~i,j = ~i,j ~i,j e = ~i,j

or zero,

I(n,r)

element

sum over all

of

SK(N,r ) , according

or zero, according

,

then

i e I(n,r)

.

is that it is the set of those

which lie in

Now define the following

B E A(n,r)

as as

of

.

SK(N,r)

B e A(n,r)*

and it is clear i c I(n,r) j ~ I(n,r)

once that

eSK(N,r)e

:

= SK(n,r ) •

(using

(2.3c))

that

or not, and or not.

It follows

at

104

From 6.2

(taking

S = SK(N,r))

which takes each some agreeable

(6.5c) which

If

V ~ MK(N,r)

properties,

V c MK(N,r)

lie in

A(n,r)

to

~V

to

then

Z

in

B e A(N,r)

E A(n,r) statement

eV = ZV B ,

N

in

or not;

This functor has

.

n

variables

~eV

XI, .... X N)

eV $ = e ~$V = V $

the first statement

~ ~ A(N,r)

(which is a

X I , . . . , X n)

is related

by the formula

= ~v(Xl ..... Xn,O ..... O)

then

comes from this,

summed over those

the character

variables

~ev(Xl,...,Xn)

If

Y~(n,r)

which we now describe.

over

(a polynomial

Proof.

eVe

Consequently

symmetric polynomial

a:MK(N,r)÷~(n,r)

we have a functor

.

or zero,

according

of (6.5c) follows.

together with the definition

as

The second

of a character

(see 3.4).

Next we look to see how our functor behavas and

VX,K

(6.5d) the

FX, K .

Lemma. X-tableau

Proof.

X

n+l

places

be distinct,

Let T. i

X s A+ (N,r)\

x,K '

A(n,r)*

and

i ~ I(n,r)

.

Then

is not standard.

we must have

satisfies Xn+ 1 # 0 .

in its first column. since

D

For this we need a lemma.

= (XI,X2,...,X N)

X ~ A(n,r)

on the modules

i ~ I(n,r)

.

X I ~ X 2 ~ ... ~ XN , So the

X-tableau

But the entries Therefore

T. i

Ti

and since has at least

in this column cannot all is not standard

(see (4.5a)).

105

(6.5e)

Theorem.

D%, K ~ VI, K

or

other words

Proof. if

FI, K

Then

eXl = 0

Suppose

X

= Dl, K

or

Vl, K ,

such that the

i E B

implies

Conversely, eXl # 0

by

eX l # 0

,

T. i

~

denote

,

and that

of

is standard

Since

n

parts.

~ s i(n,r)

Then

to the number of

((4.5a),

(6.5d) that

FI, K

In

non-zero

is equal

X%

any one of

~ s i(n,r)

has more than

we deduce from

eX l = O .

X%

if and only if

~ ~ i(n,r)

l-tableau

then from (6.5c) that

and let

the dimension

i s I(n,r)

eF%, K = 0

,

if and only if

first that

i ~ B

we have

+ % s A (N,r)

Let

(5.3b)).

Since

Xl = O ,

and

is a factor module

of

Vl, K ,

also.

suppose

(6.5c),

that

~ c i(n,r) *

for any of the modules

We know that XI

X ~l # 0 ,

in question.

hence

This completes

the proof of (6.5e)°

If

~ e A+(N,r)

I ~ A(n,r)

,

is that

then a necessary ~ = p

for some

and sufficient ~ s A+(n,r)

.

condition

that

We know from (5.4b)

+ that

{FI,K:I

s A (N,r)

is a full set of irreducible

modules

in

So (6.2g) and the theorem just proved give the first statement

MK(N,r)

in (6.5f)

below.

(6.5f) MK(n,r) character

p

{eF ,,K : p s A+(n,r) .

In fact formula,

(including

is a full set of irreducible modules

eF , K ~ F ,K valid

p = O)

for

(isomorphism

all

+ ~ e A (n,r)

,X n)

=

,

in

MK(n,r))

and for

.

every

...

~p,,p(X I,

...,Xn,O

,...,0)

Hence the characteristic

:

n,p(X I,

in

.

.

106

Proof.

The second and third statements

eF , , K

and

F

K are both i r r e d u c i b l e modules in

character with leading

Remarks

i.

following

~I ~n X I ...X

term

A direct proof lines.

Let

that

E(n)

The inclusion

E(n) mr C

of a

E(N) mr

and to induce an isomorphism

2. onto

N > n > r ,

+ A (N,r)

have at most e A+(n,r)

(6.5g)

categories.

hd ,

E(N)

el,...,e n ,

with basis V ~,K

el,...,e N . to

eV~*,K

'

Similarly we may show

eF ,,K

.

~ ~ ~*

.

For any

+ I g A (N,r)

r

non-zero

parts.

Then

(6.5f)

N > n > r ,

are naturally

MK(N,r)

equivalent

the functor described to

r ,

for some uniquely

d = dN, n

= MK(r,r)

a functor

take for

W e MK(n,r)

of

dN,n:b~(N,r) classes

can defined

~ M K(n,r)

of irreducible

In fact we have the stronger result:

[Co, p.7]).

h

% = ~*

+ A (n,r)

from

being a partition

the sets of isomorphism

(6.5g), we must produce

dh

,

shows that the functor

the functor

In particular

gives a b i j e c t i o n

Hence

for example

each

K-space with basis

can be shown to take

in these two categories.

If

To prove

(i), 3.5).

can be made along the

F ,K

K-space

F ,K

the map

induces a bijection between modules

a

having

eD ~*,K

D, K

If

NK(n,r)

(see Remark

eF ,,K

denote

which we regard as subspace

that

follow from the fact that

defines

for all

an equivalence

N ~ r .

h:MK(n,r ) ÷ MK(N,r)

to the appropriate

of

identity

We leave it to the reader to verify

such that functors

(see

that we may

in 6.2, which in the present

case takes

107

h(W) = SK(N,r)e

We might mention then the

that

(6.5g)

K-algebras

Do,

p.34]) .

6.6

Application

has another

SK(N,r)

IV.

Some

mSK(n,r )

and

on decomposition MQ(n,r)

context,

numbers.

÷ MK(n,r)

and prove

a general

If

V ~ mod S

and

of

V%

to

V% ,

(6.6a)

in

V .

in mod

% e A ,

in any composition

series

V = VO b V I ' ) . . .

Here

Lemma. n%(eV)

If

result

theory

of

of T. Martins

EMa]

reduction

S

n%(V)

{V% : % g A}

he a full set

is an algebra

over any field.

the composition

is the number of

multiplicity

of factors

isomorphic

V

-)Vz = 0 .

be an idempotent

to be the set of those

Let

denote by n%(V)

e

eSe"

in 2.5.

where

That means,

Now let

(6.6b)

S ,

(see

numbers.

Then we apply this to the modular

We start with a piece of notation. modules

N > n > r ,

are Mot ira equivalent

our "mod S--)mod

which was described

of irreducible

If

on decomposition

In this section we first extend 6.2 to a "modula#"

formulation:

SK(n,r)

theorems

W .

% g A

of

S ,

such that

I e A' ,

is the composition

then

and define

A' ,

as in (6.2g),

eV% # O .

n%(eV)

multiplicity

= n%(V) of

,

eV%

for any

V ~ mod S .

in the eSe-module

eV .

108

Proof.

From

By (6.2a),

(6.6a) we get a series of eSe-modules

e(Vj_I/V j) m eVj_i/eV j

Removing

those terms

Vj_I/V j

is isomorphic

composition

series

Now let properties IC)

C

K

,

element of

SC

~:R ÷ K .

SK = SR ~ K

i.e.

,

for

f.k = ~(r)k

theory of algebras

(ii)

reduction of

R

Let

SC

SK

mR ,

(see

~,

VR

K

and

with the

(iii)

SR

as

is an

If

{U~, K : d s A}

VK

via

w ,

representation

the categories (cf. 2.5),

as

or "admissible

is the R-span of some C-basis

SK-mOdule;

"R-order"

{u I m iK,..,u b R IK}

reduction"

[~o, 48.1(iv),

the identity

R-module

connects

can be found an "R-form"

as left

contains

there is a

R. Brauer's m o d u l a r

of "modular

or

R

contains

thus

is regarded

~ANT, p.lll])

§6, p.256],

{Vx, C : X s A} ,

C

C-algebra with finite basis

closed:

r c R , k ~ K).

can be regarded VC .

and

for which

(in particular, R ,

be a

1 .

at once.

K-algebra with K-basis

by the process

(i)

of

j = 1 .....

we are left with a

S R = Ru I * ... e Ru b

is a

V C ~ mod S C

,

be a subring of

ideal domain

(IB]; see also

V R , i.e.

SRV R ~ V R

VK = VR m K

(6.6c)

mod

In each

R-lattice"

i.e.

~ s A\A'

and is m u l t i p l i c a t i v e l y

means

follows.

and

such that the set

Then

and

eV.j_l = eV.j ,

The lemma follows

be fields,

m

SC

for

for some

eV .

(here and below,

mod

Vw

eSe-modules),

is the field of fractions

{Ul,...,Ub}

SC .

for which

is a principal

ring-homomorphism

in

to

for

C ,

(i) R

, (ii)

eVj_ 1

(as

eV = eV 0 ~ e V l ~ ... D~eVz = O.

of

p.299]).

VC ,

and

Then

is called a modular

109

are full sets of irreducible modules we define for each

% ~ A ,

Let

e = eR

in

n~(V%,K)

be an idempotent

m o d eSce

Similarly,

, namely

e K = e R m 1K

{eKU~, K : ~ e A'}

eSRe

R-module,

,

eSRe

reduction

from

decomposition

the

with

numbers by .

Proof.

be an

eSce-module

eKSKe K . to

in a modular

VR

e ~ SC ,

A' = {% e A : eV~, C

K-algebra

in

C-algebra SR .

mod eKSKe K ; .

#0}

S K = SR m K ,

mod eKSKe K ,

namely

eSce

-

this is because,

as

For the same reason, we can we have a process

let us denote

Of course

of modular

the corresponding

these are defined only for

these decomposition

numbers,

and

is very simple and satisfactory.

[Ma]).

R-form of

V R = eV R • (l-e)V R eV C ,

Since

where

in the

Therefore

d%~(eSe)

(T. Martins .

,

modules

The connection between

d%~(S) = d%~(eSe)

Let

in the

defined previously,

Theorem

sun,hand of

U6, K

dx6 = d%6(S)

A' = {6 ~ A : eKU~, K # 0}

R-order

mod eSce

~ e A'

dx6 (S)

(6.6d)

number

SR .

{eV~, C : ~ c A'}

is a direct summand of

eSRe ~ K

,

respectively,

By (6.2g) we get a full set of irreducible

is an idempotent

where

is an

identify

X e A'

of

in the ring

and so we get a full set of irreducible

Now

mod S K

VX, C

we can apply the theory of 6.2. modules

SC ,

6 g A the dec£mposition

to be the composition m u l t i p l i c i t y reduction of

in mod

Let

% e A' ,

V C = V%, C This implies

and also that the

6 e A'

Then that

eKSKeK-mOdule

eV R eV R

.

Then

is a direct is an

eV R m K

R - f o r m of the can be

110

identified

with

multiplicity of

U~, K

of

in

eKV K ,

where

eKU6, K

in

VK .

R = Z ,

and define

K

~:Z ÷ K

any infinite ~(n) = n.l K •

given in 2.3.

the categories

and

MK(n,r)

Identify

respectively,

Corresponding

C = Q

for all SK

to the sets

with mod SQ

Denote

the decomposition

numbers

which

appear

(6.6c)

Fix

SK(n,r)

by the isomorphism

and

(6.6e) map from (i)

Theorem. A(n,r)

mod

n, r

p ,

SK

with

and let

MQ(n,r)

in the general

case we take

+

these

,

{F%, K : ~. g A (n,r)}

.

sets are indexed by the same set

numbers

by

d%u = dl

(GLn)

.

These

A+(n,r).)

are the same

in the formulae

~%,o(XI,...,Xn)

of 3.5, Remark

characteristic

n s Z .

+

in this case,

multiplicity

as in 2.4.

{V>~,Q : ~ e A (n,r)}

(It happens

the composition

(field of rational

field of finite

S--SQ(n,r), S Z = Sz(n,r) Identify

(6.6b)

(6.6d).

of (6.6d) we take

by

By

is the same as the composition

eKV K

This proves

For our applications numbers),

VK = VR m K .

=

E+ s A (n,r)

d%~,p(Xl,''',X n)

(i).

Suppose into

that

A(N,r)

da6(GL n) = d ,B,(GLN)

,

N > n , given

for any

in

and let 6.5.

~ ÷

Then

~,B ~ A+(n,r)

•

be the injective

111

(ii)

d~,~(GL N) = O ,

Proof.

(i)

and let

(6.5e),

X e A+(N,r)k~(n,r) *

is a direct application

e = eZ

e e SZ ,

for any

and

be the element of e K = e m 1K

the sets

A',A'

A+(N,r) /~ A(n,r) *

of (6.6d). SQ

Take

defined

is the element of

and

p e A(n,r)*

SQ = SQ(N,r)

as in (6.5b). SK

defined by

, etc.,

Clearly (6.5b).

By

which appear in (6.6d) are both equal to

So we take

~ = ~ , , p = B*

in (6.6d),

and then use

(6.5f). (ii)

Since

% ~ A(n,r)

,

composition m u l t i p l i c i t y eF ,K

in

eKV%, K ,

Part

decomposition

(6.6f)

Here

Fp, K

A+(N,r) . identical

V%, K

numbers

In other words

dx~(GLN)

= O ,

for all

n

r and

are contained

K ,

the

in the m a t r i x

E A(r)

which can be identified with the set of all

~ = (~l,...,%r) and the map

So (6.6e)(i)

the

is complete.

d% (GLn)

,

By (6.6b)

is the same as that of

(i) of this theorem shows that, with fixed

+ A(r) = A (r,r)

N > r ,

in

eKV%, K = 0 .

p c A(n,r)

(dx~(GLr))%,D

partitions Then

of

because

and the proof of (6.6e)

Remark.

then by (6.5e),

of ~ ÷ ~

r .

Assume induces

first

n = r

a bijection

of

in (6.6e). A(r)

shows that the d e c o m p o s i t i o n m a t r i x for

(up to this bijection)

with

(6.6f).

Next take

N = r .

onto GL N Then

is

112 *

n ~ r

and the map

those

% c A(r)

matrix

the submatrix

of (6.6f)

to partitions

Theorem

for

than

bijectively n

non-zero

n

in this case we merely

Here

partitions

r .

in

of

~%,K

"

E SQ ,

(see (6.3e),

(6.6g)

A(P)(r)

which are column

Let

eK = ~ (6.4%)

showing

dx6(G(r))

~ SK .

,

~a',

the matrix

given in James's

article

[Ja'~.

3.~).

S = SQ(n,r)

If and

those columns

and

,

p-regular multiplicity (r # n) ,

The result

then

~ ~ A(P)(r)

•

and char

of etc.,

D~ ~ eKF6, K

r ! n ,

2 < r < 6 ,

To see this,

KG(r)-modules

the composition

each).

shows

G(r) is

p-singular.

6 c A(P)(r)}

S%,Q e eV%,Q

% c A(r)

for

which

which

group

suppress

the set of all column

preceding

Theorem

(6.6f)

:

(6.6d) with

We have

for all

QG(r)-,

{D 6

denote

and the remarks

(James

= d%6(G(r))

,

denotes

Now we may apply

Theorem

d%6(GLn)

~ s A(r)}

with

all rows and columns

for the symmetric

that we have full sets of irreducible

respectively.

So the

parts.

numbers

to partitions

:

parts.

(up to this bijection)

by repressing than

onto the set of

a simple proof of a theorem of James,

of (6.6f)

{~.,Q

Tables

obtained

having more

of (6.6f), which refer

e = ~

i (n,r)

is identical

n

of decomposition

also a submatrix

D6

GL

(6.6d) gives

that the matrix

recall

takes

which have not more

decomposition

refer

+

~ * ~

is as follows.

K = 2,3

are

6 Representation theory of the symmetric group

6.1 The functor f : MK (n, r) → mod KG(r) (r ≤ n) In this chapter we shall apply our results on the representations of the general linear group ΓK = GLn (K), to the representation theory over K of the symmetric group G(r). The method is to use a process invented by Schur in his dissertation [47]. Suppose ﬁrst that r ≤ n. Then there exists a weight ω = (1, 1, . . . , 1, 0, . . . , 0) in Λ(n, r) containing r 1’s. We shall see that for any module V ∈ MK (n, r), the ω-weight-space V ω can be regarded as a left KG(r)-module. The correspondence V → V ω determines a functor f : MK (n, r) → mod KG(r). Schur proved (see [47, sections III, IV]) that in case K = C this functor gives an equivalence between the categories MK (n, r) and mod KG(r); by this means he showed that modules in MC (n, r) are completely reducible, hence are determined up to isomorphism by their characters (see [47, p. 35]). The proof which we have given of this fact, see (2.6e), is essentially Schur’s later proof in [48, p. 77]. Then Schur was able to handle the case n < r by an argument [47, pp. 61–63] which uses another functor, this time from MK (r, r) to MK (n, r). This second functor will be described in 6.5. Of course Schur used his functor f , and its “inverse” (see 6.2), to make deductions about MK (n, r)—his starting point was the known representation theory of G(r). But since we have already got some knowledge of MK (n, r) by the “combinatorial” methods of §4, §5, it is also sometimes proﬁtable to work in the other direction. Let us keep K, n, r ﬁxed for the moment, and write S = SK (n, r). Any module V ∈ MK (n, r) can be regarded as left S-module, and therefore for any weight α ∈ Λ(n, r), the weight-space V α = ξα V (see 3.2) can be regarded as left S(α)-module, where S(α) denotes the algebra ξα Sξα . We get then a functor (6.1a)

fα : MK (n, r) → mod S(α),

54

6 Representation theory of the symmetric group

which takes each module V ∈ MK (n, r) to V α ∈ mod S(α), and each morphism θ : V → V in MK (n, r) to its restriction θα : V α → (V )α . S(α) is a K-algebra with ξα as identity element. If we choose some element i ∈ I(n, r) which belongs to α, for example (6.1b)

i = (1, 1, . . . , 1, 2, 2, . . . , 2, . . . , n, n, . . . , n), α1

α2

αn

we may use the multiplication rules in 2.3 to show that S(α) is spanned, as K-space, by the elements ξiπ,i , π ∈ G. From the equality rule in 3.2 follows that, for any elements π, π ∈ G, ξiπ,i = ξiπ ,i if and only if π, π belong to the same double coset with respect to the subgroup Gα = { π ∈ G : iπ = i } of G. So S(α) has K-basis {ξiπ,i }, π running over a set of representatives of the double-coset space Gα \G/Gα . Now suppose that r ≤ n, and that ω is the weight described above. The element (6.1b) corresponding to α = ω is written (6.1c)

u = (1, 2, . . . , r) ∈ I(n, r).

Since the stabilizer in G of this element is Gω = {1}, the algebra S(ω) has Kbasis { ξuπ,u : π ∈ G }. An elementary application of multiplication rule (2.3b) shows that ξuπ,u ξuπ ,u = ξuππ ,u , for all π, π in G. We have therefore an isomorphism of K-algebras (6.1d)

S(ω) ∼ = KG(r),

which takes ξuπ,u → π for all π ∈ G = G(r). By means of this isomorphism the categories mod S(ω) and mod KG(r) can be identiﬁed. With this identiﬁcation we deﬁne the Schur functor [47, p. 22], f : MK (n, r) −→ mod KG(r) to be the functor f = fω . Remark. For the general case, where α is any weight in Λ(n, r) (and with no restriction on n, r) S(α) is isomorphic to the Hecke ring HK (G, Gα ) over K. We may follow Iwahori [25, p. 218] and deﬁne the Hecke ring H(G, H) for any subgroup H of any ﬁnite group G, as follows. H(G, H) has free Z-basis {χA }, where A runs over the set H\G/H of all double-cosets of H in G; the product of elements in this basis is given by zA, B, C χC , (6.1e) χA χB = C∈H\G/H

where if γ is any ﬁxed element of C, zA, B, C is the number of H-cosets Hπ in the set A−1 γ ∩ B. (For an explanation of this artiﬁcial-looking rule, see [25, §1].) Alternatively we may deﬁne H(G, H) to be the endomorphism ring of the subset [H]ZG of ZG, this subset being regarded as right ZGmodule. In this interpretation χA becomes the ZG-endomorphism of [H]ZG which takes [H] to [A].

6.2 General theory of the functor f : mod S → mod eSe

55

Returning now to our case G = G(r), H = Gα , we leave it as an exercise to prove that the K-linear map S(α) → HK (G, Gα ) given by ξiπ,i → χGα πGα for all π ∈ G, is an isomorphism of K-algebras.

6.2 General theory of the functor f : mod S → mod eSe It soon becomes clear that many properties of Schur’s functor belong to a much more general context. Let S be any K-algebra (it does not need to be ﬁnite-dimensional) and let e = 0 be any idempotent in S. We deﬁne a functor f : mod S → mod eSe as follows. If V ∈ mod S, clearly the subspace eV of V is an eSe-module, so we deﬁne f (V ) = eV ∈ mod eSe. If θ : V → V is a morphism in mod S, then we deﬁne f (θ) : eV → eV to be the restriction of θ; clearly f (θ) is an eSe-morphism. It is important to observe that f is an exact functor, in other words (6.2a) Suppose 0 → V → V → V → 0 is an exact sequence in mod S, then 0 → eV → eV → eV → 0 is an exact sequence in mod eSe. This is quite elementary. The next proposition, though easy and undoubtedly well known, does not seem to appear in the literature1 (a special case is given by Curtis and Fossum [12, p. 402]. Much of the present section 6.2 appears, sometimes with diﬀerent proofs, in the Ph.D. dissertation of T. Martins [41]). (6.2b) If V ∈ mod S is irreducible, then eV is either zero or is an irreducible module in mod eSe. Proof. Let W be any non-zero eSe-submodule of eV . Then W = eW , and also SW = SeW , which is a non-zero S-submodule of V , is equal to V . Hence eV = e(SeW ) = (eSe)W ⊆ W . This proves W = eV . Therefore if eV = 0, then eV is an irreducible eSe-module. This proves (6.2b). Now suppose V ∈ mod S, and deﬁne V(e) to be the sum of all the S-submodules V0 of V such that eV0 = 0—in other words, V(e) is the largest S-submodule of V which is contained in (1 − e)V . We also deﬁne a(V ) = V /V(e) . Then we can make a functor a : mod S → mod S; notice that if θ : V → V is a morphism in mod S, then θ maps V(e) into V(e) , hence θ induces a well-deﬁned map a(θ) : a(V ) → a(V ). The virtue of this functor, is that it gets rid of the part of each module V which is annihilated by f , and does this without destroying anything in f (V ). Expressed precisely, we have 1 Our functor is a special case of a functor described by M. Auslander in [2]; see p. 243. I am indebted to J. Alperin for this reference.

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6 Representation theory of the symmetric group

(6.2c) Let V ∈ mod S. Then the natural map αV : V → a(V ) = V /V(e) induces an isomorphism f (αV ) : f (V ) → f (a(V )). Proof. Clearly f (αV ), which is just the restriction of αV to f (V ) = eV , is onto f (a(V )) = e · a(V ). And Ker f (αV ) = eV ∩ V(e) = 0 since V(e) ⊆ (1 − e)V . Thus f (αV ) is an isomorphism. Our next objective is to deﬁne functors from mod eSe to mod S, which can serve, at least partially, as inverses to f . As ﬁrst attempt we employ the deﬁnition for W ∈ mod eSe. h(W ) = Se ⊗eSe W, Since Se is a left S-module (it is a left ideal of S, of course) and also a right eSe-module, h(W ) is well-deﬁned and is a left S-module. If ψ : W → W is a morphism in mod eSe, then h(ψ) = 1Se ⊗ ψ : h(W ) → h(W ) is a morphism in mod S. We get in this way a functor h : mod eSe → mod S. Moreover the next proposition shows that h is a “right-inverse” to f . (6.2d) Let W ∈ mod eSe. Then e · h(W ) = e ⊗ W , and the map w → e ⊗ w (w ∈ W ) gives an eSe-isomorphism W ∼ = e · h(W ) = f (h(W )). Proof. e · h(W ) = e(Se ⊗eSe W ) = eSe ⊗eSe W = e ⊗ W , as stated. Thus the map deﬁned above takes W onto e · h(W ); it is elementary to check that it is an eSe-map. To prove that it is injective, ﬁrst notice that there is a well-deﬁned map η : Se ⊗eSe W → W such that η(s ⊗ w) = esw, for all s ∈ Se, w ∈ W . Then if w ∈ W is such that e ⊗ w = 0, we get 0 = η(e ⊗ w) = w. This establishes the injectivity of the map w → e ⊗ w, and (6.2d) is proved. The trouble with the functor h is that it usually takes an irreducible module W to a module h(W ) which is not irreducible. However we have (6.2e) If W ∈ mod eSe is irreducible, then h(W )(e) is the unique maximal proper submodule of h(W ). Hence a(h(W )) is irreducible. Proof. Write V = h(W ). Then by (6.2d) and (6.2c), f (a(V )) ∼ = f (V ) ∼ = W. Thus a(V ) = 0, which shows that V(e) is a proper submodule of V . Now let V be any proper submodule of V . If eV = 0 then eV , being an eSe-submodule of the irreducible eSe-module V = e · h(W ) (recall e · h(W ) = e ⊗ W ∼ = W, by (6.2d)) is equal to e · h(W ). Then V ⊇ SeV = S(e ⊗ W ) = h(W ) = V , a contradiction. So eV = 0, i.e. V is contained in V(e) . This proves (6.2e). Deﬁnition. Let h∗ denote the functor ah : mod eSe → mod S, so that h∗ (W ) = h(W )/h(W )(e) for all W ∈ mod eSe. By (6.2c) and (6.2d) this functor h∗ , like h, is a right inverse to f , i.e. f (h∗ (W )) ∼ = W for all W ∈ mod eSe. By (6.2e) h∗ takes irreducibles to irreducibles. We have ﬁnally

6.3 Application I. Specht modules and their duals

57

(6.2f ) If V ∈ mod S is irreducible and if eV = 0, then h∗ (eV ) ∼ =V. Proof. There is an S-map β : h(eV ) = Se ⊗eSe eV → V , which takes s ⊗ ev to sev, for all s ∈ Se, v ∈ V . The image of β is SeV , which equals V because V is irreducible. So the kernel of S is a maximal proper submodule of h(eV ). But eV ∈ mod eSe is irreducible by (6.2b), hence the only maximal proper submodule of h(eV ) is h(eV )(e) , by (6.2e). Therefore β induces an isomorphism of h(eV )/h(eV )(e) = a(h(eV )) = h∗ (eV ) onto V . Taking together all these facts, we arrive at our main theorem. (6.2g) Theorem. Suppose { Vλ : λ ∈ Λ } is a full set of irreducible modules in mod S, indexed by a set Λ, and let Λ = {λ ∈ Λ : eVλ = 0}. Then {eVλ : λ ∈ Λ } is a full set of irreducible modules in mod eSe. Moreover if λ ∈ Λ , then Vλ ∼ = h∗ (eVλ ). ∼ eV (as K-spaces), for Remarks. 1. It is well-known that HomS (Se, V ) = any V ∈ mod S (see [11, p. 375]). Therefore if V is irreducible, eV = 0 if and only if V is a homomorphic image of Se. 2. When we come to apply the Schur functor to the Carter-Lusztig modules Vλ,K , it will be useful to notice that if any V ∈ mod S has a max max , then eV is either equal to eV unique maximal proper submodule V max ) = 0) or else it is the unique maximal proper submodule of (i.e. e(V /V the eSe-module eV . The proof is easy. 3. In the same context we shall use the following: If ( , ) is a symmetric bilinear form on V such that (eV, (1 − e)V ) = 0, and if ( , )e denotes the restriction of this form to eV , then rad ( , )e = e · rad( , ). Again, the proof is an easy exercise.

6.3 Application I. Specht modules and their duals In this section we shall apply the general theory of 6.2 to the special case of the Schur functor f : MK (n, r) → mod KG(r). Here K is any inﬁnite ﬁeld, and n, r are ﬁxed integers such that r ≤ n. We take S = SK (n, r) and e = ξω = ξu,u (see 6.1 for notation), and identify eSe with KG(r) by the isomorphism (6.1d), which takes ξuπ,u → π, for all π ∈ G = G(r). Notice that f (V ) = eV = V ω , for any V ∈ MK (n, r). Our aim is to calculate the eﬀect of f on the modules Dλ,K , Vλ,K . Notice that the elements λ of Λ+ (n, r) are in one-to-one correspondence with the partitions λ of r (because r ≤ n). We shall write Λ = Λ+ (n, r), and think of Λ as the set of all partitions of r. From now on, λ is a ﬁxed element of Λ. Recall (p. 35) that Dλ,K is the K-span of the elements (Tl : Ti ) = s(σ) cl,iσ , all i ∈ I(n, r). σ∈C(T )

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6 Representation theory of the symmetric group

We saw (p. 37) that these bideterminants (Tl : Ti ) all lie in the right λ-weightspace λAK (n, r) = AK (n, r) ◦ ξλ of AK (n, r). But by deﬁnition f (Dλ,K ) is ω of Dλ,K , and therefore lies in the (left) ω-weightthe ω-weight-space Dλ,K space (6.3a)

AK (n, r)ω = ξω ◦ AK (n, r) ◦ ξλ

λ

of λAK (n, r). Elementary calculations based on formulae (4.4a), (4.4a ), (6.3a) λ ω show that AK (n, r) = π∈G K · cl,uπ . Since cl,uπ = cl,uπ if and only if π ∈ πR(T ) (see 4.5), there is an isomorphism of K-spaces (6.3b)

AK (n, r)ω → KG[R(T )]

λ

which takes cl,uπ → π[R(T )], for all π ∈ G. Now KG[R(T )] is a left ideal of the group algebra KG, hence is a left KG-module. On the other hand λAK (n, r)ω , being the ω-weight-space of the left SK (n, r)-module λAK (n, r), becomes a left KG-module by means of (6.1d). To be explicit, the element τ ∈ G acts on the element cl,uπ to give τ cl,uπ = ξuτ,u ◦ cl,uπ , which by (4.4a) is equal to cl,uτ π . It follows at once that (6.3b) is a left KG-isomorphism. ω has K-basis consisting of all (Tl : Ti ) By (4.7a), p. 39, f (Dλ,K ) = Dλ,K such that i ∈ ω and Ti is standard. The elements i in ω can be written, uniquely, in the form i = uπ (π ∈ G). The isomorphism (6.3b) takes (Tl : Tuπ ) to s(σ) πσ [R(T )] = π {C(T )} [R(T )], σ∈C(T )

and so it takes f (Dλ,K ) to the left KG-submodule (left ideal) ST,K = KG {C(T )} [R(T )] of KG. We shall deﬁne ST,K to be the Specht module (over K) corresponding to the bijective λ-tableau T . (This is a little diﬀerent from the original deﬁnition of Specht; for an explanation of the latter, and of the equivalence of the two deﬁnitions, see [45, p. 91].) We have now the (6.3c) Theorem. The Specht module ST,K has K-basis consisting of the elements π {C(T )} [R(T )] such that Tuπ is standard. If char K = 0, then ST,K is an irreducible KG-module. If we choose for every λ ∈ Λ a bijective λ-tableau T λ , and write Sλ,K = ST λ ,K , then {Sλ,K : λ ∈ Λ} is a full set of irreducible KG-modules. Proof. The ﬁrst statement comes by applying the isomorphism (6.3b) to the basis of Dλ,K given by (4.7a), already quoted. If char K = 0, then each Dλ,K is irreducible by (4.7b), and since Dλ,K has character Sλ (by (4.7a)), { Dλ,K : λ ∈ Λ } is a full set of irreducible modules in MK (n, r). Then the last statement in (6.3c) follows at once from (6.2g), since in this present case f (Dλ,K ) ∼ = Sλ,K is non-zero for all λ ∈ Λ.

6.3 Application I. Specht modules and their duals

59

Now let’s look at the module Vλ,K . By deﬁnition (see (5.1b)) this is a ω subspace of E ⊗r , therefore f (Vλ,K ) = Vλ,K is a subspace of f (E ⊗r ) = (E ⊗r )ω . From the formula (2.6a) which gives the action of S = SK (n, r) on E ⊗r , we see that for any weight α ∈ Λ(n, r) (and with no restriction on n, r) (E ⊗r )α = ξα E ⊗r = K · ei . i∈α

So in particular (E ⊗r )ω has K-basis { euπ : π ∈ G }. The structure of (E ⊗r )ω as left KG-module is given by τ euπ = ξuτ,u euπ = euτ π , for all τ, π ∈ G. Therefore there is a left KG-module isomorphism (6.3d)

(E ⊗r )ω → KG,

which takes euπ → π, for all π ∈ G. We know that Vλ,K has K-basis { buπ : π ∈ G, Tuπ standard }, by (5.3b) and (5.4a). Recall that, for any i ∈ I(n, r), bi = ξi,l fl = ξi,l el {C(T )} (see 5.3). If we put i = uπ in formula (5.3c) we get ξuπ,l el = eu π[R(T )]. This means buπ = eu π[R(T )]{C(T )}. This element is carried by the isomorphism (6.3d) to the element π[R(T )]{C(T )} of KG. Therefore Vλ,K is carried to the left KG-submodule (left ideal) S T,K = KG [R(T )] {C(T )} of KG. We have the following theorem, whose proof is entirely analogous to that of (6.3c). (6.3e) Theorem. The module S T,K deﬁned above has K-basis consisting of the elements π [R(T )] {C(T )} such that Tuπ is standard. If char K = 0, then S T,K is an irreducible KG-module. If we choose for each λ ∈ Λ a bijective λ-tableau T λ , and write S λ,K = S T λ ,K , then {S λ,K : λ ∈ Λ} is a full set of irreducible KG-modules. The module S T,K is in fact the dual (in the usual sense) to the Specht module ST,K —this was ﬁrst proved by G. D. James [26, p. 460]. We can give another proof: the modules Vλ,K , Dλ,K are dual to each other under ω ω , Dλ,K are dual the contravariant form ( , ) described in 5.1. Therefore Vλ,K to each other under the restriction of this form (see 3.3). The contravariant property (2.7d) gives (ξuπ,u v, d) = (v, ξu,uπ d) = (v, ξuπ−1 ,u d), ω ω , d ∈ Dλ,K . But this becomes (πv, d) = (v, π −1 d) when for all π ∈ G, v ∈ Vλ,K ω ω we regard Vλ,K , Dλ,K as KG-modules by means of (6.1d), and this shows that these KG-modules are dual to each other. Naturally we can transfer this form, by means of the isomorphisms (6.3b), (6.3d), to give an invariant form S T,K × ST,K → K. We leave it to the reader to do this, and also to apply the calculation given in 5.3 to exhibit the following explicit version of the invariant form in question.

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6 Representation theory of the symmetric group

(6.3f ) Theorem. The KG-modules S T,K , ST,K are dual to each other. There is an invariant bilinear form ( , ) : S T,K × ST,K → K such that π [R(T )] {C(T )}, π {C(T )} [R(T )] = Ωπ,π for all π, π ∈ G, where Ωπ,π = s(σ), sum over all σ ∈ C(T ) such that the tableaux πT and π σT are row-equivalent. The matrix ( Ωπ,π : πT, π T standard ) is unipotent triangular, relative to a suitable ordering of the standard πT . Remarks. 1. Since u = (1, 2, . . . , r), the tableau Tu can be identiﬁed with the basic tableau T , and Tuπ with πT . 2. The matrix (Ωπ,π ) appearing in (6.3f) is just that part of the D´esarm´enien matrix (Ω(i, j)) corresponding to i, j ∈ ω ; in fact Ωπ,π = Ω(uπ, uπ ). 3. All the results in this section remain true when K is replaced by Z. For if we take K = Q, then we may check that the isomorphisms (6.3b), (6.3d) ω ω , Vλ,Z to ST,Z = ZG {C(T )} [R(T )], S T,Z = ZG [R(T )] {C(T )}, take Dλ,Z respectively. Therefore these last are Z-forms of the QG-modules ST,Q and S T,Q , respectively, with Z-bases { π {C(T )} [R(T )] : Tuπ standard } and { π [R(T )] {C(T )} : Tuπ standard }.

6.4 Application II. Irreducible KG(r)-modules, char K = p Throughout this section we assume that K has ﬁnite characteristic p, and that r, n are positive integers satisfying r ≤ n. In 5.4 we constructed a full set { Fλ,K : λ ∈ Λ+ (n, r) = Λ } of irreducible modules in MK (n, r). Apply the Schur functor f , and we have by (6.2g) the theorem (6.4a) Let Λ be the set of all partitions of r, and let Λ be the subset of Λ ω ω = 0. Then { Fλ,K : λ ∈ Λ } is a full set consisting of those λ such that Fλ,K of irreducible KG(r)-modules. Of course this still leaves open the crucial question: what is the set Λ ? The answer is contained in the next theorem. (6.4b) Theorem (Clausen2 , James3 ). The set Λ of (6.4a) consists of those partitions λ = (λ1 , λ2 , . . . , λr , 0, . . .) of r which are “column p-regular” i.e. for which all the integers λ1 − λ2 , λ2 − λ3 , . . . , λr lie between 0 and p − 1.

2 3

[8, Lemma 6.4, p. 184] [28, Theorem 3.2]

6.4 Application II. Irreducible KG(r)-modules, char K = p

61

Proof. It will be convenient to work, not with Fλ,K , but with the isomormin phic module Dλ,K (see (5.4c)). We denote this module by X. We must show that X ω = 0 if and only if λ is column p-regular. By (5.4d), X is generated as SK (n, r)-module by (Tl : Tl ). Therefore it is spanned as K-space by the elements ξi,j ◦ (Tl : Tl ), for all i, j ∈ I = I(n, r). By (4.4b) ξi,j ◦ (Tl : Tl ) = ξi,j (ch,l ) (Tl : Th ), h∈I

which is zero unless j ∼ l. Therefore X is K-spanned by the elements (6.4c) ξi,l ◦ (Tl : Tl ) = ξi,l (ch,l ) (Tl : Th ) = (Tl : Th ), all i ∈ I. h∈I

h∈iR(T ) α

The element (6.4c) lies in the α-weight-space X , where α is the weight containing i. So X ω is K-spanned by those elements (6.4c) such that i ∈ ω. If i ∈ ω, then G acts regularly on iG = ω i.e. iπ = iπ implies π = π , for all π, π ∈ G. In particular the elements iτ (τ ∈ R(T )) are all distinct. So (6.4d) If i ∈ ω, then ξi,l ◦ (Tl : Tl ) = τ ∈R(T ) (Tl : Tiτ ). Suppose that H is the group of all elements θ of R(T ) which preserve the set of columns of the basic λ-tableau T . Such a permutation θ can be speciﬁed by a sequence θ1 , θ2 , . . ., where for each q ≥ 1, θq is a permutation of the set Wq of all t ≥ 1 such that column t of T has length q. In the notation of (4.2a), θ maps x(s, t) to x(s, θq (t)), for all s ≥ 1, and all t ∈ Wq . Since |Wq | = λq −λq+1 , the order of H is (λ1 − λ2 )! (λ2 − λ3 )! · · · . Now it follows from the expression of (Tl : Ti ) as product of determinants (see 4.3) that (Tl : Ti ) = (Tl : Tiθ ), for all i ∈ I and all θ ∈ H. So by breaking up the sum in (6.4d) into H-orbits, we see that it is divisible by |H|. If λ is column p-singular, |H| is divisible by p, hence every term (6.4d) is zero, i.e. X ω = 0. This proves one half of (6.4b). To prove the other half, we assume that λ is column p-regular, and show that X ω = 0. For this it is enough to show that ξu,l ◦ (Tl : Tl ) = 0. By (6.4d) and (4.3a), ξu,l ◦ (Tl : Tl ) is equal to s(σ) clσ,uτ . (6.4e) σ∈C(T ) τ ∈R(T )

There is a unique element π ∈ C(T ) which reverses the order of the entries in each column of Tl , namely π : x(s, t) → x(q + 1 − s, t),

for s ≥ 1, and t ∈ Wq .

For example if λ = (7, 5, 2, 2) we have 1111111 22222 Tl = , 33 44

Tlπ

4422211 33111 = . 22 11

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6 Representation theory of the symmetric group

We shall prove that the coeﬃcient of clπ,u in (6.4e) is not zero. If σ ∈ C(T ) and τ ∈ R(T ) are such that clπ,u = clσ,uτ , then there is some γ ∈ G such that lπγ = lσ, uγ = uτ . This implies γ = τ , hence lπτ = lσ. Consider the maximum entry in Tl , say M (in the example, M = 4). In Tl , and hence also in Tlσ , all entries M are in the columns t ∈ WM . But in Tlπ , hence also in Tlπτ , all entries M are in the ﬁrst row. Since Tlπτ = Tlσ , all entries M in Tlσ are in the same places as the entries M in Tlπ . Next we consider the places occupied by entries M − 1, M − 2, . . . in turn, and by arguments similar to that given conclude that Tlπ = Tlσ , hence that π = σ. The group of elements τ ∈ R(T ) satisfying lπτ = lπ clearly has order q

(λq − λq+1 )! , w= q≥1

which is prime to p. The argument just given shows that the coeﬃcient of clπ,u in (6.4e) is s(π) w · 1K = 0, and so the proof of (6.4b) is complete. This theorem has some interesting consequences. First we need a lemma concerning the left ideal Sξω of S, which is also a right module for the algebra S(ω) = ξω Sξω , and hence, by (6.1d), a right KG-module. (6.4f ) Sξω has K-basis { ξi,u : i ∈ I(n, r) }. The K-isomorphism Sξω → E ⊗r given by ξi,u → ei for all i ∈ I(n, r) is a left S = SK (n, r)-map and a right KG-map. The proof of (6.4f) is routine. Now let V ∈ MK (n, r) be irreducible. By 6.2, remark 1, V ω = 0 if and only if V is a homomorphic image of Sξω , and hence of E ⊗r . But both E ⊗r and V are self-dual (by 2.7, Example 1, and (5.4c), proof). We have therefore (6.4g) If V ∈ MK (n, r) is irreducible, then V ω = 0 if and only if V is isomorphic to a submodule of E ⊗r . (Notice, we assume r ≤ n.) Corollary (James [28, Theorem 3.2]). Fλ,K is isomorphic to a submodule of E ⊗r if and only if λ is column p-regular. Next we have a theorem concerning the “dual” Specht module S T,K = KG [R(T )] {C(T )} of 6.3. In 5.5 was deﬁned a contravariant form

, on the space E ⊗r {C(T )}. Restrict this to the ω-weight-space (E ⊗r )ω {C(T )}, and then transfer it to KG{C(T )} by means of the isomorphism (6.3d). The result is a symmetric, invariant form on KG{C(T )} which we denote by ( , ), and which is speciﬁed by the formula: if π, π ∈ G, then s(π −1 π ) if π −1 π ∈ C(T ), (6.4h) π {C(T )}, π {C(T )} = / C(T ). 0 if π −1 π ∈

6.4 Application II. Irreducible KG(r)-modules, char K = p

63

In 5.5 we considered the form obtained by restricting

, to Vλ,K , and showed (see (5.5c)) that the radical of this form is the unique maximal submax module Vλ,K of Vλ,K . It is a routine matter now to apply the Schur functor, and use remarks 2, 3 of 6.2 to prove the following. (6.4i) Let ( , ) be the invariant form on S T,K obtained by restricting the form given by (6.4h). Then ( , ) is non-zero on S T,K if and only if λ is column p-regular. If λ is column p-regular, then the radical of ( , ) is the max max unique maximal submodule S T,K of S T,K , and S T,K / S T,K ∼ = f (Fλ,K ). From (6.4i) we may deduce a well-known theorem of James (see (6.4k), below), by the following elementary device. Let β denote the K-algebra automorphism of KG given by β(π) = s(π) π, for all π ∈ G. Let Ks denote the ﬁeld K, regarded as one-dimensional KG-module by the action πk = s(π) k, for π ∈ G, k ∈ K. Then if M is any left ideal of KG, β(M ) is also a left ideal of KG, and there is a KG-isomorphism β(M ) ∼ = M ⊗K Ks which takes m ⊗ 1K → β(m), for all m ∈ M . It is trivial to check that β maps {C(T )}, [R(T )] to [R(T )], {C(T )} respectively, where T is the λ tableau (λ is the partition of r conjugate to λ) obtained by “transposing” the λ-tableau T . The bilinear form (6.4h) on KG{C(T )} is translated by β to a symmetric, invariant bilinear form ( , ) on KG[R(T )] speciﬁed by the formula: if π, π ∈ G, then 1 if π −1 π ∈ R(T ), (6.4j) π [R(T )], π [R(T )] = / R(T ). 0 if π −1 π ∈ Moreover β(S T,K ) = ST ,K —which shows incidentally that ST ,K ∼ = S T,K ⊗K Ks —and so (6.4i) translates as follows. (6.4k) Theorem (James [27, Theorems 11.1, 11.5]). Let T be a bijective µ-tableau, where µ is a partition of r. Let ( , ) be the invariant form on ST ,K obtained by restricting the form given by (6.4j). Then ( , ) is nonzero on ST ,K if and only if µ is p-regular (by deﬁnition, µ is p-regular if µ is column p-regular). If µ is p-regular, then the radical of ( , ) is the unique max max maximal submodule ST ,K of ST ,K , and ST ,K /ST ,K ∼ = f (Fµ ,K ) ⊗K Ks . Remark. Comparison with the notation of James in [27], shows that the module Dµ [27, p. 39] is isomorphic to f (Fµ ,K ) ⊗K Ks . The module Dλ in [27, §1] is isomorphic to f (Fλ,K ). So the connection between James’s two families of irreducible KG-modules is (6.4l) Dλ ∼ = Dλ ⊗K Ks , for all column p-regular λ.

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The importance of James’s theorem, or of the equivalent theorem (6.4i), is that it gives a satisfactory “natural” labelling of the (isomorphism classes of) irreducible KG-modules: for each column p-regular λ, Dλ is isomorphic to the unique irreducible quotient of a dual Specht module S λ,K . We have also seen that Schur’s functor f gives an independent connection, Dλ ∼ = f (Vλ,K ). The Schur functor has a one-sided inverse, namely the functor h∗ deﬁned in 6.2. It is interesting that h∗ is related to a construction used by James in [28]. First we re-deﬁne h, h∗ as functors from mod KG(r) → MK (n, r), using the isomorphism Sξω ∼ = E ⊗r of (6.4f), and the isomorphism ξω Sξω ∼ = KG(r) of (6.1d). This means that for each W ∈ mod KG(r) we deﬁne h(W ) = E ⊗r ⊗KG W,

h∗ (W ) = h(W )/h(W )(ω) ,

where for any V ∈ MK (n, r), V(ω) is the sum of all S-submodules U of V such that U ω = 0. Since the isomorphism (6.4f) takes ξω to eu , it follows from (6.2d) that h(W )ω = eu ⊗ W . (6.4m) Let W be any left ideal of KG, regarded as left KG-module. Deﬁne the S-map γ : h(W ) → E ⊗r W by γ(x ⊗ w) = xw, for all x ∈ E ⊗r , w ∈ W . Then Ker γ = h(W )(ω) . Hence γ induces an S-isomorphism h∗ (W ) ∼ = E ⊗r W . Proof. Suppose that V = Ker γ. Any element v of V ω can be written v = eu ⊗w for some w ∈ W , since V ω ⊆ h(W )ω = eu ⊗ W . But we have 0 = γ(v) = eu w. This implies w = 0, since the elements eu π = euπ (π ∈ G) form a K-basis of eu KG. Hence v = 0, i.e. V ⊆ h(W )(ω) . If γ(h(W )(ω) ) is not zero, it contains some irreducible submodule M . Since the ω-weight-space of h(W )(ω) is zero, the same must be true for M . But M , as irreducible submodule of E ⊗r , satisﬁes M ω = 0 by (6.4g). This contradiction implies h(W )(ω) ⊆ V , and the rest of the proof of (6.4m) is immediate. Since KG is a Frobenius algebra [11, p. 420], every irreducible left KG-module is isomorphic to some left ideal W of KG [11, p.417, (61.6)]. If we combine this with (6.4m) and (6.2g), we have a way of constructing some irreducible SK (n, r)-modules (this method is due to James [28]). Of course we can get in this way only modules isomorphic to Fλ,K for λ column p-regular. Example. W = K[G] is an ideal of KG, and as left KG-module W affords the trivial representation of G. By (6.4m), h∗ (W ) is isomorphic to the S-module E ⊗r W = E ⊗r [G]. If i ∈ α ∈ Λ(n, r), then ei [G] = |Gα | vα , where vα is the basis element of Vr,K described in 5.2, Example 1. So E ⊗r W is a submodule of Vr,K , and has weights α, for all α such that p does not divide |Gα | = α1 ! · · · αn !. So h∗ (W ) ∼ = Fλ,K , where λ is the highest such weight. It is easy to see that λ = (p − 1, . . . , p − 1, s, 0, . . .), where there are q terms p − 1, and the non-negative integers q, s are given by r = q(p − 1) + s.

6.5 Application III. The functor f : MK (N, r) → MK (n, r) (N ≥ n)

65

6.5 Application III. The functor f : MK (N, r) → MK (n, r) (N ≥ n) In this section we ﬁx our inﬁnite ﬁeld K, and also the integer r ≥ 0. We consider connections between categories MK (n, r), as n varies. Suppose that N , n are positive integers such that N ≥ n. We shall produce a functor d = dN,n : MK (N, r) → MK (n, r), which can be viewed as special case of our general “mod S → mod eSe” functor of 6.2. The functor d gives a very easy way of passing from the irreducible modules in MK (N, r) to those in MK (n, r), and behaves in a very satisfactory way with regard to characters (see (6.5b)). It is based on a construction given by Schur in his dissertation [47, p. 61]. Since N ≥ n, we may regard I(n, r) as a subset of I(N, r), namely I(N, r) consists of all i = (i1 , . . . , ir ) with components i ∈ N , and I(n, r) is the subset of those i whose components all lie in n. With this convention, SK (n, r) can be regarded as a subalgebra of SK (N, r). For SK (N, r) has basis (6.5a)

{ ξi,j : i, j ∈ I(N, r) },

and we can identify SK (n, r) with the K-subspace spanned by those ξi,j for which i, j ∈ I(n, r). For the rule (2.3b) for multiplying two elements ξi,j , ξk,l of type (6.5a), has the consequence that if i, j, k, l ∈ I(n, r), then the coeﬃcient of any ξp,q in the product ξi,j ξk,l is zero unless both p, q ∈ I(n, r), while if p, q do belong to I(n, r), then this coeﬃcient is the same as it would be if ξi,j ξk,l were computed in SK (n, r). We deﬁne an injective map α → α∗ of Λ(n, r) into Λ(N, r) as follows. If α = (α1 , . . . , αn ) ∈ Λ(n, r), we deﬁne α∗ ∈ Λ(N, r) by α∗ = (α1 , . . . , αn , 0, . . . , 0). Then the image of Λ(n, r) under this map is the set Λ(n, r)∗ = { β ∈ Λ(N, r) : βn+1 = · · · = βN = 0 }. Notice that if i belongs to a weight β ∈ Λ(n, r)∗ , then i ∈ I(n, r). So another description of Λ(n, r)∗ is that it is the set of those G(r)-orbits of I(N, r) which lie in I(n, r). Now deﬁne the following element of SK (N, r): (6.5b) e= ξβ , sum over all β ∈ Λ(n, r)∗ . β

This is an idempotent of SK (N, r), and it is clear (using (2.3c)) that eξi,j = ξi,j or zero, according as i ∈ I(n, r) or not, and ξi,j e = ξi,j or zero, according as j ∈ I(n, r) or not. It follows at once that

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eSK (N, r)e = SK (n, r). From 6.2 (taking S = SK (N, r)) we have a functor d : MK (N, r) → MK (n, r) which takes each V ∈ MK (N, r) to eV ∈ MK (n, r). This functor has some agreeable properties, which we now describe. β V , summed over those β ∈ Λ(N, r) (6.5c) If V ∈ MK (N, r) then eV = which lie in Λ(n, r)∗ . Consequently the character ΦeV (which is a symmetric polynomial over Z in n variables X1 , . . . , Xn ) is related to ΦV (a polynomial in N variables X1 , . . . , XN ) by the formula ΦeV (X1 , . . . , Xn ) = ΦV (X1 , . . . , Xn , 0, . . . , 0). Proof. Suppose β ∈ Λ(N, r), then eV β = eξβ V = V β or zero, according as β ∈ Λ(n, r)∗ or not; the ﬁrst statement of (6.5c) follows. The second statement comes from this, together with the deﬁnition of a character (see 3.4). Next we look to see how our functor behaves on the modules Dλ,K , Vλ,K and Fλ,K . For this we need a lemma. (6.5d) Lemma. Let λ ∈ Λ+ (N, r)\Λ(n, r)∗ and i ∈ I(n, r). Then the λ-tableau Ti is not standard. Proof. Clearly, λ = (λ1 , λ2 , . . . , λN ) satisﬁes λ1 ≥ λ2 ≥ · · · ≥ λN , and since λ ∈ / Λ(n, r)∗ we must have λn+1 = 0. So the λ-tableau Ti has at least n+1 places in its ﬁrst column. But the entries in this column cannot all be distinct, since i ∈ I(n, r). Therefore Ti is not standard (see (4.5a)). (6.5e) Theorem. Let λ ∈ Λ+ (N, r), and let Xλ denote any one of Dλ,K , Vλ,K or Fλ,K . Then eXλ = 0 if and only if λ ∈ Λ(n, r)∗ . In other words eXλ = 0 if and only if λ has more than n non-zero parts. Proof. Suppose ﬁrst that λ ∈ / Λ(n, r)∗ , and that β ∈ Λ(n, r)∗ . If Xλ = Dλ,K or Vλ,K , then the dimension of Xλβ is equal to the number of i ∈ β such that the λ-tableau Ti is standard ((4.5a), (5.3b)). Since i ∈ β implies i ∈ I(n, r), we deduce from (6.5d) that Xλβ = 0, and then from (6.5c) that eXλ = 0. Since Fλ,K is a factor module of Vλ,K , we have eFλ,K = 0 also. Conversely, suppose that λ ∈ Λ(n, r)∗ . For any of the modules Xλ in question, we know that Xλλ = 0, hence eXλ = 0 by (6.5c). This completes the proof of (6.5e). If λ ∈ Λ+ (N, r), then a necessary and suﬃcient condition that λ ∈ Λ(n, r)∗ is that λ = µ∗ for some µ ∈ Λ+ (n, r). However, we know from (5.4b) that { Fλ,K : λ ∈ Λ+ (N, r) } is a full set of irreducible modules in MK (N, r). So (6.2g) and the theorem just proved give the ﬁrst statement in (6.5f) below.

6.6 Application IV. Some theorems on decomposition numbers

67

(6.5f ) {eFµ∗ ,K : µ ∈ Λ+ (n, r)} is a full set of irreducible modules in MK (n, r). In fact eFµ∗ ,K ∼ = Fµ,K (isomorphism in MK (n, r)). Hence there is the character formula, valid for all µ ∈ Λ+ (n, r), and for every characteristic p (including p = 0): Φµ,p (X1 , . . . , Xn ) = Φµ∗ ,p (X1 , . . . , Xn , 0, . . . , 0). Proof. The second and third statements follow from the fact that eFµ∗ ,K and Fµ,K are both irreducible modules in MK (n, r) having character with leading term X1µ1 · · · Xnµn (see remark (i), 3.2). Remarks. (i) A direct proof that eFµ∗ ,K ∼ = Fµ,K can be made along the following lines. Let E(n) denote a K-space with basis e1 , . . . , en , which we regard as subspace of a K-space E(N ) with basis e1 , . . . , eN . The inclusion E(n)⊗r ⊆ E(N )⊗r can be shown to take Vµ,K to eVµ∗ ,K , and to induce an isomorphism Fµ,K ∼ = eFµ∗ ,K . Similarly we may show ∼ ∗ that Dµ,K = eDµ ,K . (ii) If N ≥ n ≥ r, then the map µ → µ∗ gives a bijection from Λ+ (n, r) onto Λ+ (N, r). For any λ ∈ Λ+ (N, r), being a partition of r, can have at most r non-zero parts. Hence λ = µ∗ for some uniquely deﬁned element µ ∈ Λ+ (n, r). Then it follows from (6.5f) that the functor dN,n : MK (N, r) → MK (n, r) induces a bijection between the sets of isomorphism classes of irreducible modules in these two categories. In fact we have the stronger result: (6.5g) If N ≥ n ≥ r, the functor d = dN,n deﬁnes an equivalence of categories. In particular MK (N, r) MK (r, r) for all N ≥ r. To prove (6.5g), we must produce a functor h : MK (n, r) → MK (N, r) such that hd, dh are naturally equivalent to the appropriate identity functors (see for example [10, p. 7]). We leave it to the reader to verify that we may take for h the functor described in 6.2, which in the present case takes each W ∈ MK (n, r) to h(W ) = SK (N, r)e ⊗SK (n,r) W. We might mention that (6.5g) has another formulation: If N ≥ n ≥ r, then the K-algebras SK (N, r) and SK (n, r) are Morita equivalent (see [10, p. 34]).

6.6 Application IV. Some theorems on decomposition numbers In this section we ﬁrst extend our “mod S → mod eSe” theory of 6.2 to a “modular” context, and prove a general result of T. Martins [41] on decomposition

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numbers. Then we apply this to the modular reduction MQ (n, r) → MK (n, r) which was described in 2.5. We start with a piece of notation. Let { Vλ : λ ∈ Λ } be a full set of irreducible modules in mod S, where S is an algebra over any ﬁeld. If V ∈ mod S and λ ∈ Λ, denote by nλ (V ) the composition multiplicity of Vλ in V . That means, nλ (V ) is the number of factors isomorphic to Vλ , in any composition series of V (6.6a)

V = V0 ⊃ V1 ⊃ · · · ⊃ Vl = 0.

Now let e be an idempotent of S, and deﬁne Λ , as in (6.2g), to be the set of those λ ∈ Λ such that eVλ = 0. (6.6b) Lemma. If λ ∈ Λ , then nλ (eV ) = nλ (V ), for any V ∈ mod S. Here nλ (eV ) is the composition multiplicity of eVλ in the eSe-module eV . Proof. From (6.6a) we get a series of eSe-modules eV = eV0 ⊇ eV1 ⊇ · · · ⊇ eVl = 0. By (6.2a), e(Vj−1 /Vj ) ∼ = eVj−1 /eVj (as eSe-modules), for j = 1, . . . , l. Removing those terms eVj−1 for which eVj−1 = eVj , i.e. for which Vj−1 /Vj is isomorphic to Vπ for some π ∈ Λ\Λ , we are left with a composition series for eV . The lemma follows at once. Now let C, K be ﬁelds, and R be a subring of C with the properties (i) R is a principal ideal domain (in particular, R contains 1C ), (ii) C is the ﬁeld of fractions of R, and (iii) there is a ring-homomorphism π : R → K. Suppose SC is a C-algebra with ﬁnite basis {u1 , . . . , un }, such that the set SR = Ru1 ⊕ · · · ⊕ Rub contains the identity element of SC and is multiplicatively closed: thus SR is an “R-order” in SC . Then SK = SR ⊗ K is a Kalgebra with K-basis {u1 ⊗ 1K , . . . , un ⊗ 1K } (here and below, ⊗ means ⊗R , and K is regarded as R-module via π, i.e. r · k = π(r)k, for r ∈ R, k ∈ K). R. Brauer’s modular representation theory of algebras ([5]; see also [1, p. 111]) connects the categories mod SC and mod SK by the process of “modular reduction” (cf. 2.5), as follows. In each VC ∈ mod SC can be found an “R-form” or “admissible R-lattice” VR , i.e. (i) VR is the R-span of some C-basis of VC , and (ii) SR VR ⊆ VR (see [5, §6, p. 256], or [16, 48.1(iv), p. 299]). Then VK = VR ⊗ K can be regarded as left SK -module; VK is called a modular reduction of VC . If (6.6c)

{ Vλ,C : λ ∈ Λ },

{ Uδ,K : δ ∈ ∆ }

6.6 Application IV. Some theorems on decomposition numbers

69

are full sets of irreducible modules in mod SC , mod SK respectively, we deﬁne for each λ ∈ Λ, δ ∈ ∆ the decomposition number dλδ = dλδ (S) to be the composition multiplicity nδ (Vλ,K ) of Uδ,K in a modular reduction of Vλ,C . Let e = eR be an idempotent in the ring SR . Since e ∈ SC , we can apply the theory of 6.2. By (6.2g) we get a full set of irreducible modules in mod eSC e, namely { eVλ,C : λ ∈ Λ }, where Λ = { λ ∈ Λ : eVλ,C = 0 }. Similarly, eK = eR ⊗ 1K is an idempotent in the K-algebra SK = SR ⊗ K, and so we get a full set of irreducible modules in mod eK SK eK , namely { eK Uδ,K : δ ∈ ∆ }, where ∆ = { λ ∈ ∆ : eK Uδ,K = 0 }. Now eSR e is an R-order in the C-algebra eSC e—this is because, as Rmodule, eSR e is a direct summand of SR . For the same reason, we can identify eSR e⊗K with eK SK eK . Therefore we have a process of modular reduction from mod eSC e to mod eK SK eK ; let us denote the corresponding decomposition numbers by dλδ (eSe). Of course these are deﬁned only for λ ∈ Λ , δ ∈ ∆ . The connection between these decomposition numbers, and the dλδ (S) deﬁned previously, is very simple and satisfactory. (6.6d) Theorem (T. Martins [41]). Let λ ∈ Λ , δ ∈ ∆ . Then dλδ (S) = dλδ (eSe). Proof. Let VR be an R-form of VC = Vλ,C . Then eVR is a direct summand of VR = eVR ⊕ (1 − e)VR . This implies that eVR is an R-form of the eSC emodule eVC , and also that the eK SK eK -module eVR ⊗ K can be identiﬁed with eK VK , where VK = VR ⊗ K. By (6.6b) the composition multiplicity of eK Uδ,K in eK VK is the same as the composition multiplicity of Uδ,K in VK . This proves (6.6d). For our applications of (6.6d) we take C = Q (ﬁeld of rational numbers), R = Z, K any inﬁnite ﬁeld of characteristic p > 0, and deﬁne π : Z → K by π(n) = n · 1K for all n ∈ Z. Fix n, r and let S = SQ (n, r), SZ = SZ (n, r). Identify SK with SK (n, r) by the isomorphism given in 2.3. Identify the categories mod SQ and mod SK with MQ (n, r) and MK (n, r) respectively, as in 2.4. Corresponding to the sets (6.6c) in the general case we take { Vλ,Q : λ ∈ Λ+ (n, r) },

{ Fλ,K : λ ∈ Λ+ (n, r) }.

(It happens in this case, these sets are indexed by the same set Λ+ (n, r).) Denote the decomposition numbers by dλµ = dλµ (GLn ). These are the same numbers which appear in the formulae dλµ Φλ,p (X1 , . . . , Xn ) Φλ,0 (X1 , . . . , Xn ) = µ∈Λ+ (n,r)

of 3.5, remark (1). (6.6e) Theorem. Suppose that N ≥ n, and let α → α∗ be the injective map from Λ(n, r) into Λ(N, r) given in 6.5. Then

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(i) dαβ (GLn ) = dα∗ β ∗ (GLN ), for any α, β ∈ Λ+ (n, r). (ii) dµν (GLN ) = 0, for any λ ∈ Λ+ (N, r)\Λ(n, r)∗ and µ ∈ Λ(n, r)∗ . Proof. (i) is a direct application of (6.6d). Take SQ = SQ (N, r), etc., and let e = eZ be the element of SQ deﬁned as in (6.5b). Clearly e ∈ SZ , and eK = e ⊗ 1K is the element of SK deﬁned by (6.5b). By (6.5e), the sets Λ , ∆ which appear in (6.6d) are both equal to Λ+ (N, r) ∩ Λ(n, r)∗ . So we take λ = α∗ , µ = β ∗ in (6.6d), and then use (6.5f). (ii) Suppose λ ∈ / Λ+ (n, r)∗ , then by (6.5e), eK Vλ,K = 0. By (6.6b) the composition multiplicity of Fµ,K in Vλ,K is the same as that of eFµ,K in eK Vλ,K , because µ ∈ Λ(n, r)∗ . In other words dλµ (GLN ) = 0, and the proof of (6.6e) is complete. Remark. Part (i) of this theorem shows that, with ﬁxed r and K, the decomposition numbers dλµ (GLn ) for all n are contained in the matrix (6.6f )

(dλµ (GLr ))λ,µ∈Λ(r) .

Here Λ(r) = Λ+ (r, r), which can be identiﬁed with the set of all partitions λ = (λ1 , . . . , λr ) of r. Assume ﬁrst n = r in (6.6e). Then N ≥ r, and the map α → α∗ induces a bijection of Λ(r) onto Λ+ (N, r). So (6.6e)(i) shows that the decomposition matrix for GLN is identical (up to this bijection) with (6.6f). Next take N = r. Then n ≤ r and the map α → α∗ takes Λ+ (n, r) bijectively onto the set of those λ ∈ Λ(r) which have not more than n non-zero parts. So the decomposition matrix for GLn is identical (up to this bijection) with the submatrix of (6.6f) obtained by repressing all rows and columns which refer to partitions having more than n parts. Theorem (6.6d) gives a simple proof of a theorem of James, which shows that the matrix of decomposition numbers for the symmetric group G(r) is also a submatrix of (6.6f) — in this case we merely suppress those columns of (6.6f) which refer to partitions which are column p-singular. To see this, recall that we have full sets of irreducible QG(r)-, KG(r)-modules { S λ,Q : λ ∈ Λ(r) },

{ Dδ : δ ∈ Λ(p) (r) },

respectively. Here Λ(p) (r) denotes the set of all column p-regular partitions of r. Let dλδ (G(r)) denote the composition multiplicity of Dδ in S λ,K . Now we may apply (6.6d) with S = SQ (n, r) (r ≤ n), etc., e = ξω ∈ SQ , eK = ξω ∈ SK . We have Sλ,Q ∼ = eVλ,Q and Dδ ∼ = eK Fδ,K (see (6.3e), (6.4l) and the remarks preceding each). The result is as follows. (6.6g) Theorem (James [28, Theorem 3.4]). If r ≤ n, then dλδ (GLn ) = dλδ (G(r)), for all λ ∈ Λ(r) and δ ∈ Λ

(p)

(r).

Tables showing the matrix (6.6f) for 2 ≤ r ≤ 6, and char K = 2, 3 are given in James’s article [28].

Appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J.A. Green and M. Schocker

A Introduction

A.1 Preamble These lectures describe some combinatorial properties of the set I(n, r) of all “words” i1 i2 . . . ir of length r, whose “letters” i1 , i2 , . . . , ir are drawn from the “alphabet” n = {1, . . . , n}. Clearly I(n, r) is a ﬁnite set, with nr elements. Let Λ(n, r) be the set of all vectors β = (β1 , . . . , βn ) whose coeﬃcients are non-negative integers satisfying ν∈n βν = r. The elements β ∈ Λ(n, r) are sometimes called weights (see section 3.1). Let Λ+ (n, r) be the subset of Λ(n, r) consisting of all β which are dominant, i.e. which satisfy β1 ≥ · · · ≥ βn (≥ 0). A dominant weight in this sense is often referred to as a partition of r with no more than n parts. Example. Λ+ (2, 4) = {(4, 0), (3, 1), (2, 2)}. The set I(n, r) plays a humble rˆ ole in the representation theory of the general linear group GL(n, K) (see section 2.6), because it indexes the basis { vi = vi1 ⊗ · · · ⊗ vir : i ∈ I(n, r)} of the r-fold tensor power V ⊗r of a vector space V of dimension n, with respect to a given basis {v1 , . . . , vn } of V . But the present work is not based on linear algebra. We shall see that I(n, r) has a rich combinatorial structure in its own right, based on two operations which may be performed on any word i ∈ I(n, r); namely (A.1a) the Robinson–Schensted algorithm, and (A.1b) the application of maps e˜c , f˜c which are essentially Littelmann’s “root operators” eα , fα (see [35] and (A.3g)(2)). Peter Littelmann uses the root operators as foundation of a remarkable theory [35], sometimes called the “path model” of the classical representation theory of GLn ; this is more combinatorial, and simpler in some ways, than the classical theory. Our work is an attempt to understand this “protorepresentation theory” of GLn .

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A Introduction

A striking feature of Littelmann’s theory is that it applies to arbitrary complex, symmetrizable Kac-Moody algebras. Our work, which applies only to sln , is therefore restricted to the special case of algebras of type An−1 . But there is some advantage in this restriction; Littelmann’s “paths” become “words”, and we may work in the familiar combinatorial context of this set of lecture notes. (A.1a) and (A.1b) will be described brieﬂy in §A.2, §A.3, and discussed in more detail later.

A.2 The Robinson-Schensted algorithm This algorithm (henceforth referred to as the Schensted process) turns a word i ∈ I(n, r) into a triple (λ(i), P (i), Q(i)), where (A.2a) λ(i) = (λ1 (i), . . . , λn (i)) is a dominant weight; i.e. λ(i) is a partition of r into at most n parts, (A.2b) P (i) is a standard tableau of “shape” λ(i) (see section 4.2 and (4.5a)). The entries in the tableau P (i) are the letters i1 , i2 , . . . , ir in the word i, permuted in such a way that P (i) is standard, i.e. so that the entries in each row of P (i) are weakly increasing (≤) from left to right, and the entries in each column are strictly increasing (<) from top to bottom. (A.2c) Q(i) is a standard tableau of “shape” λ(i), whose entries are the integers 1, . . . , r permuted in such a way that Q(i) is standard1 . Schensted calls P (i) and Q(i) the P -symbol and the Q-symbol (respectively) of the word i (see [46, p. 181]). Schensted’s rules which deﬁne λ(i), P (i) and Q(i) will be given in §B.2. But it may be useful to look at the special case r = 2, where λ(i), P (i) and Q(i) are easy to describe. Take r = 2, and n any integer ≥ 2. A typical word in I(n, 2) is i1 i2 . The only dominant weights are (2, 0, 0, . . . , 0) and (1, 1, 0, . . . , 0). The only values 1 for Q(i) are the tableaux 1 2 and . 2 Table A.1 below (which is made using the rules in §B.2; see (B.4b)) produces a partition I(n, 2) = I 1 2 ∪˙ I 1 , where 2

(A.2d) I 1 2 = { i ∈ I(n, 2) : i1 ≤ i2 } and I 1 = { i ∈ I(n, 2) : i1 > i2 }. 2

1

Standard tableaux were ﬁrst deﬁned by A. Young [59] in his representation theory of the symmetric group Sym{1, . . . , r}. For this reason, standard tableaux are often called Young tableaux, or generalized Young tableaux [34].

A.3 The operators e˜c , f˜c

75

From this table we see that the set I 1 2 is the set of all i ∈ I(n, 2) such 1 that Q(i) = 1 2 , and I 1 is the set of all i ∈ I(n, 2) such that Q(i) = . 2 2 These sets are therefore the equivalence classes for the equivalence ≈ which we shall deﬁne in §A.4. The general case will be discussed in §C.1.

i

λ(i)

P (i)

Q(i)

i1 ≤ i2

(2, 0, 0, . . . , 0)

i1 i2

1 2

i1 > i2

(1, 1, 0, . . . , 0)

i2 i1

1 2

Table A.1. The Schensted process in case n ≥ r = 2.

A.3 The operators ˜ ec , ˜ fc Let a, b ∈ n, a = b, and let αa,b = (0, . . . , 0, 1, 0, . . . , 0, −1, 0, . . . , 0) denote the element of Zn which has 1, −1 at the places a, b respectively, and zero at all other places. These n(n − 1) vectors are called the roots of a system of type An−1 . Deﬁne Σ = {α1,2 , α2,3 , . . . , αn−1,n }. This is a subset of the set of all roots; its elements are called the simple roots.2 Choose an element c ∈ {1, 2, . . . , n−1}. To deﬁne Littelmann’s operators e˜c and f˜c we need some preliminary deﬁnitions. • Deﬁne the map ω = ωc,c+1 : n → Z by the rule ω(ν) = 1, −1 or zero, according as ν = c, ν = c + 1, or ν ∈ / {c, c + 1}. • Deﬁne the map hic : {0, 1, . . . , r} → Z by the rule: (A.3a) hic (0) = 0, and hic (t) = ω(i1 ) + · · · + ω(it ) for all t ∈ {1, . . . , r}. This means for any t ∈ {1, . . . , r}, (A.3b) hic (t) is the number of c’s in the initial segment i1 i2 . . . it of the word i, minus the number of c + 1’s in this segment.3 • Next let M = Mci denote the largest of the integers hic (0), hic (1), . . . , hic (r). Notice that Mci is always ≥ 0 since hic (0) = 0. 2

To read this Appendix, it is not necessary to know the theory of roots and root systems! 3 i hc is sometimes called the height function.

76

A Introduction

• There may be several values of t ∈ {0, 1, . . . , r} such that hic (t) = Mci ; let q = qci be the least of these values, and let q = q ci be the greatest. (A.3c) Lemma. (i) If q = 0, then iq = c. (ii) If q = r, then iq+1 = c + 1. Proof. (i) Suppose q = 0. We know that hic (q) = M . Let µ = hic (q − 1). By (A.3a) M = hic (q) = µ + ω(iq ). The possible values for ω(iq ) are 1, −1 and 0. But if ω(iq ) = −1 then M = µ − 1, hence µ > M against the deﬁnition of M . If ω(iq ) = 0, then µ = M , against the deﬁnition of q, which says that q is the least value of t for which hic (t) = M . Hence ω(iq ) = 1, which implies that iq = c. The proof of (ii) is similar, and is left to the reader. (A.3d) Deﬁnition (see [35, §1]). With the notation given above, deﬁne maps e˜c , f˜c : I(n, r) → I(n, r) ∪ {∞} as follows. (A.3e) If M i = 0, deﬁne f˜c (i) = ∞ (or say “f˜c (i) is undeﬁned”). If M i = 0, deﬁne f˜c (i) to be the word s ∈ I(n, r) given by st = it if t = q, and sq = c + 1. (A.3f ) If M i = hic (r), deﬁne e˜c (i) = ∞ (or say “˜ ec (i) is undeﬁned”). If M i = hic (r), deﬁne e˜c (i) to be the word s ∈ I(n, r) given by st = it if t = q + 1, and sq+1 = c. (A.3g) Remarks. (1) We have labelled these operators with the index c, rather than with the corresponding simple root α = αc,c+1 . (2) Let B : I(n, r) → I(n, r) be the operator which turns each word i1 i2 . . . ir into its “reverse” ir ir−1 . . . i2 i1 . Then the maps just deﬁned are related to Littelmann’s “root operators” fα , eα (see [35, §1]) as follows: f˜c = Bfα B, e˜c = Beα B. (3) Let i ∈ I(n, r). Then each of f˜c , e˜c takes i either to ∞, or to a word which is identical to i except at one place. At this “critical place”, f˜c (i) changes the entry from c to c + 1, and e˜c (i) changes the entry from c + 1 to c (see (A.3c)). (4) The weight wt(i) of a word i ∈ I(n, r) is the vector β ∈ Zn deﬁned as follows: for each ν ∈ n, βν is the number of places ∈ r for which i = ν (see section 3.1). Then (3) shows that wt(f˜c (i)) = wt(i) − αc,c+1 , if f˜c (i) = ∞. Similarly wt(˜ ec (i)) = wt(i) + αc,c+1 , if e˜c (i) = ∞. ˜ (5) The maps fc , e˜c are “inverse” to each other in the sense: if f˜c (i) = ∞, then e˜c f˜c (i) = i, while if e˜c (i) = ∞, then f˜c e˜c (i) = i. (6) Concatenation. If i ∈ I(n, r) and j ∈ I(n, s), deﬁne the concatenation of i and j to be the word i | j = (i1 , . . . , ir , j1 , . . . , js ) ∈ I(n, r + s). Then for any c ∈ {1, . . . , n − 1} we have f˜c (i) | j if Mci ≥ hic (r) + Mcj , and f˜c (i | j) = i | f˜c (j) if Mci < hic (r) + Mcj ,

A.3 The operators e˜c , f˜c

and e˜c (i | j) =

77

e˜c (i) | j if Mci > hic (r) + Mcj , and i | e˜c (j) if Mci ≤ hic (r) + Mcj .

All of the statements in (A.3g) are due (and in much greater generality) to Littelmann; see [35, §2]. However these statements are also easily veriﬁed directly from the deﬁnitions above. As an example, we prove the second of the two statements in (A.3g)(5), namely (5*) If i ∈ I(n, r) and c ∈ {1, . . . , n−1} such that e˜c (i) = ∞, then f˜c e˜c (i) = i. Proof. To calculate e˜c (i), we ﬁrst calculate the height function hic . This function was deﬁned in (A.3a): hic (0) = 0, and hic (t) = ω(i1 ) + · · · + ω(it ) for all t ∈ {1, . . . , r}; it is given as the third line of table A.2 below. Let M = Mci ; recall the deﬁnition of q = q ci (see (A.3b) and (A.3c)), and notice that in our case q < r, because e˜c (i) = ∞. By (A.3c)(ii), iq+1 = c + 1. In the fourth row of table A.2 are inequalities (e.g. hic (t) ≤ M ) which, taken together, express that q is the largest value of t such that hic (t) = M .

t

0

it hic (t)

0

1

2

···

q = q ci

q+1

···

r

i1

i2

···

iq

iq+1 = c + 1

···

ir

ω(i1 )

ω(i1 ) + ω(i2 )

···

M = Mci

M −1

···

hic (r)

≤M st hsc (t) f˜c (s)t

i1 0

M i2

···

iq

<M +1 i1

i2

···

<M c M +1

iq

c+1

···

ir

≤M +1 ···

ir

Table A.2. The height functions of i and s = e˜c (i).

According to deﬁnition (A.3f), the word s = e˜c (i) coincides with i except at the place q + 1. At this place iq+1 = c + 1, and sq+1 = c. To calculate f˜c (s), we must know the function hsc . Clearly hsc (t) = hic (t) for all t ∈ {0, . . . , q}, and it is easy to see that hsc (t) = hic (t) + 2 for all t ∈ {q + 1, . . . , r}. Now hic (q+1) = hic (q)+ω(iq+1 ) = M −1, hence hsc (q+1) = M +1. We may now check the inequalities in the penultimate line of table A.2. These show ec (i)) = i, as that Mcs = M + 1 > 0, and that qcs = q + 1. But then f˜c (s) = f˜c (˜ required.

78

A Introduction

A.4 What is to be done The Schensted process associates to each i ∈ I(n, r) a triple (λ(i), P (i), Q(i)). In §C.1 we shall deﬁne two equivalences on the set I(n, r). Deﬁnitions. (A.4a) i ∼ j means that P (i) = P (j), and (A.4b) i ≈ j means that Q(i) = Q(j). Donald Knuth [34] introduced the relation ∼, and proved that ∼ is the equivalence on I(n, r) generated by a collection of basic moves i → j; see §C.3. The main result of these lectures is the following analogue to Knuth’s theorem: (A.4c) Theorem A. Let i, j ∈ I(n, r). Then i ≈ j if and only if there is a ﬁnite sequence of words (elements of I(n, r)): i(1), i(2), . . . , i(s) such that i(1) = i, i(s) = j and for each adjacent pair i(ν), i(ν + 1) either there exists an element c ∈ {1, . . . , n − 1} such that f˜c (i(ν)) = i(ν + 1), or there exists an element c ∈ {1, . . . , n − 1} such that e˜c (i(ν)) = i(ν + 1). Expressed less formally, Theorem A says that ≈ is the equivalence relation on I(n, r) generated by a collection of basic moves i ⇒ j, where i ⇒ j means that e˜c (i) = j for some c ∈ {1, 2, . . . , n − 1}, or that f˜c (i) = j. Theorem A will be proved in §D.2. Chapter D also contains some notes on the representation theory of the “Littelmann algebra” L(n, r), which is an analogue of the Schur algebra S(n, r). The proof of Theorem A depends on the following (A.4d) Proposition B. Let c ∈ {1, 2, . . . , n−1}. Then the operation f˜c commutes with the Schensted process, in the following sense: (A.4e) KP (f˜c (i)) = f˜c (KP (i)) for all i ∈ I(n, r). To explain the symbols KP which appear in (A.4e), we need the Deﬁnition. Suppose given a standard tableau

P =

y1,1

y1,2

···

···

y2,1

y2,2

···

y2,λ2

···

ym,λm

.. . ym,1

y1,λ1

A.4 What is to be done

79

of shape λ ∈ Λ+ (n, r), with all its entries ya,b ∈ n. Deﬁne the “Knuth unwinding” of P to be the following word: KP = ym,1 · · · ym,λm ym−1,1 · · · ym,λm−1 · · · y1,1 · · · y1,λ1 ; see §C.2, or [34, page 173]. This is an element of I(n, r), therefore we can apply the operators f˜c , e˜c to it, and (A.4e) makes sense4 . Proposition B will be proved in §C.4.

4 Some authors identify the tableau P with the word KP , but to be cautious, we shall not make this identiﬁcation in this Appendix.

B The Schensted Process

B.1 Notations for tableaux Choose a dominant weight λ ∈ Λ+ (n, r). The shape of λ is the following subset of Z × Z (see, for example, section 4.2): (B.1a) [λ] := { (a, b) : 1 ≤ a ≤ n, 1 ≤ b ≤ λa }. A λ-tableau, or a tableau of shape λ, is a map U : [λ] → Z. We may think of U as a “partial matrix”, with U ((a, b)) as the entry in U at the place (a, b). These entries are elements of Z, and U has entries only at the places (a, b) ∈ [λ]. The entry U ((a, b)) is often denoted ua,b . We usually assume that a tableau U = (ua,b )(a,b)∈[λ] is standard ; i.e. that each row (a) is weakly increasing from left to right: ua,1 ≤ ua,2 ≤ · · · ≤ ua,λa , and that each column (b) is strictly increasing from top to bottom: u1,b < u2,b < · · · < uβb ,b (βb is the length of column (b)). Notice that the latter condition implies that if the entries ua,b of a tableau U all lie in n, then the number of rows of U cannot exceed n.

B.2 The map Sch : I(n, r) → T(n, r) Let T (n, r) be the set of all triples (λ, P, Q) such that λ ∈ Λ+ (n, r), P is a λ-tableau whose entries are drawn from the set n = {1, 2, . . . , n}, and Q is a λ-tableau whose entries are 1, 2, . . . , r in some order. (The r entries of Q are distinct; the r entries of P may include repetitions.) The subject of this chapter is Schensted’s map [46, pp. 180–181] (B.2a) Sch : I(n, r) → T (n, r). We shall deﬁne Sch in §B.3, and prove in §B.6 that it is bijective [46, p. 182]. The map Sch is deﬁned by induction on r. For r = 1, let

82

B The Schensted Process

(B.2b) Sch(i) = (1, 0, . . . , 0), i1 , 1 for any one-letter word i = i1 in I(n, 1). Note that i1 , 1 are tableaux of shape (1, 0, . . . , 0), which means that each can be regarded as a 1 × 1 matrix. From now on we assume r > 1. Insertion. Fundamental for Schensted’s work [46] is a process (or algorithm) which “inserts” a given element x1 of n into a given tableau U . The result of this process is a tableau U ← x1 whose entries are the entries of U (although perhaps in a diﬀerent order), together with one extra entry x1 . Example. Using the methods to be explained in §B.4, we shall show that if 1 1 1 2 U= and x1 = 1, then U ← x1 is the tableau 2 (see (B.4b)). 4 4 The insertion process will be described in the next section; see (B.3b) and (B.3d). Now suppose U is the middle term of an element (µ, U, V ) of T (n, r − 1). As soon as we have calculated P = U ← x1 , we shall be able (see (B.3e)) to construct a dominant weight λ ∈ Λ+ (n, r), and also a λ-tableau Q, such that (λ, P, Q) is an element of T (n, r). We shall denote this element (µ, U, V ) ← x1 . See also [46, p. 181] and [34, pp. 712, 713]. The process provides the inductive step needed to deﬁne Sch(i) for any i = i1 i2 . . . ir−1 ir ∈ I(n, r) (r > 1), namely (B.2c) Deﬁnition. Sch(i) := Sch(i ) ← ir , where i = i1 i2 . . . ir−1 . Therefore we have a formula (which can also be used as a deﬁnition of Sch(i), see [46, p. 181]). (B.2d) Formula. Sch(i) := (· · · ((Sch(i1 ) ← i2 ) ← i3 )

···

) ← ir .

B.3 Inserting a letter into a tableau Suppose we have (1) an element (µ, U, V ) of T (n, r − 1), where r > 1, and (2) an element x1 of n. In this section we deﬁne the element (µ, U, V ) ← x1 of T (n, r). To do this, we ﬁrst deﬁne the tableau U ← x1 . Schensted does this by modifying U row by row. For this purpose, it is convenient1 to supplement each row (a) of U with two “virtual entries” ua,0 = 0 and ua,µa +1 = ∞. Note that (a, 0) and (a, µa + 1) are not elements of [µ], and therefore ua,0 and ua,µa +1 are not true entries in row (a). Take any y ∈ n. Even though y may not be equal to any of the entries ua,k of row (a), we may “position” y into row (a), using the following elementary lemma. 1

See [34, p. 711].

B.3 Inserting a letter into a tableau

83

(B.3a) Lemma. For any y ∈ n and any a ∈ n, there is a unique element k(a) = k(a, y) in {1, 2, . . . , µa , µa + 1} such that ua,k(a,y)−1 ≤ y < ua,k(a,y) . Proof. Deﬁne k(a, y) to be the smallest k ∈ {1, 2, . . . , µa , µa + 1} such that y < ua,k . Example. Suppose µa = 5, and that row (a) (including the virtual entries) is (0) 2 2 2 3 7 (∞) . If y = 2, then k(a) = 4, because ua,3 ≤ 2 < ua,4 . If y = 1, then k(a) = 1, because ua,0 ≤ 1 < ua,1 . If y = 4, then k(a) = 5, because ua,4 ≤ 4 < ua,5 . The situation k(a, y) = µa + 1 = 6 occurs if and only if ua,5 ≤ y < ∞, that is, if and only if y ≥ 7. The insertion sequence. Let µ ∈ Λ+ (n, r), let U be a µ-tableau whose entries all lie in n, and let x1 ∈ n. In order to deﬁne P := U ← x1 ﬁrst make the “insertion sequence” (B.3b) x1 , k(1), x2 , k(2), ..., xz , k(z), which contains all the data needed to construct U ← x1 . Deﬁnition of the insertion sequence. Step 1. • x1 is the given element of n. • k(1) is the smallest k ∈ {1, . . . , µ1 , µ1 + 1} such that x1 < u1,k . Equivalently, k(1) is the unique element of {1, . . . , µ1 , µ1 + 1} such that u1,k(1)−1 ≤ x1 < u1,k(1) . The case k(1) = µ1 + 1 occurs if and only if u1,µ(1) ≤ x1 (< x1,µ(1)+1 = ∞), i.e. if and only if x1 ≥ u1,µ1 (hence x1 is ≥ all entries in row (1) of U ). • If k(1) = µ1 + 1, the sequence is ended. Step 2. Now assume that k(1) = µ1 + 1. Then continue the deﬁnition of the insertion sequence. • x2 := u1,k(1) . • k(2) is the smallest k ∈ {1, . . . , µ2 , µ2 + 1} such that x2 < u2,k . Equivalently, k(2) is the unique element of {1, . . . , µ2 , µ2 + 1} such that u2,k(2)−1 ≤ x2 < u2,k(2) . The case k(2) = µ2 + 1 occurs if and only if x2 ≥ u2,µ(2) . • If k(2) = µ2 + 1, the sequence is ended. Step 3. Now assume that k(2) = µ2 + 1. Then continue • x3 := u2,k(2) , etc. Inductive Step. The general step is as follows: after xa−1 (:= ua−2,k(a−2) ) and k(a − 1) have been deﬁned, then • If k(a − 1) = µa−1 + 1, the sequence is ended. Now assume that k(a − 1) = µa−1 + 1, and proceed to deﬁne

84

B The Schensted Process

• xa := ua−1,k(a−1) , • k(a) is the least k ∈ {1, . . . , µa , µa + 1} such that xa < ua,k . Equivalently, k(a) is the unique element of {1, . . . , µa , µa + 1} such that (B.3c) ua,k(a)−1 ≤ xa < ua,k(a) . Deﬁnition of z. For each a such that k(a − 1) = µa−1 + 1, (B.3c) shows that xa < ua,k(a) = xa+1 . Therefore the sequence x1 < x2 < · · · is ﬁnite (x1 , x2 , . . . are all elements of n). Deﬁne z to be the largest element of n such that k(z − 1) = µz−1 + 1. Then we must have k(z) = µz + 1 (otherwise we could go on to deﬁne k(z + 1)), and uz,µz ≤ xz (< ∞), i.e. xz is ≥ every entry in row (z) of U . (B.3d) Deﬁnition of U ← x1 . Let λ = µ + εz , where εz is the n-vector with 1 in place z, and zero at all other places. We shall show in (B.5b) that λ ∈ Λ+ (n, r). Deﬁne U ← x1 to be the λ-tableau P = (pa,b )(a,b)∈[λ] whose entry pa,b is identical with the corresponding entry ua,b of U , except 1◦ at the places (a, k(a)) for a = 1, 2, . . . , z − 1. At these places we deﬁne pa,k(a) = xa (whereas ua,k(a) = xa+1 ), and 2◦ at place (z, µz + 1), where U has no entry, we deﬁne P to have entry pz,µz +1 = xz . The shape of P is λ = µ + εz = (µ1 , . . . , µz−1 , µz + 1, µz+1 , . . . , µn ), because row (a) of P has the same length as row (a) of U , for all a = z, while the length of row (z) of P is one more than the length of row (z) of U . Note that, for all a > z, the row (a) of P is identical to row (a) of U . (B.3e) Deﬁnition of (µ, U, V) ← x1 . Let (µ, U, V ) ∈ T (n, r − 1), and let x1 be an element of n. Let P be the tableau U ← x1 deﬁned in (B.3d). Let λ be the weight µ + εz . Deﬁne Q by enlarging the µ-tableau V , giving it a new entry r in place (z, µz + 1). Then (µ, U, V ) ← x1 is by deﬁnition the triple (λ, P, Q). (B.3f ) Exercise. Prove that k(1) ≥ k(2) ≥ · · · ≥ k(z) in any case. [Hint. Let a ∈ {2, . . . , z}. We must prove that k(a) ≤ k(a − 1). By deﬁnition k(a) lies in {1, . . . , µa + 1}, therefore k(a) ≤ µa + 1, which is ≤ k(a − 1) if k(a − 1) ≥ µa + 1. But if k(a − 1) < µa + 1, i.e. k(a − 1) ≤ µa , then there exists an entry ua,k(a−1) in row (a) of U . Column standardness of U shows that ua,k(a−1) > ua−1,k(a−1) . Therefore xa = ua−1,k(a−1) < ua,k(a−1) . But k(a) is the least k ∈ {1, . . . , µa + 1} such that xa < ua,k . It follows that k(a) ≤ k(a − 1).] Note. We have not yet proved that the triple (λ, P, Q) belongs to T (n, r). For this we must show that λ ∈ Λ+ (n, r), and that P , Q are standard. These things will be proved in §B.5, but we ﬁrst look at some examples.

B.4 Examples of the Schensted process

85

B.4 Examples of the Schensted process The basic operation for the Schensted process is the insertion of a letter into a tableau. So suppose r > 1, let µ be an element of Λ+ (n, r − 1), let U be a µ-tableau, and let x1 be any element of n. We want to ﬁnd the tableau P = U ← x1 . The tableau P = U ← x1 (see (B.3d)) can be made by modifying the rows (1), (2), . . . of U , in turn. First “position” x1 (which may or may not be equal to one of the entries of U ) into row (1) of U (see Lemma (B.3a)). This means, ﬁnd the (unique) element k(1) such that u1,k(1)−1 ≤ x1 < u1,k(1) . Assume that k(1) = µ1 + 1. Let x2 := u1,k(1) . Now let x1 “bump”2 x2 into row (2), which means: (i) change the entry x2 = u1,k(1) in place (1, k(1)) to x1 = p1,k(1) , and then (ii) “position” x2 into row (2), that is: ﬁnd the unique index k(2) such that u2,k(2)−1 ≤ x2 < u2,k(2) . Then row (1) of U , changed by (i), is row (1) of P . Now we are ready to change row (2) of U into row (2) of P . In general, when row (a − 1) of U has been changed into row (a − 1) of P , we deﬁne xa := ka−1,k(a−1) and “bump” xa into row (a). This process goes on until we reach row (z), where k(z) = µz + 1. Then row (z) of P is made by adjoining an entry xz to row (z) of U , in the new place (z, µz + 1) (which was not a place for U ). All subsequent rows of P are the same as the corresponding rows of U . It is sometimes better to use a slightly diﬀerent “technology”, to construct U ← x1 from U . Here one makes the parameters x1 , k(1), x2 , k(2), . . . as before, and records these on the tableau U ; for each a we put bracketed (xa ) between the entries ua,k(a−1) and ua,k(a) of row (a). We do not change any of the entries of U . The resulting diagram (it is not a tableau in our sense) is called “U prepared for insertion of x1 ”. We pass from this diagram to P = U ← x1 by replacing . . . (xa ) xa+1 . . . by . . . xa . . ., for each a ∈ {1, . . . , z − 1}; for row (z), replace . . . uz,µz (xz ) by . . . uz,µz xz . (B.4a) Example. Suppose we want to insert x1 = 2 into the tableau U shown in the left-hand column of table B.1 below. The second column shows U “prepared” for this insertion. This means that we have put bracketed (xa ) between the entries ua,k(a)−1 , ua,k(a) for each a = 1, 2, . . . , z; we have not yet changed any of the entries of U . Once U has been prepared, make P = P (i) from U by changing the entry ua,k(a) = xa+1 to pa,k(a) = xa , for all a = 1, 2, . . . , z − 1, i.e. the term xa in the bracketed (xa ) replaces its right-hand neighbour xa+1 = ua,k(a) . This procedure determines row (z), namely it is the ﬁrst row where (xz ) does not have a right-hand neighbour. The row (z) of P , is 2 The verb “bump” was introduced, in this context, by Knuth [34, p. 713]. Alternatives would be “dump”, or even “jump”.

86

B The Schensted Process

found by replacing (xz ) by xz . In the example shown, we have x1 = 2, x2 = 3, z = 2, k(1) = 5, k(2) = 3. Note that 3 in row (1) of U , is bumped into row (2).

U

1

1

3

3

4

2

P =U ←2

U prepared for insertion of 2

2

3

1

1

3

3 (3)

2

2 (2) 3

4

1

1

2

3

3

3

2

2

4

Table B.1. Example for the insertion process.

(B.4b) Example. Consider two “extreme” possibilities. (i) It can happen that, for some a, the element xa < ua,1 . In that case k(a) = 1, and (B.3a) shows that 0 ≤ xa < ua,1 . When U is “prepared” as above, row (a) looks like this: (xa ) ua,1 ua,2 · · · ua,µa . (ii) It can happen that, for some a, we have k(a) = µa + 1, and µa+1 = 0. This means that we must “bump” xa+1 (= ua,k(a) ) into an empty row (a + 1). We have 0 < xa+1 < ∞, which allows us to say that k(a + 1) = 1. Then k(a + 1) = µa+1 + 1. So a + 1 = z, and P has an entry xa+1 in place (a + 1, 1). The following example illustrates both possibilities (i) and (ii). Suppose we insert x1 = 1 into the tableau U =

1 2 . Prepared for this insertion, U 4

1 (1) 2 1 1 becomes (2) 4 . Hence P = U ← 1 = 2 . In this example x1 , x2 , x3 (4) 4 are 1, 2, 4, respectively, z = 3; and k(1) = 2, k(2) = 1, k(3) = 1. We are now in a position to calculate the P -symbol P (i) and the Q-symbol Q(i) of a given word i ∈ I(n, r). To ﬁnd P (i) = P (i1 i2 , · · · ir ), we must calculate, successively, the P -symbols of the words i1 , i1 i2 , . . . , i1 i2 · · · ir , starting with P (i1 ) = i1 , and using the insertion process P (i1 i2 · · · it ) = P (i1 i2 · · · it−1 ) ← it .

B.4 Examples of the Schensted process

87

It follows that the entries of P (i) are the entries i1 , . . . , ir of i, in some order. The construction of Q(i) is diﬀerent, Q(i1 · · · it ) is not made by inserting it into Q(i1 · · · it−1 ); the construction follows deﬁnition (B.3e), see Example (B.4c) below. (B.4c) Example. Calculate P (i), where i = 1 4 2 1 2. Calculate also λ(i) and Q(i). First we must work out the successive tableaux Pt (i) = P (i1 . . . it ), for y t = 1, 2, . . . , 5. We ﬁnd (the operator −→ means “insert y into the tableau on the left”) 4 2 P1 (i) = 1 −→ P2 (i) = 1 4 −→ P3 (i) = 1 2 4

1 1 2 1 1 2 1 −→ P4 (i) = 2 −→ P5 (i) = P (i) = 2 . 4 4 At the same time we get the dominant weight at each stage, namely λ(i1 · · · it ) is just the shape of the tableau Pt (i). In particular, λ(i) = (3, 1, 1, 0, . . . , 0). Now we make the tableaux Qt (i) = Q(i1 . . . it ) as follows: if we know Qt−1 (i), then Qt (i) is got by putting “t” in the place which was new, when Pt (i) was constructed from Pt−1 (i). Thus Q1 (i) = 1 ,

Q2 (i) = 1 2 ,

Q3 (i) = 1 2 , 3 1 2 , Q4 (i) = 3 4

1 2 5 Q5 (i) = Q(i) = 3 . 4

(B.4d) Example. Calculate λ(i), P (i), Q(i) for any word i = i1 i2 ∈ I(n, 2), and so verify the table given in §A.2. To ﬁnd P (i), we must insert i2 into the tableau U = i1 . When U is prepared for this insertion, it becomes i1 (i2 ) in case i1 ≤ i2 , and it becomes

(i2 ) i1

in case i1 > i2 . Therefore P (i) = i1 i2

(i1 )

in case i1 ≤ i2 ,

i2

in case i1 > i2 . It follows that λ(i) is (2, 0, 0, . . . , 0) or i1 (1, 1, 0, . . . , 0), in these respective cases. To ﬁnd Q(i), we must add 2 to V = Q(i1 ) = 1 in the place (z, µz + 1).

and P (i) =

In the case i1 ≤ i2 , we have z = 1, so Q(i) = 1 2 . In case i1 > i2 , we have z = 2, so Q(i) =

1 . 2

88

B The Schensted Process

B.5 Proof that (µ, U, V) ← x1 belongs to T(n, r) We keep the notations of §§B.2, B.3, B.4. Suppose that r > 1 and that (µ, U, V ) ∈ T (n, r − 1). Let x1 ∈ n. The triple (λ, P, Q) = (µ, U, V ) ← x1 is deﬁned in (B.3e). In this section we shall prove that (λ, P, Q) ∈ T (n, r). For this we must show that λ ∈ Λ+ (n, r), and that P , Q are both standard. Write ua,b for the (a, b)-entry of U , and pa,b for the (a, b)-entry of P . (B.5a) Proposition. The weight λ = µ + εz = (µ1 , . . . , µz−1 , µz + 1, µz+1 , . . . , µn ) is dominant. It follows that Q is standard. Proof. We already know that µ1 ≥ · · · ≥ µz−1 ≥ µz ≥ µz+1 ≥ · · · ≥ µn because µ is dominant. If λ is not dominant, it must be that µz−1 = µz . But this leads to a contradiction. We know that xz ≥ uz,µz , and that uz,µz > uz−1,µz because U is standard. But uz−1,µz ≥ uz−1,k(z−1) = xz (the last equality is the deﬁnition of xz ), and putting these inequalities together gives the contradiction xz > xz . By the deﬁnition (B.3e), Q is made by adding an entry r at the end of row (z) of V . It is clear that Q is a standard λ-tableau, whose entries are 1, 2, . . . , r in some order. (B.5b) Proposition. The λ-tableau P is standard. Proof. First we shall show that P is “row standard”, i.e. that (i) pa,h−1 ≤ pa,h for all adjacent pairs (a, h − 1), (a, h) of places in any row (a) of [λ]. If (a, k(a)) is not one of (a, h − 1), (a, h) then by (B.5a) pa,h = ua,h and pa,h−1 = ua,h−1 , therefore (i) follows from the corresponding fact for row (a) of U . If (a, h) = (a, k(a)) then pa,h−1 = ua,h−1 ≤ xa , and xa = pa,k(a) = pa,h ; thus (i) holds. There remains the case (a, h − 1) = (a, k(a)). Then (i) says pa,k(a) ≤ pa,k(a)+1 . But pa,k(a) = xa and pa,k(a)+1 = ua,k(a)+1 . Thus (i) follows from xa < xa+1 = ua,k(a) ≤ ua,k(a)+1 . To complete the proof of Proposition (B.5b), we must show that P is “column standard”, i.e. that if (a, h) and (a + 1, h) are adjacent places in the same column of [λ], then (ii) pa+1,h > pa,h . If h = k(a) and h = k(a + 1) then pa,h = ua,h and pa+1,h = ua+1,h , hence (ii) follows from ua+1,h > ua,h , which holds because U is column standard.

B.6 The inverse Schensted process

89

If h = k(a), h = k(a + 1) then ua,h = ua,k(a) = xa+1 > xa . But ua+1,k(a) > ua,k(a) because U is column standard. Therefore pa+1,k(a) = ua+1,k(a) > xa = pa,k(a) , which proves (ii) in this case. Now suppose that h = k(a + 1), h = k(a). Then pa+1,k(a+1) = xa+1 , and pa,k(a+1) = ua,k(a+1) . In place (a, k(a)) of P we have xa (see (B.3d)). Since P is “row standard” (just proved, above) and k(a + 1) ≤ k(a) (see (B.3f)) we have ua,k(a+1) ≤ xa ; also xa < xa+1 by (B.3b). So pa,k(a+1) = ua,k(a+1) ≤ xa < xa+1 = pa+1,k(a+1) . This proves (ii) in case h = k(a + 1), h = k(a). There remains only the case h = k(a) = k(a + 1). In this case pa+1,h = pa+1,k(a+1) = xa+1 and pa,h = pa,k(a) = xa . But xa+1 > xa , therefore (ii) holds. The proof of Proposition (B.5b) is now complete.

B.6 The inverse Schensted process This section and the next are devoted to Schensted’s fundamental (B.6a) Theorem (see [46, p. 182]; [34, pp. 715–716]). The map Sch : I(n, r) → T (n, r) is bijective. This will be proved by constructing a map M : T (n, r) → I(n, r) which is a two-sided inverse to Sch (see (B.7b)). If r = 1, it is easy to make a map M inverse to Sch. The only element + (n, 1) is λ = (1, 0, . . . , 0), hence any in Λ T (n, 1) has the form element in to be x (regarded as λ, , λ, x , 1 for some x ∈ n. We deﬁne M x 1 a 1-letter word). By (B.2b), Sch(x) = λ, x , 1 . It is easy to check now that M : T (n, 1) → I(n, 1) is a two-sided inverse to Sch : I(n, 1) → T (n, 1). It follows that Sch is bijective in case r = 1. From now on in this section, assume that r > 1. The process given in §B.3 delivers a map, which we call insertion, (B.6b) J : T (n, r − 1) × n → T (n, r), which takes a pair ((µ, U, V ), x1 ) to the element (µ, U, V ) ← x1 of T (n, r). Next deﬁne another map, called extrusion, (B.6c) E : T (n, r) → T (n, r − 1) × n. To make E, we need an “inverse Schensted process”, which will turn any (λ, P, Q) ∈ T (n, r) into a pair consisting of a triple (µ, U, V ) ∈ T (n, r − 1) and an element w1 ∈ n. How to deﬁne E. Let (λ, P, Q) ∈ T (n, r). Let (a, b) ∈ [λ] be the (unique) place where qa,b = r. Since Q is standard, r must be at the end of its row. Therefore if a = z, then b must be λz , so that qz,λz = r. But r is also at the end of its column, which implies that λz > λz+1 . This proves

90

B The Schensted Process

(B.6d) The weight µ = λ − εz is dominant. Hence µ ∈ Λ+ (n, r − 1). Deﬁnition of the extrusion sequence. We shall next deﬁne the extrusion sequence (B.6e) l(z), wz , l(z − 1), wz−1 , . . . , l(1), w1 . To make this sequence, we must know Q (which determines z) as well3 as the λ-tableau P . Step 1. • l(z) = λz ; • wz := pz,λz (this is the entry in P , at the place (z, λz ) where Q has entry r). Step 2. • l(z − 1) is the largest l ∈ {1, 2, . . . , λz−1 } such that pz−1,l < wz . Equivalently, l(z − 1) is the unique element in {1, 2, . . . , λz−1 } such that pz−1,l(z−1) < wz ≤ pz−1,l(z−1)+1 . • wz−1 := pz−1,l(z−1) . Inductive Step. When l(a + 1) and wa+1 := pa+1,l(a+1) have been deﬁned, we go on to deﬁne • l(a) is the largest l ∈ {1, . . . , λa } such that pa,l < wa+1 . Equivalently, l(a) is the unique element in {1, . . . , λa } such that (B.6f ) pa,l(a) < wa+1 ≤ pa,l(a)+1 . • wa := pa,l(a) . Note that if a < z there is always at least one l ∈ {1, 2, . . . , λa } such that pa,l < wa+1 , namely l = l(a + 1); this is because P is column standard, hence pa,l(a+1) < pa+1,l(a+1) = wa+1 . So the extrusion sequence (B.6e) always ends with . . . , l(1), w1 . Final Step. The last two terms are as follows. • l(1) is the largest l ∈ {1, 2, . . . , λ1 } such that p1,l < w2 , and • w1 := p1,l(1) . We say that the element w1 ∈ n has been “extruded”4 from P (or more precisely from the given element (λ, P, Q) in T (n, r)). But the extrusion process also deﬁnes an element (µ, U, V ), see below. (B.6g) Deﬁnition of E. Let (λ, P, Q) ∈ T (n, r). Then E((λ, P, Q)) is the pair ((µ, U, V ), w1 ), where • µ = λ − εz = (λ1 , . . . , λz−1 , λz − 1, λz+1 , . . . , λn ), To deﬁne P = U ← x1 , we did not need to know Q, because z is deﬁned by the insertion sequence; see (B.3c), (B.3d). 4 In the way that a small amount (w1 ) of toothpaste is extruded from its tube (λ, P, Q). 3

B.6 The inverse Schensted process

91

• V is Q, with the entry qz,λz = r removed, • w1 is the extruded element of n deﬁned above, and • U = (ua,b )(a,b)∈[µ] is the following µ-tableau: ua,b = pa,b for all (a, b) ∈ [µ], except 1◦ at the places (a, l(a)) for a = 1, 2, . . . , z − 1. At these places we deﬁne ua,l(a) = wa+1 (whereas pa,l(a) = wa ), and 2◦ there is no entry in U at the place (z, λz ), because U is a µ-tableau and / [µ]. (z, λz ) ∈ To complete the deﬁnition of E, the following lemma is required. (B.6h) Lemma. The triple (µ, U, V ) belongs to T (n, r − 1). Proof. From (B.6d) we know that µ ∈ Λ+ (n, r − 1). It is clear that V is a µ-tableau whose entries are 1, 2, . . . , r − 1, in some order. It remains only to show that U is standard. The proof of this is very similar to that of Proposition (B.5b), and we leave to the reader. (B.6i) Proposition. The maps J, E are inverse to each other. Proof. We shall ﬁrst prove that (i) E ◦ J = idT (n,r−1)×n . Take any element ((µ, U, V ), x1 ) in T (n, r − 1) × n. Let (B.6j) x1 , k(1), . . . , xz , k(z), be the insertion sequence used to deﬁne (λ, P, Q) = J((µ, U, V ), x1 ) = (µ, U, V ) ← x1 . Here z is such that k(z) = µz + 1 = λz , where λ = µ + εz . Note that Q has r in place (z, λz ), and pz,λz = xz (see (B.3d) and (B.3e)). To prove (i) it is enough to show that E((λ, P, Q)) = ((µ, U, V ), x1 ). Now E((λ, P, Q)) is determined by the extrusion sequence (see (B.6e)) (B.6k) l(z), wz , l(z − 1), wz−1 , . . . , l(1), w1 . The “z” which appears in (B.6k) indexes the row of Q which contains the entry r, see (B.6d). Therefore this “z” is the same as the z in (B.6j). From (B.6e) we have l(z) = λz and wz = pz,λz . But from the deﬁnition of P , pz,λz = pz,µz +1 = xz . Therefore (B.6l) l(z) = k(z) and wz = xz . Our ambition is to prove (B.6m) l(a) = k(a) and wa = xa for all a ∈ {z, z−1, . . . , 1}. Suppose a < z and that (using “upward” induction) (B.6n) l(a + 1) = k(a + 1) and wa+1 = xa+1 .

92

B The Schensted Process

By (B.3c), there holds xa < xa+1 = ua,k(a) ≤ ua,k(a)+1 . However the deﬁnitions in §B.3 show that xa = pa,k(a) , and (B.6n) gives wa+1 = xa+1 . So xa = pa,k(a) < wa+1 ≤ ua,k(a)+1 = pa,k(a)+1 . But comparing this with (B.6f), we see that l(a) = k(a). Hence wa = pa,l(a) = pa,k(a) = xa ; thus (B.6m) holds for all a. In particular, w1 = x1 , and we ﬁnd easily that E(J(((µ, U, V ), x1 )))) = E((λ, P, Q)) = ((µ, U, V ), x1 ); in other words we have proved (i). To complete the proof of (B.6i) we must prove (ii) J ◦ E = idT (n,r) . Take any element (λ, P, Q) ∈ T (n, r). Let (B.6k) be the extrusion sequence which deﬁnes E((λ, P, Q)) = ((µ, U, V ), w1 ) (see (B.6g)). Let (B.6o) (w1 =) x1 , k(1), x2 , k(2), . . . , xz , k(z) be the insertion sequence which deﬁnes J((µ, U, V ), w1 ). In order to prove (ii) we must show that J((µ, U, V ), w1 ) = (λ, P, Q). The ﬁrst step is to prove (B.6p) wa = xa for all a ∈ {1, 2, . . . , z}. This holds for a = 1, by deﬁnition. Suppose that (B.6p) holds for some a. By (B.6f), l(a) is the unique element of {1, 2, , . . . , z} such that (B.6q) pa,l(a) = wa < wa+1 ≤ pa,l(a)+1 . From this follows that pa,l(a)−1 ≤ wa < wa+1 . Hence, using the deﬁnition (B.6g) of U , we have ua,l(a)−1 ≤ wa < ua,l(a) , and since wa = xa , there holds ua,l(a)−1 ≤ xa < ua,l(a) . However this proves that l(a) = k(a), from (B.3c). Consequently wa+1 = pa,l(a) = pa,k(a) = xa+1 (see (B.3b); we are here using the insertion of x1 into (µ, U, V )). Now we can prove, by induction on a, that (B.6r) wa = xa and l(a) = k(a), for all a ∈ {1, 2, . . . , z}. Using (B.6r) and the deﬁnitions (B.3d) and (B.3e) (applied to the insertion of x1 into (µ, U, V )), it is quite easy to show that J((µ, U, V ), w1 ) = (λ, P, Q). This concludes the proof of Proposition (B.6i).

B.7 The ladder We shall deﬁne a map M : T (n, r) → I(n, r) inverse to Sch : I(n, r) → T (n, r), and hence prove Schensted’s Theorem (B.6a). M will be given as the product of maps E0 , E1 , . . . , Er−1 displayed in table B.2 (“The ladder”) below.

B.7 The ladder

Set

Typical element of set i1 i2 . . . is is+1 . . . ir−1 ir

I(n, r) J1

6

Er−1

?

T (n, 1) × I(n, r − 1) J2

(λ1 , P1 , Q1 ), i2 . . . is is+1 . . . ir−1 ir

6

Er−2

? .. .

Js−1

.. .

6

Er−s+1

?

T (n, s − 1) × I(n, r − s + 1) Js

(λs , Ps , Qs ), is+1 . . . ir−1 ir

6

Er−s−1

?

.. .

6

E1

?

T (n, r − 1) × I(n, 1) Jr

(λs−1 , Ps−1 , Qs−1 ), is is+1 . . . ir−1 ir

Er−s

?

.. .

Jr−1

6 T (n, s) × I(n, r − s)

Js+1

93

(λr−1 , Pr−1 , Qr−1 ), ir

6

E0

? T (n, r)

(λr , Pr , Qr )

Table B.2. The ladder.

94

B The Schensted Process

Notations and Explanations. To deﬁne Es : T (n, r − s) × I(n, s) −→ T (n, r − s − 1) × I(n, s + 1), ﬁrst apply E to a typical element (λr−s , Pr−s , Qr−s ) of the set T (n, r − s): this gives a pair ((λr−s−1 , Pr−s−1 , Qr−s−1 ), ir−s ) where ir−s is some element of n. By deﬁnition, Es takes the element ((λr−s , Pr−s , Qr−s ), ir−s+1 . . . ir−1 ir ) of T (n, r − s) × I(n, s) to ((λr−s−1 , Pr−s−1 , Qr−s−1 ), ir−s ir−s+1 . . . ir−1 ir ). The map Jr−s : T (n, r − s − 1) × I(n, s + 1) −→ T (n, r − s) × I(n, s) : takes (by deﬁnition) ((λr−s−1 , Pr−s−1 , Qr−s−1 ), ir−s ir−s+1 . . . ir−1 ir ) −→ ((λr−s , Pr−s , Qr−s ), ir−s+1 . . . ir−1 ir ), where (λr−s , Pr−s , Qr−s ) = (λr−s−1 , Pr−s−1 , Qr−s−1 ) ← ir−s . Note. To explain the top step of the ladder, take T (n, 0) to be the 1-element set which contains only the triple (λ, P, Q), where λ = (0, 0, . . . , 0) and P , Q are empty tableaux. Then identify T (n, 0) × I(n, r) with I(n, r). In the same way, the bottom step is T (n, r) × I(n, 0) = T (n, r), where I(n, 0) consists of the empty word only. (B.7a) Exercise. Prove that Jr−s = E−1 s . [Hint: use Proposition (B.6i).] As we go up the ladder, the successive operators Es erode T (n, r), step by step, until it becomes I(n, r). This progress is inverted as we go down from I(n, r) to T (n, r), using the operators Js . But this “going down” is exactly described by the formula (B.2c), which means that Sch = Jr ◦ Jr−1 ◦ · · · ◦ J1 . Deﬁne (B.7b) M := Er−1 ◦ · · · ◦ E0 . By (B.7a), M is a two-sided inverse to Sch. This proves Theorem (B.6a).

C Schensted and Littelmann operators

C.1 Preamble Schensted (see §§B.3, B.6) associates to every word i ∈ I(n, r) a unique triple (λ(i), P (i), Q(i)) ∈ T (n, r). This provides the following decomposition (disjoint union) of the set I(n, r): (C.1a) I(n, r) = Iλ (n, r), λ∈Λ+ (n,r)

where Iλ (n, r) is the set of all i ∈ I(n, r) such that λ(i) = λ, for each dominant weight λ ∈ Λ+ (n, r). We deﬁne the shape of a word i to be the shape of P (i) (which is also the shape of Q(i)). So Iλ (n, r) is the set of all words of shape λ. In a case where n, r are supposed known, we may write Iλ (n, r) = Iλ . Example. The set I(3, 3) is decomposed into three subsets I(300) , I(210) and I(111) ; this decomposition of I(3, 3) is illustrated in §E.1. Assume from now on that λ ∈ Λ+ (n, r) is ﬁxed. Deﬁnition. Deﬁne two equivalence relations ∼ and ≈ on Iλ : if i, j ∈ Iλ then (C.1b) i ∼ j means that P (i) = P (j), and (C.1c) i ≈ j means that Q(i) = Q(j). We will use the following notation. (C.1d) For any (standard) λ-tableau P whose entries are drawn from the set n, let Iλ (P, ∼) be the ∼ equivalence class { i ∈ Iλ : P (i) = P }, and (C.1e) For any (standard) λ-tableau Q whose entries are 1, 2, . . . , r (in some order), let Iλ (Q, ≈) be the ≈ equivalence class { i ∈ Iλ : Q(i) = Q }.

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C Schensted and Littelmann operators

Remark. If either of the tableaux P , Q is given, its shape λ is known. For this reason we will usually omit the suﬃx λ, and write Iλ (P, ∼) = I(P, ∼) and Iλ (Q, ≈) = I(Q, ≈). The equivalence relation ∼ was introduced by Knuth, who proved that ∼ is the equivalence relation on Iλ generated by a certain collection of basic (or “elementary”) moves i → j, each of which aﬀects only two places in i and j. Knuth’s theorem will be proved in §§C.3, C.4. This proof is based on Knuth’s paper [34, Theorem 6, p. 723]. Littelmann deﬁnes a graph G, in a wider context than here [35, p. 504]. Theorem A (see (A.4c) and Chapter D) will show that (in our present context) the equivalence relation determined by G is equal to ≈. We regard Theorem A as an analogue to Knuth’s theorem; it says that ≈ is the equivalence relation on Iλ generated by a certain collection of elementary moves i ⇒ j, where i ⇒ j means that there exists c ∈ {1, 2, . . . , n − 1} such that f˜c (i) = j or such that e˜c (i) = j. Notice that if i ⇒ j, then the words i and j diﬀer in exactly one place; see (A.3g)(2). Example. The tables in §E.1 show the ∼ and ≈ classes for the case n = r = 3. The ≈ classes are given as vertical columns in these tables; for example I 1 3 , ≈ = {211, 212, 311, 213, 312, 313, 322, 323}, 2 1 and I 2 , ≈ is the one-word set {321}. The ∼ classes are given as hori3 zontal rows in the tables in §E.1; for example I 1 3 , ∼ = {231, 213}, 2 and I 1 1 2 , ∼ is the one-word set {112}. The one-word set {321} is both a ∼ and a ≈ class.

C.2 Unwinding a tableau To each tableau Y we shall associate a word KY , which may be called the (Knuth) unwinding of Y , as follows (see [34, p. 723] or [18, p. 17]). Let λ ∈ Λ+ (n, r) be a dominant weight, and let m be the number of rows of [λ], so that λ1 ≥ λ2 ≥ · · · ≥ λm > 0. Deﬁne the Knuth ordering < on [λ] as follows (see [34, p. 723]): (C.2a) (m, 1) < (m, 2) < · · · < (m, λm ) < (m − 1, 1) < (m − 1, 2) < · · · < (m − 1, λm−1 ) < ··· < (2, 1) < (2, 2) < · · · < (2, λ2 ) < (1, 1) < (1, 2) < · · · < (1, λ1 ).

C.2 Unwinding a tableau

97

Now let Y = (ya,b )(a,b)∈[λ] be any λ-tableau. Deﬁne KY to be the word (C.2b) of length r obtained by writing out the entries ya,b according to the order (C.2a): (C.2b) KY := ym,1 ym,2 . . . ym,λm ym−1,1 ym−1,2 . . . ym−1,λm−1 . . . y2,1 y2,2 . . . y2,λ2 y1,1 y1,2 . . . y1,λ1 . The word KY is (by deﬁnition) the unwinding of the tableau Y . So KY is the word obtained by writing out the entries of each row of Y from left to right, starting with the bottom row, and working up to the ﬁrst row. 1 1 2 2 , then KY = 3231122. Example. If Y = 2 3 3 Suppose that i ∈ I(n, r). The Schensted process (see §B.3) constructs an element (λ(i), P (i), Q(i)) of T (n, r), where λ ∈ Λ+ (n, r) and P (i) is a λ-tableau. Then the “unwinding” KP (i) of P (i) is a word, an element of I(n, r). Thus we have an operation KP : I(n, r) → I(n, r), which takes each i in I(n, r) to KP (i). However, if we apply the Schensted process to KP (i), we just get P (i) again; this follows from Proposition (C.2c) below. (C.2c) Proposition. Let λ ∈ Λ+ (n, r) with λ1 ≥ · · · ≥ λm > 0, and let Y be a λ-tableau. (i) P (KY ) = Y , and (ii) Q(KY ) is completely determined by the shape λ of Y ; it is the same for all λ-tableaux Y . The tableau Q(KY ) is described in (C.2h). Proof of part (i) of Proposition (C.2c). We shall prove (i) by induction on the number m of rows of Y . If m = 1, then Y is a one-rowed tableau y1,1 y1,2

···

y1,λ1 of shape (λ1 , 0, . . . , 0), and KY = y1,1 y1,2 . . . y1,λ1 .

We make P (KY ) by successively inserting y1,2 , . . . , y1,λ1 into the tableau y1,1 (see (B.2d) and (B.3d)). But since y1,1 ≤ y1,2 ≤ · · · ≤ y1,λ1 , each insertion simply adds a new entry to the ﬁrst row. Therefore P (KY ) = y1,1 y1,2

···

y1,λ1 = Y . Thus (i) holds if m = 1. By the deﬁnition (B.3e),

we have Q(KY ) = 1 2 · · · λ1 ; this proves that (ii) also holds. Now suppose that m > 1 and that Proposition (C.2c) holds for any tableau with m−1 rows. In particular it holds for the tableau X made by removing the ﬁrst row of Y ; therefore P (KX) = X. It is clear that KY = KX | y1,1 · · · y1,λ1 , hence P (KY ) = P (KX) ← y1,1 ← · · · ← y1,λ1 = X ← y1,1 ← · · · ← y1,λ1 . Diagram C.1 shows X, and above it, in parentheses, are the entries of row (1) of Y ; these are not entries of X.

98

C Schensted and Littelmann operators (y1,1 ) y2,1 y3,1 .. . .. . .. . X= .. . .. . .. . .. . yβ1 ,1

(y1,2 ) · · · (y1,t−1 ) y2,2 · · · y2,t−1 y3,2 · · · y3,t−1 .. .. . . .. .. . . .. .. . . .. .. . . .. . yβt−1 ,t−1 .. . yβ2 ,2 0 ···

0

(y1,t ) · · · (y1,λ2 ) · · · (y1,λ1 ) y2,t · · · y2,λ2 · · · 0 y3,t · · · y3,λ2 · · · 0 .. .. . . .. . yβλ2 ,λ2 .. . yβt ,t 0

···

0

···

0

0

···

0

···

0

Diagram C.1. The tableau X, made by removing the ﬁrst row from the tableau Y .

In diagram C.1, the number βs denotes the length of column s, including the term (y1,s ). Therefore β1 = m, and β = (β1 , β2 , . . . , βλ1 , 0, . . . , 0) can be regarded as the partition of r conjugate to λ = (λ1 , λ2 , . . . , λβ1 , 0, . . . , 0). There holds β1 ≥ β2 ≥ · · · ≥ βt−1 ≥ βt ≥ · · · ≥ βλ1 , but for ease of drawing, diagram C.1 illustrates a case where the βs are distinct. (C.2d) Deﬁnition. Let t ∈ {0, 1, 2, . . . , λ1 }. Let X[0] := X, and if t ≥ 1, deﬁne X[t] to be the diagram obtained from diagram C.1 by removing the parentheses from (y1,1 ), (y1,2 ), . . . , (y1,t ), and then pushing columns (1), (2), . . . , (t) down by one place. (C.2e) Remark. For each s ∈ {1, . . . , t}, column (s) of X[t] is the same as column (s) of Y . In particular, X[λ1 ] = Y . (C.2f ) Lemma. Let t ∈ {0, 1, 2, . . . , λ1 }. Then P (KX | y1,1 y1,2 . . . y1,t ) = X[t]. In particular, P (KY ) = P (KX | y1,1 y1,2 . . . y1,λ1 ) = X[λ1 ] = Y . Thus (C.2f) will complete the proof of part (i) of (C.2c). Proof of Lemma (C.2f ). We use induction on t. If t = 0, the lemma claims that X = X[0], which is true. So let t ∈ {1, 2, . . . , λ1 }, and suppose that the lemma is true when t is replaced by t − 1; that is P (KX | y1,1 y1,2 . . . y1,t−1 ) = X[t − 1]. Now P (KX | y1,1 y1,2 . . . y1,t ) = P (KX | y1,1 y1,2 . . . y1,t−1 ) ← y1,t . So to prove Lemma (C.2f), it will be enough to prove that X[t − 1] ← y1,t is the tableau X[t] deﬁned in (C.2d). The tableau X[t − 1] is displayed in

C.2 Unwinding a tableau

X[t − 1] =

y1,1 y2,1 .. . .. . .. . .. . .. .

y1,2 y2,2 .. . .. . .. . .. .

··· ···

99

(y1,t ) · · · (y1,λ2 ) · · · (y1,λ1 ) y2,t · · · y2,λ2 · · · 0 y3,t · · · y3,λ2 · · · 0 .. .. . . .. . yβλ2 ,λ2 .. .

y1,t−1 y2,t−1 .. . .. . .. . .. .

yβt ,t

yβ2 −1,2 · · · yβt−1 ,t−1 yβ1 −1,1 yβ2 ,2 yβ1 ,1 0 ··· 0

0

···

0

···

0

0

···

0

···

0

Diagram C.2. The tableau X[t − 1] = P (KX | y1,1 y1,2 . . . y1,t−1 ).

diagram C.2. We calculate X[t − 1] ← y1,t by the general insertion procedure given in §B.3. To make the present notation conform with that in §B.3, take U := X[t − 1], x1 := y1,t and µ to be the shape of U . Calculate the insertion parameters x1 , k(1), x2 , k(2), . . . by the inductive rule: given the element xa = ya,t for some a ≥ 1, deﬁne k(a) to be the unique element k in {1, . . . , µa , µa + 1} such that (1) ua,k−1 ≤ xa < ua,k . Then deﬁne xa+1 := ua,k(a) . It is very easy to ﬁnd k(a) in our case. There holds (2) ya,t−1 ≤ ya,t < ya+1,t , for any a ∈ {1, . . . , βt −1}; the inequalities in (2) follow from the fact that Y is standard. But (2) is the same as (1), if we take k = t and xa+1 = ya+1,t . This shows that k(a) = t and xa+1 = ya+1,t . So starting with x1 = y1,t , which is given, we may ﬁnd all insertion parameters for the insertion X[t − 1] ← y1,t . The result is given in table C.1 below. The parameter z (see the line under (B.3c)) is equal to βt . Notice that all the k(a) are equal to t, which means

2

···

a

1

xa

y1,t y2,t · · ·

k(a)

t

t

···

a

···

ya,t · · · t

···

βt yβt ,t t

Table C.1. Insertion parameters for the insertion X[t − 1] ← y1,t .

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C Schensted and Littelmann operators

that all the xa lie in column t. Use (B.3d) to ﬁnd X[t − 1] ← y1,t ; the result is X[t] as claimed in Lemma (C.2f). Proof of part (ii) of Proposition (C.2c). We are dealing with the word j = KY , whose letters are indexed by the set [λ]. Fix an element (a, s) ∈ [λ]. Then the segment j(m,1) j(m,2) · · · j(a,s) of j has length (C.2g) ψ (λ) (a, s) := λm + · · · + λa+1 + s. This gives a bijective, order-preserving map ψ = ψ (λ) : [λ] → r. We may regard ψ = ψ (λ) as the λ-tableau whose “unwinding” Kψ is the word1 1 2 3 · · · (r−1) r. Notice that the tableau ψ = ψ (λ) is, in general, not standard. 6 7 8 Example. If λ = (3, 3, 2, 0, . . . , 0), then ψ = ψ (λ) = 3 4 5 . 1 2 We shall prove part (ii) of Proposition (C.2c) by proving the following much stronger result: (C.2h) Proposition (see [3, Appendix C]). Let Y be any λ-tableau. Then Q(KY ) = Q(λ) , where Q(λ) is the λ-tableau given by Q(λ) (a, s) := ψ (λ) (βs + 1 − a, s), for all (a, s) ∈ [λ]. Expressed in words: Q(λ) is obtained by reversing each column of the tableau ψ (λ) . 1 2 5 Example. If λ = (3, 3, 2, 0, . . . , 0), then Q(λ) = 3 4 8 . 6 7 If Proposition (C.2h) is true, the tableau Q(λ) must be standard, since it is the Q-symbol of a word KY (see §B.5). Proof of Proposition (C.2h). We use induction on m. The case m = 1 is easy; if Y = y1,1 y1,2

···

y1,λ1 , then Q(KY ) = 1 2

· · · λ1 , which is the

same as Q(λ) (in this case we have Q(λ) = ψ (λ) ). So now suppose that m > 1 and that Proposition (C.2h) is true when Y is replaced by any tableau with m − 1 rows. In particular it is true for the ∗ tableau X obtained by removing the ﬁrst row from Y , so that Q(KX) = Q(λ ) , where λ∗ = (λ2 , . . . , λm , 0, . . . , 0) ∈ Λ+ (n, r − λ1 ) is the shape of X. 1

This word belongs to I(r, r). To deﬁne ψ (λ) , we should regard λ as an element of Λ+ (r, r), but notice that [λ] = [λ ] if λ is obtained from λ by adding zeros: λ = (λ1 , . . . , λm , 0, 0, . . . , 0).

C.2 Unwinding a tableau

101

We proved part (i) of Proposition (C.2c), namely that P (KY ) = Y , by calculating in turn the P -symbols of the words KX,

KX | y1,1 ,

KX | y1,1 y1,2 ,

...,

KX | y1,1 y1,2 · · · y1,λ1 = KY.

So we shall do the same for the Q-symbols. The ﬁrst step is to ﬁnd Q(KX | y1,1 ). Use the procedure described in (B.3e), taking U = X, x1 = y1,1 and r = ψ(1, 1). (Notice that r is the length of the word KX | y1,1 ). To go from X = P (KX) to P (KX | y1,1 ) = X ← y1,1 is very easy; push down the ﬁrst column of X by one place, and then put y1,1 into the top place of that column (see proof of part (i) of Proposition (C.2c)). The tableaux X and X ← y1,1 are shown in table C.2. To ﬁnd Q(KX | y1,1 ), ∗ use the recipe in (B.3e). Our induction hypothesis gives Q(KX) = Q(λ ) . The λ∗ -tableau X y2,1 y3,1 .. . .. . .. . .. . yβ1 ,1

y2,2 y3,2 .. . .. . .. .

··· ···

yβ2 ,2 0

··· ···

The (λ∗ + ε1 )-tableau X ← y1,1

y2,λ2 y3,λ2 .. .

0 0 .. .

yβλ2 ,λ2 .. .

0 .. .

0 0

0 0

y1,1 y2,2 y3,2 y2,1 .. .. . . .. .. . . .. .. . . .. . yβ2 ,2 yβ1 −1,1 0 0 yβ1 ,1

··· ···

y2,λ2 y3,λ2 .. .

0 0 .. .

yβλ2 ,λ2 .. .

0 .. .

0 0 0

0 0 0

··· ··· ···

Table C.2. Inserting y1,1 into the tableau X (P -symbol).

∗

∗

Diagram C.3 shows ψ (λ ) . To make Q(λ

ψ (λ

∗

)

)

ψ(2, 1) ψ(2, 2) · · · ψ(2, λ2 ) ψ(3, 1) ψ(3, 2) · · · ψ(3, λ2 ) .. .. .. . . . .. .. . . · · · ψ(βλ2 , λ2 ) = .. .. .. . . . .. 0 . ψ(β2 , 2) 0 ··· 0 ψ(β1 , 1) ∗

∗

we reverse each column of ψ (λ ) ; this 0 0 .. . 0 . 0 0 0

Diagram C.3. The tableau ψ (λ ) , where λ∗ = (λ2 , . . . , λm , 0, . . . , 0).

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C Schensted and Littelmann operators

gives the left-hand pane in table C.3. Now follow the instructions in (B.3e): to

The Q-symbol of the word KX

The Q-symbol of the word KX | y1,1

ψ(β1 , 1) ψ(β2 , 2) · · · ψ(βλ2 , λ2 ) 0

ψ(β1 , 1) ψ(β2 , 2) · · · ψ(βλ2 , λ2 ) 0

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

ψ(4, 1)

ψ(3, 2) · · ·

ψ(3, λ2 ) 0

ψ(4, 1)

ψ(3, 2) · · ·

ψ(3, λ2 ) 0

ψ(3, 1)

ψ(2, 2) · · ·

ψ(2, λ2 ) 0

ψ(3, 1)

ψ(2, 2) · · ·

ψ(2, λ2 ) 0

ψ(2, 1)

0

···

0

0

ψ(2, 1)

0

···

0

0

ψ(1, 1)

0

···

0

0

Table C.3. Inserting y1,1 into the tableau X (Q-symbol).

go from the left-hand pane to the right-hand pane, we adjoin a new place—this must be the place which is new when we go from P (KX) to P (KX) ← y1,1 , namely the place (β1 , 1). And in this place we must put “r”, which in our case is ψ(1, 1). This gives Q(KX | y1.1 ) shown in the right-hand pane of table C.3. We go on to insert y1,2 , . . . , y1,λ1 in turn. At each insertion, say y1,t , we adjoin ψ(1, t) to the bottom of column (t). But the new column (t) so made is the same as column (t) of Q(λ) . When we have inserted y1,λ1 we have the complete tableau Q(λ) . This ﬁnishes the proof of Proposition (C.2h). Hence we have proved Proposition (C.2c). (C.2i) Exercise. Let λ ∈ Λ+ (n, r), and let i be any element of Iλ (n, r). Then i = KP (i) if and only if Q(i) = Q(λ) . In other words, the ≈ class of Q(λ) consists of all i ∈ Iλ (n, r) which satisfy i = KP (i). [Hint. Let i ∈ I(n, r). Schensted’s Theorem (B.6a) tells us that (i) i = KP (i) if and only if Sch(i) = Sch(KP (i)). Now assume that i ∈ Iλ (n, r), which means that λ(i) = λ (see (C.1a)). Hence (ii) Sch(i) = (λ, P (i), Q(i)). To calculate Sch(KP (i)) we take Y = P (i) in (C.2c). This shows that P (KP (i)) = P (i). Since P (i) has shape λ, it follows that λ(KP (i)) = λ. But (C.2c)(ii) and (C.2h) tell us that Q(KP (i)) = Q(λ) . Therefore (iii) Sch(KP (i)) = (λ, P (i), Q(λ) ). Now (i), together with (ii) and (iii), give the desired result: i = KP (i) if and only if Q(i) = Q(λ) .]

C.3 Knuth’s theorem

103

C.3 Knuth’s theorem (C.3a) Theorem (see [34, p. 723]). Let i, j be words in I(n, r). Then i ∼ j (i.e. P (i) = P (j)) if and only if there is a ﬁnite sequence of words (C.3b) i(1), i(2), . . . , i(s) such that i(1) = i, i(s) = j and each consecutive pair of words i(σ − 1), i(σ) is connected by a basic (or elementary) move of type K or K . These basic moves are as follows [34, p. 723]. Deﬁnition. A move of type K changes a word (C.3c) . . . b c a . . .

to

. . . b a c . . .,

where a, b, c are letters (i.e. elements of n) such that a < b ≤ c. A move of type K changes a word (C.3d) . . . a c b . . .

to

. . . c a b . . .,

where a, b, c are letters (i.e. elements of n) such that a ≤ b < c. Remarks. (i) Each basic move is assumed to be symmetric, i.e. if a move takes a word w to another word w , then it also takes w to w. (ii) In (C.3c) and (C.3d), the symbol . . . stands for a word (possibly empty) which is not changed in the move. For example the type K move (C.3c) changes BbcaC to BbacC, where B, C are ﬁxed words. (By (i), this move also takes BbacC to BbcaC.) The “only if” part of Knuth’s theorem will be proved in this section, and the “if” part will be proved in §C.4. So in this section (§C.3) we must prove (C.3e) If i, j ∈ I(n, r) are such that P (i) = P (j), then i can be connected to j by a ﬁnite sequence of basic moves. The essence of this is that every insertion operation U to U ← x can be broken down into a sequence of basic moves. The next proposition puts this fact in a form suitable for our purposes; in (C.3p) it will be shown that (C.3f) implies (C.3e). (C.3f ) Proposition. Let r > 1, µ ∈ Λ+ (n, r − 1). Let U be any µ-tableau and x any element of n. Regard KU and x as words of lengths r − 1 and 1 respectively, so that the “concatenation” w = KU | x of these may be regarded as an element in I(n, r). Then there is a ﬁnite sequence of basic moves in I(n, r) which takes w to the word K(U ← x).

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C Schensted and Littelmann operators

Proof. We shall give in (C.3i)–(C.3k) an explicit sequence of basic moves which takes w to K(U ← x)2 . It is desirable to ﬁx ﬁrst some notation for the words which will be used in the proof of (C.3f). (C.3g) Notation for words and places. All the words in this section have length r, and their entries are labelled by the set of places [µ]∪{(r)}. The r − 1 elements of [µ] are arranged according to the Knuth order (see (C.2a)), and (r) is the last place. Therefore if µ= (µ1 , . . . , µm , 0, . . . , 0) ∈ Λ+ (n, r − 1) µj = r − 1), then a typical word looks (with µ1 ≥ · · · ≥ µm > 0 and like this: y = ym,1 . . . ym,µm . . . y1,1 . . . y1,µ1 yr . To resume the proof of (C.3f), write x = x1 and w = KU | x, so that w = um,1 . . . um,µm . . . u1,1 . . . u1,µ1 x1 . Recall from (B.3b) the “parameters” of the insertion of x1 into the tableau U : for each a ∈ {1, . . . , z}, deﬁne xa := ua−1,k(a−1) if a > 1, or if a = 1 deﬁne x1 = x. Deﬁne k(a) to be the smallest k ∈ {1, 2, . . . , µa , µa + 1} such that xa < ua,k . If k(a) = µa + 1 (which means that xa ≥ ua,µa ), then the insertion sequence stops at this stage. Deﬁne z to be the ﬁrst a such that xa ≥ ua,µa . The tableau U ← x1 is denoted P = (pa,b )(a,b)∈[λ] , where λ = µ + εz (see (B.3d)). According to (B.3d), each row a > z of the tableau U , is identical to the corresponding row of P = U ← x1 ; and (B.3d)(1◦ ) shows that also uz,t = pz,t for all t ∈ {1, . . . , µz }. Therefore (C.3h) ua,t = pa,t for all places (a, t) ≤ (z, µz ). Next deﬁne a sequence of words ξ(a, t), one for each place (a, t) ∈ [µ], which will “interpolate” between the words w = KU | x1 and KP = K(U ← x1 ). Use the following notation: if τ ∈ [µ], then τ + (respectively τ −) denotes the place immediately after (respectively immediately before) τ in the order (C.3g) of [µ] ∪ {(r)}. For example, if a ∈ {2, . . . , z}, then (a, t)+ is (a, t + 1) for all 1 ≤ t < µa , and (a, µa )+ = (a − 1, 1). Deﬁnition of the words ξ(a, t). (C.3i) If (a, t) ≤ (z, µz ), then deﬁne ξ(a, t) := KP . (C.3j) If (a, t) > (z, µz ) and k(a)+1 ≤ t ≤ µa , then deﬁne ξ(a, t)(a,t)+ := xa , ξ(a, t)τ := uτ if τ ≤ (a, t), and ξ(a, t)τ := pτ − if τ > (a, t)+. (C.3k) If (a, t) > (z, µz ) and 1 ≤ t ≤ k(a), then deﬁne ξ(a, t)(a,t) := xa+1 , ξ(a, t)τ := uτ if τ < (a, t), and ξ(a, t)τ := pτ − if τ > (a, t).

2 In fact the construction of these basic moves is an essential part of Knuth’s proof of his theorem; see [34, end of p. 723, and ﬁrst 7 lines of p. 724].

C.3 Knuth’s theorem

105

(C.3l) Pivot of ξ(a, t). Assume that (a, t) > (z, µz ). Then the word ξ(a, t) can be described as follows: deﬁne the pivot of ξ(a, t) to be the pair (xa , (a, t)+) in case (C.3j), and to be (xa+1 , (a, t)) in case (C.3k). Then in both cases the rule is: at every place τ left of the pivot, let ξ(a, t)τ = uτ , and at every place τ right of the pivot, let ξ(a, t)τ = pτ − . At the pivot itself, ξ(a, t) has entry xa in case (C.3j), or xa+1 in case (C.3k). If a word is among the ξ(a, t), it is completely determined by its pivot. (C.3m) Proposition. ξ(1, µ1 ) = w = KU | x1 . ξ(z − 1, 1) = KP = K(U ← x1 ). ξ(a, 1) = ξ(a + 1, µa+1 ), for all a ∈ {1, . . . , m − 1}. If k(a) + 1 ≤ t ≤ µa , there is a basic move of type K which takes ξ(a, t) to ξ(a, t − 1). (v) If 2 ≤ t ≤ k(a), there is a basic move of type K which takes ξ(a, t) to ξ(a, t − 1).

(i) (ii) (iii) (iv)

Proof. (i) The pivot of ξ(1, µ1 ) is (x1 , (r)), therefore ξ(1, µ1 ) has uτ in each place τ ∈ [µ], and x1 in place (r). Hence ξ(1, µ1 ) = KU | x1 . (ii) The pivot of ξ(z −1, 1) is (xz , (z −1, 1)) since 1 ≤ t ≤ k(z −1) for t = 1. Thus ξ(z − 1, 1) has xz at place (z − 1, 1). At a place τ < (z − 1, 1), the entry in ξ(z − 1, 1) is uτ , and this equals pτ , by (C.3h). At t > (z − 1, 1), the entry is pτ − . Therefore ξ(z − 1, 1) = KP . (iii) All that is needed, is to show that both ξ(a, 1) and ξ(a + 1, µa+1 ) have the same pivot (xa+1 , (a, 1)). We leave this as an exercise for the reader. (iv) Suppose that k(a) + 1 ≤ t ≤ µa . By deﬁnition (C.3j), the entries of ξ(a, t) at the places (a, t)−, (a, t), (a, t)+ are u(a,t)− , ua,t , xa , respectively. If these entries are denoted b, c, a, then we shall prove that a < b ≤ c. The inequality b ≤ c, i.e. u(a,t)− ≤ u(a,t) , follows from (a, t)− = (a, t − 1) and the standardness of U (note that k(a)+1 ≤ t implies that 2 ≤ t). To see that a < b, use the inequality ua,k(a)−1 ≤ xa < ua,k(a) (see (B.3c)). Standardness of U gives ua,k(a) ≤ ua,t−1 , since k(a) + 1 ≤ t. Therefore xa < ua,k(a) ≤ ua,t , hence a < b. We may now make a move of type K which interchanges a and c, and leaves ξ(a, t) otherwise unchanged. At the places (a, t), (a, t)+, the word K ξ(a, t) has entries xa , ua,t . However ua,t = pa,t since t = k(a). It is now easy to see that K ξ(a, t) = ξ(a, t − 1). (v) The proof is on the same lines as that of (iv). Suppose 2 ≤ t ≤ k(a). By (C.3k) the entries in ξ(a, t) at the places (a, t)− = (a, t − 1), (a, t), (a, t)+ are ua,t−1 , xa+1 , pa,t , respectively. If these entries are denoted a, c, b, we leave it to the reader to prove that a ≤ b < c. This shows that there is a move of type K which interchanges a, c, and leaves ξ(a, t) otherwise unchanged. Therefore the entries in K ξ(a, t) at places (a, t − 1), (a, t), are xa+1 , ua,t−1 . But ua,t−1 = pa,t−1 because t−1 = k(a). It follows that K ξ(a, t) = ξ(a, t−1). This completes the proof of Proposition (C.3m).

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And this proves Proposition (C.3f). (C.3n) Example. Take µ = (4, 2, 1, 1) ∈ Λ+ (4, 8), and U as given in (C.3o). Then U is a µ-tableau. Now take x1 = 1. We calculate P = U ← x1 by the methods of §B.4. This tableau also is given in (C.3o). It is a λ-tableau, where λ = (4, 2, 2, 1) ∈ Λ+ (4, 9).

(C.3o)

1 1 2 2 U= 2 4 , 3 4

1 1 1 2 P = 2 2 . 3 4 4

The parameters for the insertion of x1 = 1 are as follows (see (B.3b)): z = 3 and x1 = 1 (= p1,k(1) ), x2 = 2 (= u1,k(1) = p2,k(2) ), x3 = 4 (= u2,k(2) = p3,k(3) ), k(1) = 3,

k(2) = 2,

k(3) = 2.

It is rather easy to display the words ξ(a, t), for all (a, t) ∈ [µ] (see table C.4). By (C.3m)(i),(ii) we know that ξ(1, 4) = KU | x1 , and ξ(2, 1) = KP ; so write in these words. If (a, t) ≤ (z, µz ) then ξ(a, t) = KP by (C.3i), and this gives us ξ(3, 1) and ξ(4, 1). If (a, t) > (z, µz ), determine the pivot of ξ(a, t), using (C.3j) or (C.3k) as appropriate. For example the pivot of ξ(1, 4) is (x1 , (1, 4)+) = (x1 , (9)), and the pivot of ξ(1, 2) is (x2 , (1, 2)). For each ξ(a, t), we have underlined the ﬁrst term (xa or xa+1 ) in the pivot of ξ(a, t) in table C.4. Proof of the “only if ” part of Knuth’s theorem. Suppose i ∈ I(n, r), and for each s ∈ {0, 1, . . . , r} deﬁne Ps (i) = P (i1 . . . is ) (take P0 (i) to be the empty tableau). Let s ∈ {1, 2, . . . , r} and let U = Ps−1 (i) and x = is . Then Proposition (C.3f) provides a sequence of words ξ(a, t), and hence a sequence of basic moves taking the word KPs−1 (i) | is to the word K(Ps−1 (i) ← is ) = KPs (i). Using the notation of §B.7 (the “ladder”), we may now construct a sequence of basic moves taking KPs−1 | is is+1 . . . ir to KPs | is+1 . . . ir ; we simply use the sequence of words ξ(a, t) | is+1 . . . ir in place of the ξ(a, t). Since we can do this for each s = 1, 2, . . . , r, we deduce the following fundamental proposition: (C.3p) Proposition. Given i ∈ I(n, r), there is a sequence of basic moves in I(n, r) which takes i to KP (i). It is now easy to prove (C.3e), which is the “only if” part of Knuth’s theorem (C.3a). For given i, j ∈ I(n, r) such that P (i) = P (j), we use (C.3p) to make two sequences of basic moves, one taking i to KP (i) and one taking j to KP (j) = KP (i). Then the ﬁrst of these sequences, followed by the “reverse” of the second, takes i to j.

C.4 The “if” part of Knuth’s theorem

Place

(4, 1)

(3, 1)

(2, 1)

(2, 2)

(1, 1)

(1, 2)

(1, 3)

(1, 4)

(9)

Move

ξ(1, 4)

u4,1

u3,1

u2,1

u2,2

u1,1

u1,2

u1,3

u1,4

ξ(1, 3)

u4,1

u3,1

u2,1

u2,2

u1,1

u1,2

x2

x1 p1,4 = p1,3 = u1,4

K

ξ(1, 2)

u4,1

u3,1

u2,1

u2,2

u1,1

x2

p1,2

p1,3

p1,4

K

ξ(1, 1)

u4,1

u3,1

u2,1

u2,2

x2

p1,1 p1,2 = u1,1

p1,3

p1,4

−

ξ(2, 2)

u4,1

u3,1

u2,1

x3

x2 p1,1 = p2,2

p1,2

p1,3

p1,4

K

p4,1 p3,1 x3 p2,1 = u4,1 = u3,1 = p3,2

p2,2

p1,1

p1,2

p1,3

p1,4

−

ξ(3, 1)

p4,1

p3,1

p3,2

p2,1

p2,2

p1,1

p1,2

p1,3

p1,4

−

ξ(4, 1)

p4,1

p3,1

p3,2

p2,1

p2,2

p1,1

p1,2

p1,3

p1,4

−

ξ(2, 1)

x1

107

K

Table C.4. The sequence of words associated to Schensted insertion.

C.4 The “if ” part of Knuth’s theorem Let n, r be positive integers. In this section we will prove the “if” part of Knuth’s theorem (C.3a), that is: if i, j ∈ I(n, r) can be connected by a sequence of basic moves as in (C.3b), then i ∼ j (i.e. P (i) = P (j)). Clearly it will be enough to prove: (C.4a) If i and j in I(n, r) are connected by a basic move, then P (i) = P (j). For a standard tableau U and letters x1 , . . . , xk , we write U ← x1 x2 · · · xk = (· · · ((U ← x1 ) ← x2 ) · · · ) ← xk to ease the reading. We will prove the following (C.4b) Proposition (see [34, pp. 721, 722]). Let U be a standard tableau with entries drawn from n, and let a, b, c ∈ n. (1) If a < b ≤ c, then U ← bac = U ← bca. (2) If a ≤ b < c, then U ← acb = U ← cab.

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This proposition implies (C.4a), by means of a simple induction on the length r of i and j. Our proof of the proposition builds on Schensted’s original description of the insertion process U ← x, which reads as follows. Let U be a µ-tableau and x be a letter. If U is the empty tableau or u1,µ1 ≤ x (so that the insertion sequence (B.3b) has length z = 1), then U ← x is obtained from U by appending the letter x to the ﬁrst row of U : x U ←x=

If U is not empty and x < u1,µ1 (so that the insertion sequence (B.3b) has length z > 1), then choose k ≤ µ1 minimal with x < u1,k and set y = u1,k . The tableau U ← x has ﬁrst row (u1,1 , . . . , x, . . . , u1,µ1 ) (with x in column k), ˜ ← y. Here U ˜ is the while the remaining rows of U ← x are given by U “sub-tableau” of U obtained from U by removing the ﬁrst row. In illustrative terms: k x U ←x=

˜ ←y U

Of course, this description of U ← x follows directly from our description (B.3d). The idea of proof for the proposition is this. In either case (1) or (2), check that the ﬁrst rows of the two tableaux shown coincide. Then consider the tableaux obtained by removing the ﬁrst rows. If any of the three letters a, b, c does not bump a letter into the second row, then it is fairly easy to see that these “sub-tableaux” are equal. If all three letters a, b, c bump letters into the second row—x, y, z, say—then the letters x, y, z can be shown to satisfy either (1) or (2). We can then conclude by induction on the number of rows of U . Before we give the proof of (C.4b), let’s look at two examples. Example 1. Let n = 5, a = 1, b = 1 and c = 3 (so that part (2) of the proposition applies), and consider the tableau

C.4 The “if” part of Knuth’s theorem

109

1 1 2 3 3 4 U= 2 2 3 4 3 4 4 5 We have ˜= 2 2 3 4 . U 3 4 4 5 Let us concentrate on the ﬁrst rows of U ← acb and U ← cab. We get 1 1 1 1 3 3 U ← acb = U ← 131 = ˜ U ← 243 from Schensted’s inductive description, because a = 1 bumps x = 2 from the ﬁrst row of U , c = 3 bumps z = 4 from the ﬁrst row of U ← a, and b = 1 bumps y = 3 from the ﬁrst row of U ← ac. Similarly, we get 1 1 1 1 3 3 U ← cab = U ← 311 = ˜ U ← 423 ˜ ← 243 = U ˜ ← 423 (by induction on the Applying (C.4b)(2), it follows that U number of rows), hence also U ← 131 = U ← 311. Example 2. Let n = 5 again, a = 1, b = 1 and c = 2 (so that part (2) of proposition (C.4b) applies again), and consider the tableau 1 1 3 3 3 3 U= 2 3 4 4 4 4 5 5 5 We have ˜= 2 3 4 4 4 . U 4 5 5 5 In this case, we get 1 1 1 1 3 3 U ← acb = U ← 121 = ˜ U ← 332 from Schensted’s inductive description, because a = 1 bumps x = 3 from the ﬁrst row of U , c = 2 bumps z = 3 from the ﬁrst row of U ← a, and b = 1 bumps y = c = 2 from the ﬁrst row of U ← ac. Similarly, we get 1 1 1 1 3 3 U ← cab = U ← 211 = ˜ U ← 323 ˜ ← 332 = U ˜ ← 323 (by This time applying (C.4b)(1), it follows that U induction on the number of rows), hence also U ← 121 = U ← 211.

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Proof of Proposition (C.4b). The proof is done by induction on the number m of rows of U . If m = 0, then U is the empty tableau and (1)

U ← bac =

a c = U ← bca if a < b ≤ c, b

(2)

U ← acb =

a b = U ← cab if a ≤ b < c. c

Thus (C.4b) holds in case m = 0. Suppose m > 0. Let µ denote the shape of U and write U = (ux,y )(x,y)∈[µ] . ˜ be the tableau obtained from U by removing the ﬁrst row. Furthermore, let U We shall prove the parts (1) and (2) separately. Proof of part (1) of Proposition (C.4b). Assume we have a, b, c ∈ n such that a < b ≤ c. We want to prove that U ← bac = U ← bca. To ﬁnd the tableau W := U ← b, let b bump u1,l into row (2), where l is the smallest element of {1, . . . , µ1 , µ1 + 1} such that b < u1,l . This means: (i)

row (1) of W is the same as row (1) of U except at place (1, l), where w1,l = b, and ˜ =U ˜ ← y, (ii) the tableau obtained by removing the ﬁrst row of W , is W where y := u1,l . (If l = µ1 + 1, so that y = ∞, we make the convention ˜ .) that inserting ∞ into row (2) has no eﬀect on U

W =U ←b l b ˜ ←y U

U ← ba k a ˜ ← yx U

U ← bc

l b

l b

p c

˜ ← yz U

Table C.5. Inserting b, a and b, c into U when a < b ≤ c.

To ﬁnd the tableaux U ← ba = W ← a and U ← bc = W ← c, deﬁne k to be the smallest element of {1, . . . , µ1 , µ1 + 1} and p to be the smallest element of {1, . . . , µ1 + 1, µ1 + 2} such that a < w1,k and c < w1,p , respectively. (The case p = µ1 + 2 only occurs when k = µ1 + 1.) Then (iii) k ≤ l (because a < b = w1,l ), and

C.4 The “if” part of Knuth’s theorem

111

(iv) l < p (because w1,l = b ≤ c < w1,p ). Make U ← ba from W by letting a bump x := w1,k into row (2); make U ← bc from W by letting c bump z := w1,p into row (2). The resulting tableaux are shown in table C.5. To ﬁnd U ← bac, insert c into the tableau W := U ← ba. First ﬁnd the smallest p in {1, . . . , µ1 , µ1 + 1, µ1 + 2} such that c < w1,p . But any p such that c < w1,p is > l (because w1,l = b ≤ c), and all the entries w1,s in row (1) of W such that s ≥ l, coincide with the corresponding entries w1,s in row (1) of W , because the process which takes W to W = W ← a aﬀects only the part of row (1) to the left of (1, l). Therefore p = p , and U ← bac is shown in the left pane of table C.6. An entirely similar argument gives U ← bca, using the fact that the process which takes W to W = W ← b aﬀects only the part of row (1) to the right of (1, l); this tableau is shown in the right pane of table C.6. We next prove that x < y ≤ z. First, to prove x < y, observe U ← bac k a ˜ ← yxz U

l b

U ← bca p c

k a

l b

p c

˜ ← yzx U

Table C.6. Inserting b, a, c and b, c, a into U when a < b ≤ c.

that (iii) gives x = w1,k ≤ w1,l = b < u1,l = y. Second, to prove y ≤ z, observe that (iv) gives y = u1,l ≤ u1,p = z. This argument is also valid when y = ∞, because then z = ∞ as well. ˜ ← yxz = U ˜ ← yzx by the induction hypothesis; and It follows that U table C.6 gives the desired result U ← bac = U ← bca. There is still one “loose end” to be tidied up! Namely it can happen that k = l; the argument above still works, but we must re-draw the ﬁrst rows of the tableaux shown in table C.6. Each of these rows (which are equal) looks like k=l p c a This concludes the proof of Proposition (C.4b)(1). Proof of part (2) of Proposition (C.4b). We now assume that a ≤ b < c and show that U ← acb = U ← cab. To ﬁnd the tableaux U ← a and U ← c, deﬁne k and p to be the smallest elements of {1, . . . , µ1 , µ1 + 1} such that a < u1,k and c < u1,p , respectively.

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C Schensted and Littelmann operators

As usual, make V := U ← a from U by letting a bump x := u1,k into row (2), and make W := U ← c from U by letting c bump z := u1,p into row (2). Then k ≤ p (because a < c). We consider two cases: Case 1. k < p. Then k ≤ µ1 , hence the ﬁrst row of V has µ1 entries. To ﬁnd the tableau U ← ac = V ← c deﬁne p to be the smallest element of {1, . . . , µ1 , µ1 + 1} such that c < v1,p . But any p with c < v1,p is > k (because v1,k = a < c), and all the entries v1,s in row (1) of V with s > k, coincide with the corresponding entries u1,s of U (because the ﬁrst rows of U and V diﬀer only at place (1, k)). It follows that p = p , and c bumps the letter z = u1,p of V into row (2). An entirely similar argument shows that a bumps the letter x = u1,k = w1,k of W into row (2). The tableaux V ← c and W ← a are shown in table C.7. V ← c = U ← ac k a

p c

˜ ← xz U

W ← a = U ← ca k a

p c

˜ ← zx U

Table C.7. Inserting a, c and c, a into U when a ≤ b < c and k < p.

The ﬁrst rows of V ← c and W ← a coincide. Hence b bumps the same letter y = v1,l = w1,l into row (2) of V ← c and W ← a. Furthermore, it follows from a ≤ b < c that k < l ≤ p. The tableaux U ← acb = V ← cb and U ← cab = W ← ab are displayed in table C.8. U ← acb k a ˜ ← xzy U

l b

U ← cab p c

k a

l b

p c

˜ ← zxy U

Table C.8. Inserting a, c, b and c, a, b into U when a ≤ b < c and k < p.

C.4 The “if” part of Knuth’s theorem

113

We next prove that x ≤ y < z. First, we have x = u1,k ≤ u1,l = y if l < p, and x = v1,k ≤ v1,p−1 ≤ c = y if l = p. Second, y = v1,l ≤ v1,p = c < u1,p = z. ˜ ← xzy = U ˜ ← zxy by the induction hypothesis; and It follows that U table C.8 gives U ← acb = U ← cab in case k < p. Case 2. k = p. Then x = u1,k = u1,p = z. To ﬁnd the tableau V ← c in this case, deﬁne p to be the smallest element of {1, . . . , µ1 + 1, µ1 + 2} such that c < v1,p . But v1,k = a < c < u1,k ≤ u1,k+1 = v1,k+1 , hence p = k + 1. Therefore c bumps the letter w = v1,k+1 = u1,k+1 of V into row (2). To ﬁnd the tableau W ← a, note that w1,p−1 = u1,p−1 ≤ a < c = w1,p . Therefore a bumps the letter c = w1,p of W into row (2). The tableaux V ← c and W ← a are shown in table C.9. V ← c = U ← ac

W ← a = U ← ca

k a c ˜ ← zw U

k a ˜ ← zc U

Table C.9. Inserting a, c and c, a into U when a ≤ b < c and k = p.

From v1,k = a ≤ b < c = v1,k+1 it follows that b bumps the letter c (in column k + 1) of V ← c into row (2). From w1,k = a ≤ b < c < u1,k ≤ u1,k+1 = w1,k+1 it follows that b bumps the letter w = u1,k+1 into row (2) of W ← a. The resulting tableaux V ← cb = U ← acb and W ← ab = U ← cab are shown in table C.10. U ← acb

U ← cab

k a b

k a b

˜ ← zwc U

˜ ← zcw U

Table C.10. Inserting a, c, b and c, a, b into U when a ≤ b < c and k = p.

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C Schensted and Littelmann operators

It is clear that c < z ≤ w, because c < u1,p = z ≤ u1,p+1 = u1,k+1 = w. This argument is also valid when z = ∞, for then w = ∞ as well. Therefore ˜ ← zwc = U ˜ ← zcw, and part (1) of Proposition (C.4b) implies that U so U ← acb = U ← cab. This concludes the proof of Proposition (C.4b), hence of Knuth’s theorem (C.3a).

C.5 Littelmann operators on tableaux Suppose we have λ ∈ Λ+ (n, r), c ∈ {1, 2, . . . , n−1} and a λ-tableau P . The operator f˜c does not act on P , but it does act on the word KP . Theorem (C.5b) below will show that there is a unique tableau P˜ such that K P˜ = f˜c (KP ). It is reasonable to deﬁne f˜c (P ) to be P˜ . In (C.3g), we regarded the entries in the words KU | x1 and K(U ← x1 ) as indexed by the r-element set [µ] ∪ {(r)}. More generally, we can take any ordered r-element set T = {τ1 , τ2 , . . . , τr } such that τ1 < τ2 < · · · < τr , and use T to index the entries in a word i of length r. This means that if i is a word of length r, we write i = iτ1 iτ2 . . . iτr . In this section, it will be convenient to take T = [λ], because [λ] indexes the entries of the word KP (see §C.2). For the moment, let T = {τ1 , τ2 , . . . , τr } be an arbitrary r-element set with τ1 < τ2 < · · · < τr . Then all the deﬁnitions for f˜c given in §A.3 translate into deﬁnitions for words indexed by T in a trivial manner (we leave it to the reader to make the analogous translations for e˜c ). In case T = [λ] these deﬁnitions appear as follows. First deﬁne ωc,c+1 = ω : n → Z as in §A.3, so that ω(ν) = 1, −1 or 0 according as ν = c, c + 1 or ν ∈ / {c, c + 1}. = hKP : [λ] ∪ {0} → Z is given so that hKP (0) = 0, while The map hKP c for any t ∈ [λ] we deﬁne (C.5a) hKP (t) := ω(pa,b ), (a,b)≤t

the order ≤ being that given by (C.2a). : t ∈ [λ] }. Let M = McKP be the largest element of the set {0} ∪ { hKP c If M = 0 deﬁne f˜c (KP ) := ∞, or say that “f˜c (KP ) is undeﬁned”. If M = 0, let q = qcKP be the least element t of [λ] such that hKP (t) = M . Then there must hold pa,b = c, where q = (a, b); see (A.3c). In this case we deﬁne f˜c (KP ) to be the word obtained from KP by changing the entry pa,b = c to c + 1; all other entries in KP are left unchanged. The next theorem shows that if f˜c (KP ) = ∞, it is possible to deﬁne a tableau f˜c (P ) in such a way that K(f˜c P ) = f˜c (KP ) 3 3

Some authors identify the tableau P with the word KP , and view theorem (C.5b) as justiﬁcation of this practice. But in this Appendix we will be cautious (perhaps over-cautious!) and we do not make this identiﬁcation.

C.5 Littelmann operators on tableaux

115

(C.5b) Theorem. Let λ ∈ Λ+ (n, r), c ∈ {1, 2, . . . , n − 1} and P be a λ-tableau. Using the deﬁnitions above, assume M = 0, and deﬁne q = (a, b) to be the least place (in the order (C.2a)) such that h((a, b)) = M . We know from (A.3c) that pa,b = c. Then we have also (1) If (a, b + 1) ∈ [λ], then pa,b+1 ≥ c + 1. (2) If (a + 1, b) ∈ [λ], then pa+1,b > c + 1. (3) If we change P to P˜ by changing the entry pa,b = c to p˜a,b = c + 1, and leaving unchanged all the other entries in P , we get a λ-tableau P˜ which is standard. (4) K P˜ = f˜c (KP ). Proof. (1) Since P is standard, pa,b+1 ≥ pa,b = c. If pa,b+1 < c + 1, we would KP have pa,b+1 = c. This gives hKP c ((a, b + 1)) = hc ((a, b)) + ω(c) = M + 1, contradicting the deﬁnition of M . So there must hold pa,b+1 ≥ c + 1. (2) Since P is standard, we must have pa+1,b > c. Unless pa+1,b > c + 1, we have pa+1,b = c+1. We shall show that this leads to a contradiction. Table C.11 shows the rows (a) and (a+1) of P , and their entries in certain columns. Let b ···

b − 1

b

···

b

b+1

···

a

···

c

···

c

≥c

···

a+1

···

pa+1,b +1 c + 1 · · ·

c + 1 > c + 1 ···

Table C.11. Rows (a) and (a + 1) of P .

denote the leftmost of all columns such that pa,b = c. Since a ≥ c + 1, entries in row (a + 1) to the right of column (b) are all > c + 1. The entries in the same row, in columns (b ), . . . , (b), are all equal to c + 1. This is because such an entry pa+1,b is left of pa+1,b = c + 1, and is also > pa,b = c. From the deﬁnition (C.5a) we deduce (C.5c) hKP (a, b) = hKP (a + 1, b − 1) + X + Y + Y ∗ + Z, where X =

ω(pa+1,x ),

Y =

b ≤x≤b

Y∗ =

1≤x≤b

ω(pa+1,x ),

b+1≤x≤λa+1

ω(pa,x )

Z =

ω(pa,x ).

b ≤x≤b

But for b + 1 ≤ x ≤ λa+1 all the entries pa+1,x are > c + 1, hence all the summands ω(pa+1,x ) = 0, therefore Y = 0. Similarly Y ∗ = 0 because all the elements pa,x (for 1 ≤ x ≤ b − 1) are < c. Finally X + Z = 0 because X + Z is a sum of pairs ω(c) + ω(c + 1) = 0. Therefore (C.5c) implies that hKP (a, b) = hKP (a + 1, b − 1). But this contradicts our deﬁnition

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C Schensted and Littelmann operators

of (a, b) as the least place in [λ] such that hKP (a, b) = M . This proves part (2) of Theorem (C.5b). Part (3) is now proved, since (1) and (2) show that P˜ is standard. Then (4) follows. (C.5d) Example. Let λ = (2, 2, 0, . . . , 0), regarded as an element of Λ+ (n, 4) for some n ≥ 4. Consider the λ-tableau P = 2 2 . Then KP = 3422. Now 3 4

r

1

2

3

4

t

(2, 1)

(2, 2)

(1, 1)

(1, 2)

3

4

2

2

−1

−1

0

1

3

4

2

3

(KP )r hKP (r) (f˜c (KP ))r

Table C.12. Illustration of Theorem (C.5b).

let c = 2. Calculate f˜c (KP ) using table C.12. Notice that we have shown two set 4 and [λ], either of which can be used to index the letters of the word KP . We see that q KP = 4, or equivalently, q KP = (1, 2). Therefore f˜c (P ) = 2 3 . 3 4

C.6 The proof of Proposition B In this section we shall prove the fact, fundamental for our work, that the operation KP commutes with all the Littelmann operators f˜c . In other words, we shall prove the Proposition B. Let i ∈ I(n, r) and c ∈ {1, 2, . . . , n−1} such that f˜c (i) = ∞. Then f˜c (KP (i)) = ∞ and (C.6a) f˜c (KP (i)) = KP (f˜c (i)). For i, j ∈ I(n, r) we write iK j (respectively, iK j) if i and j are connected by a basic move of type K (respectively, K ); see (C.3c), (C.3d). The proof of Proposition B is based on the following two lemmas. (C.6b) Lemma. Let i, j ∈ I(n, r), and suppose that j is obtained from i by a basic move, say, i = (. . . , ik , ik+1 , . . .),

j = (. . . , ik+1 , ik , . . .),

ik < ik+1 .

Then M i = M j , and there are the following alternatives for q i and q j :

C.6 The proof of Proposition B

117

(a) If q i ∈ / {k, k + 1}, then q j = q i . i (b) If q = k + 1, then q j = k. (c) If q i = k, then either ik+1 = c + 1, iK j and q j = k + 2, or ik+1 = c + 1 and q j = k + 1. Proof. Set x := ik and z := ik+1 , so that x < z. We observe that (i) hjc (ν) = hic (ν) for all ν = k. This follows directly from the deﬁnition (A.3a) of hic and the fact that the words i and j are identical at all places except ν = k and ν = k + 1. Next we show that (ii) M i = M j . Suppose ﬁrst that q i = k. Then M j ≥ hjc (q i ) = hic (q i ) = M i , by (i). Assume that M j > M i . Then q j = k, by (i). This implies z = c and x < c, by (A.3c)(i). It follows that M i ≥ hic (k + 1) = hjc (k + 1) = hjc (k) = M j , a contradiction. This shows M i = M j in case q i = k. Suppose now q i = k. Then x = c, by (A.3c)(i). Hence M i ≥ M j , by (i). Assume that M i > M j . Then M i > hjc (k + 1) = hic (k + 1) = M i + ωc (z), which implies z = c + 1. If iK j, that is, x < ik−1 ≤ z, then ik−1 = c + 1 and thus hic (k) = hic (k − 2) + ωc (ik−1 ) + ωc (x) = hic (k − 2). This contradicts the minimal choice of q i . If iK j, that is, if x ≤ ik+2 < z, then ik+2 = c and thus M j ≥ hjc (k + 2) = hic (k + 2) = hic (k) = M i , again a contradiction. This shows M i = M j also in case q i = k, and (ii) is proved. / {k, k + 1}. Assume q j = q i , Now, for the proof of (a), suppose that q i ∈ then q j = k, by (i), and thus z = c. It follows that x < c and therefore, by (ii), hic (k + 1) = hjc (k) = M j = M i . This implies q i < k, since q i = k, k + 1, hence also q j < k, by (i)—a contradiction. Part (a) is proved. Now let q i = k + 1. Then z = c and x < c, and we get from (ii) that hjc (k) = hjc (k + 1) = hic (k + 1) = M i = M j . It follows that q j ≤ k. In fact, by (i), q j = k. This implies (b). Consider ﬁnally the case where q i = k. Then x = c, and q j ≥ k, by (ii). Suppose additionally that z = c + 1. Then z > c + 1, and it follows that hjc (k) < hjc (k + 1) = hic (k + 1) = hic (k) = M i = M j . But this implies q j = k + 1. To conclude, let z = c + 1. Assume iK j, so that x < ik−1 ≤ z. Then ik−1 = c + 1 and hic (k) = hic (k − 2) + ωc (ik−1 ) + ωc (x) = hic (k − 2). This contradicts the minimal choice of q i . It follows that iK j as asserted, that is x ≤ ik+2 < z. Hence ik+2 = c. Direct veriﬁcation gives hjc (k) = hic (k) − 2 = M i −2 = M j −2, hjc (k +1) = M j −1 and hjc (k +2) = M j . Therefore q j = k +2 as claimed in (c). (C.6c) Lemma. Let i, j ∈ I(n, r), and suppose j is obtained from i by a basic move. If f˜c (i) = ∞, then f˜c (j) = ∞, and f˜c (j) is obtained from f˜c (i) by a basic move. There is a corresponding statement (and proof ), with e˜c replacing f˜c .

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Proof. Thanks to symmetry in i and j, we may assume that either there exist components y = ik−1 , x = ik , z = ik+1 of i such that (K )

i = (. . . , y, x, z, . . .),

j = (. . . , y, z, x, . . .),

x < y ≤ z,

or components x = ik , z = ik+1 , y = ik+2 of i such that (K )

i = (. . . , x, z, y, . . .),

j = (. . . , z, x, y, . . .),

x ≤ y < z.

Suppose f˜c (i) = ∞. Then M j = M i > 0, by Lemma (C.6b). This implies that f˜c (j) = ∞ as well. We now consider the three cases listed in Lemma (C.6b). / {k, k + 1}. Then q i = q j , by Lemma (C.6b). Hence x and z Case (a). q i ∈ remain unchanged when we apply f˜c to i and j. The claim follows directly if y is not changed, either. Suppose that y = c, iK j and q i = k − 1. Then we get4 f˜c (i) = (. . . , c + 1, x, z, . . .),

f˜c (j) = (. . . , c + 1, z, x, . . .),

x < c ≤ z.

However, we have z ≥ c + 1, since in case z = c we get hjc (k) = hjc (k − 1) + 1, and this contradicts the maximality of hjc (q j ). Hence f˜c (i)K f˜c (j). Suppose now that y = c, iK j and q i = k + 2. Then we get f˜c (i) = (. . . , x, z, c + 1, . . .),

f˜c (j) = (. . . , z, x, c + 1, . . .),

x ≤ c < z.

However, we have z > c + 1, since in case z = c + 1 we get hic (k) = hic (k + 2), and this contradicts the minimal choice of q i . Hence f˜c (i)K f˜c (j). Case (b). q i = k + 1. Then q j = k and z = c, hence f˜c (i) = (. . . , x, c + 1, . . .),

f˜c (j) = (. . . , c + 1, x, . . .).

If iK j, then x ≤ y < z < c + 1, therefore f˜c (i)K f˜c (j). In case iK j, we get x < y ≤ z < c + 1, hence f˜c (i)K f˜c (j). Case (c). q i = k. Here x = c, and we need to consider the alternative given in Lemma (C.6b)(c). Suppose ﬁrst that z = c + 1, that iK j and q j = k + 2. Then y = c since c = x ≤ y < z = c + 1. Hence f˜c (i) = (. . . , c + 1, c + 1, c, . . .),

f˜c (j) = (. . . , c + 1, c, c + 1, . . .).

We get that f˜c (j)K f˜c (i). The case where z = c + 1 and q j = k + 1 remains. Here z > c + 1 and f˜c (i) = (. . . , c + 1, z, . . .),

f˜c (j) = (. . . , z, c + 1, . . .).

Suppose iK j, then y ≥ c + 1 since otherwise hic (k + 2) = hic (k) + 1. Therefore c + 1 ≤ y < z and f˜c (i)K f˜c (j). Similarly, if iK j, then y > c + 1, since otherwise hic (k − 2) = hic (k). Hence c + 1 < y ≤ z and f˜c (i)K f˜c (j). 4

Those values which were changed by f˜c are underlined.

C.6 The proof of Proposition B

119

We are now in a position to give the Proof of Proposition B. From Proposition (C.2c), we get P (KP (i)) = P (i). Hence, by Theorem (C.3a), there exist words i(0) , i(1) , · · · , i(k−1) , i(k) ∈ I(n, r) such that i(0) = i, i(k) = KP (i), and i(ν) is obtained from i(ν−1) by a basic move. From Lemma (C.6c), it follows that f˜c (i(ν) ) = ∞ and that f˜c (i(ν) ) is obtained from f˜c (i(ν−1) ) by a basic move, for all ν ∈ {1, . . . , k}. Applying Theorem (C.3a) again, we get P f˜c (i) = P f˜c (i(0) ) = P f˜c (i(k) ) = P f˜c (KP (i)) , hence (∗) KP (f˜c (i)) = KP (f˜c (KP (i))). There is a standard tableau P˜ such that K P˜ = f˜c (KP (i)), by Theorem (C.5b)(4). And by Proposition (C.2c)(i), KP (K P˜ ) = K P˜ . Therefore (∗) becomes KP (f˜c (i)) = K P˜ = f˜c (KP (i))).

D Theorem A and some of its consequences

In what follows, n, r are ﬁxed positive integers.

D.1 Ingredients for the proof of Theorem A We shall prove Theorem A in the next section, but we must ﬁrst study some words in I(n, r) which play a special role for the action of the Littelmann operators. To describe these words, we need the following lemma, which is an immediate consequence of the deﬁnitions in §A.3. (D.1a) Lemma. If i ∈ I(n, r) and c ∈ {1, . . . , n − 1}, then (i) f˜c (i) = ∞ if and only if #{ ν ≤ t : iν = c } ≤ #{ ν ≤ t : iν = c + 1 } for all t ∈ {1, . . . , r}, and (ii) e˜c (i) = ∞ if and only if #{ ν ≥ s : iν = c } ≥ #{ ν ≥ s : iν = c + 1 } for all s ∈ {1, . . . , r}. We are interested in the words which satisfy (i) for all c. So we set Υ := i ∈ I(n, r) : f˜c (i) = ∞ for all c ∈ {1, . . . , n − 1} . Deﬁne an operator W : I(n, r) → I(n, r) by W (i1 i2 . . . ir ) = (n + 1 − i1 , n + 1 − i2 , . . . , n + 1 − ir ). Then a word i belongs to Υ if and only if W (i) is a “lattice permutation”1 . 1 This term is rather confusing, because we shall use it for words which may not be permutations! A word j ∈ I(n, r) is called a permutation if n = r and the entries in j are 1, 2, . . . , r in some order. Lattice permutations in this sense are used by D.E. Littlewood in the character theory of the symmetric group Sym(r) (see [36, page 67]). Lattice permutations in the present sense appear in [40] and [37].

122

D Theorem A and some of its consequences

Deﬁnition. A lattice permutation, in our language, is a word j ∈ I(n, r) such that (D.1b) #{ ν ≤ s : jν = 1 } ≥ #{ ν ≤ s : jν = 2 } ≥ ··· ≥ #{ ν ≤ s : jν = n − 1 } ≥ #{ ν ≤ s : jν = n }, for all s ∈ {1, . . . , r}. For example, the word j = 11122132, an element of I(3, 8), is a lattice permutation. The word i = 33322312 belongs to Υ, because W (i) = j. Similarly, we are interested in the words which satisfy condition (ii) in Lemma (D.1a) for all c, and we set T := i ∈ I(n, r) : e˜c (i) = ∞ for all c ∈ {1, . . . , n − 1} . In (A.3g)(2), the operator B : I(n, r) → I(n, r) was deﬁned by B(i1 i2 . . . ir−1 ir ) = ir ir−1 · · · i2 i1 . Thus a word i belongs to T if and only if B(i) is a lattice permutation. Deﬁne an operator C : I(n, r) → I(n, r) by C = BW = W B. Explicitly, C(i1 i2 . . . ir−1 ir ) = (n + 1 − ir , n + 1 − ir−1 , . . . , n + 1 − i2 , n + 1 − i1 ). Remarks. (i) All these operators have square equal to the identity in I(n, r). (ii) If i ∈ I(n, r) and Sch(i) = (λ(i), P (i), Q(i)), then λ(i) is the shape of i (see §C.1). The operator C preserves shape (i.e. λ(Ci) = λ(i), see (D.3g)), but the operators B and W do not. For example, using the tables in §E.1, we see that i = 221 has shape (2, 1, 0), but B(i) = 122 and W (i) = 223 both have shape (3, 0, 0). However, C(i) = 322 has the same shape as i. (D.1c) Lemma. The operator C induces a bijection T → Υ. Hence |T| = |Υ|. Proof. Let i ∈ T. Then B(i) is a lattice permutation, and W (C(i)) = B(i). This shows that C(i) ∈ Υ. Prove similarly that i ∈ T implies that C(i) ∈ Υ. From now on in this section, we ﬁx λ ∈ Λ+ (n, r). The tableaux Tλ and Zλ . Deﬁne two λ-tableaux as follows: (D.1d) Tλ = (Ts,t )(s,t)∈[λ] where Ts,t = s for all (s, t) ∈ [λ]; we denote the word KTλ by iλ . (D.1e) Zλ = (Zs,t )(s,t)∈[λ] where Zs,t = n − βt + s for all (s, t) ∈ [λ], and βt denotes the length of column t of Zλ ; we denote the word KZλ by iλ .

D.1 Ingredients for the proof of Theorem A

123

Example. If λ = (5, 3, 2, 0, 0) ∈ Λ+ (5, 10), then 1 1 1 1 1 3 3 4 5 5 and Zλ = 4 4 5 . (D.1f ) Tλ = 2 2 2 3 3 5 5 Notice that Tλ is our old friend from (4.3b), where it is called Tl . It is useful to think of Zλ as the tableau obtained from Tλ by subjecting it to two successive operations: ﬁrst reverse each column of Tλ , and secondly replace each entry x in the tableau by n + 1 − x. In our example, 1 1 1 1 1 −→ Tλ = 2 2 2 3 3

3 3 2 1 1 −→ 2 2 1 1 1

3 3 4 5 5 = Zλ . 4 4 5 5 5

Notation. Deﬁne Q(λ) to be the set of all (standard) λ-tableaux whose entries are 1, 2, . . . , r in some order. Recall from (C.1e) that I(Q, ≈) is the set of all words i ∈ I(n, r) such that Q(i) = Q, for each Q ∈ Q(λ). These sets are the equivalence classes for ≈. (D.1g) Theorem. Let Q ∈ Q(λ). Then: (i) There is a unique word i ∈ I(n, r) such that Q(i) = Q and i belongs to T. Moreover P (i) = Tλ . (ii) There is a unique word i ∈ I(n, r) such that Q(i) = Q and i belongs to Υ. Moreover P (i) = Zλ . (D.1h) Notation. For each Q ∈ Q(λ), denote the word i in (i) by iQ , and the word i in (ii) by iQ . Proof of Theorem (D.1g). (i) By Schensted’s Theorem (B.6a) there is a unique word i such that P (i) = Tλ and Q(i) = Q. We claim that this i belongs to T. Let c ∈ {1, 2, . . . , n}. From §A.3, we know that e˜c (i) = ∞ if and only if the function hic attains its maximum Mci at the last place in the word i, i.e. hic (r) = Mci . Now hic (r) is the sum of the ω(iν ) for ν = 1, 2, . . . , r (see (A.3a)). But the r entries in the word KP (i) form a permutation of KP (i) the r entries of i. Hence hc (r) = hic (r) = Mci . By Lemma (C.6b) and Proposition (C.3p), the words i and KP (i) give the same maximum, KP (i) KP (i) i.e. Mci = Mc . We can calculate Mc = McKTλ easily; it is λc − λc+1 , and it is attained at the last place (1, λ1 ) of KP (i). Therefore the maximum Mci of hic is also attained at the last place of i. This shows that e˜c (i) = ∞ for all c, hence i ∈ T. Now we must prove uniqueness: if j ∈ I(Q, ≈) ∩ T, then j = i. It is enough ec (j)), by Proposito prove that P (j) = Tλ . We know e˜c (KP (j)) = KP (˜ KP (j) tion B, hence e˜c (KP (j)) = ∞ for all c. So the height function hc takes its maximum value at “place r”, i.e. at place (1, λ1 ) ∈ [λ].

124

D Theorem A and some of its consequences

Consider the last entry t in the ﬁrst row of P (j); we must show that t = 1. If this is false, then t > 1. Consider the height functions for c = t − 1. The entry in P (j) at place (1, λ1 ) (which corresponds to place r in the word KP (j)) KP (j) KP (j) is c + 1. So we have hc (r) < hc (r − 1), a contradiction. Hence all entries of the ﬁrst row of P (j) are equal to 1. Next consider the last entry, t say, in the sth row of P (j). We have t ≥ s since P (j) is standard. KP (j) Suppose t > s, and set c = t − 1. As before, the height function hc does not take its maximum value at r: it is constant on the letters of rows 1 up to s − 1, and its value at the last place of row s, say x, is less than its value at the place immediately preceding this last place. This is a contradiction. We have proved that P (j) = Tλ , and since we have assumed Q(j) = Q, the words j and i must be equal. If we let λ vary over all partitions in Λ+ (n, r), then this shows that |T| is equal to the number of standard tableaux having entries 1, 2, . . . , r in some order; this is also the total number of ≈-classes in I(n, r). (ii) By Schensted’s theorem (B.6a) there is a unique word i ∈ I(n, r) with Q(i) = Q and P (i) = Zλ . Using Lemma (C.6b) and Proposition B, as in the proof of (i), it is quite easy to see that i ∈ Υ. Therefore each ≈-class of words of shape λ contains at least one element of Υ. But as was noted above, if we let λ vary, the number of ≈-classes is |T|, which is equal to |Υ| by Lemma (D.1c). This implies that each ≈-class contains a unique element of Υ; this must be the word i having P (i) = Zλ and Q(i) = Q. This completes the proof of Theorem (D.1g). (D.1i) Proposition. iλ = iQ(λ) and iλ = iQ(λ) . Proof. Taking Y = Tλ in propositions (C.2c) and (C.2h), we get P (iλ ) = Tλ and Q(iλ ) = Q(λ) . However (D.1g) and (D.1h) say that P (iQ(λ) ) = Tλ and Q(iQ(λ) ) = Q(λ) . Therefore iλ = iQ(λ) , by Schensted’s theorem (B.6a). A similar proof, using Zλ in place of Tλ , gives iλ = iQ(λ) .

D.2 Proof of Theorem A We shall now prove the Theorem A described in the introduction (see (A.4a)): (D.2a) Theorem A. Let i, j ∈ I(n, r). Then i ≈ j if and only if there is a ﬁnite sequence of words (elements of I(n, r)): i(1), i(2), . . . , i(s) such that i(1) = i, i(s) = j and for each adjacent pair i(ν), i(ν + 1) either there exists an element c ∈ {1, . . . , n − 1} such that f˜c (i(ν)) = i(ν + 1), or there exists an element c ∈ {1, . . . , n − 1} such that e˜c (i(ν)) = i(ν + 1).

D.2 Proof of Theorem A

125

It is clear that the “if” part of this theorem is equivalent to the following (D.2b) Proposition. If i, j ∈ I(n, r) and c ∈ {1, . . . , n − 1} such that either f˜c (i) = j, or e˜c (i) = j, then Q(i) = Q(j). Proof. Suppose there is an element c ∈ {1, . . . , n − 1} such that j = f˜c (i). This implies that f˜c (i) = ∞. (a) We claim that Q(i) and Q(j) have the same shape. Equivalently, we claim that P (i) and P (j) have the same shape. Let P (i) have shape λ. By Proposition B (see (C.6b)) we know that KP (j) = KP (f˜c (i)) = f˜c (KP (i)). Now take P = P (i) in Theorem (C.5b). This says that there is a λ-tableau P˜ such that K P˜ = f˜c (KP ). Therefore KP (j) = K P˜ . This shows that P (j) has the same shape λ as P˜ , which is the shape of P (i). This proves claim (a). (b) We shall use induction on r to prove that Q(i) = Q(j). If r = 1 then i and j are one-letter words, and Q(i) = Q(j) follows from (B.2b). Assume now that r > 1. Write i = i ir and j = j jr , where i = i1 · · · ir−1 and j = j1 · · · jr−1 lie in I(n, r − 1). There is a place q ∈ {1, 2, . . . , r} such that iq = c, jq = c+1 and iν = jν for all ν = q (see (A.3e)). We either have q < r, or q = r. If q < r, then j = f˜c (i ), hence Q(i ) = Q(j ) by the induction hypothesis. If q = r, then j = i and clearly Q(i ) = Q(j ). It follows that Q(i ) and Q(j ) have the same shape, µ say, in either case. Let λ be the shape of Q(i). By (a), λ is also the shape of Q(j). To get Q(i) from Q(i ), one puts r into the unique place which, when added to [µ], gives [λ]. To get Q(j) from Q(j ) = Q(i ), one puts r in the unique place which, when added to [µ], gives [λ]. Hence Q(j) = Q(i). This completes the proof of Proposition (D.2b); and this proves the “if” part of Theorem A. Proof of the “only if ” part of Theorem A. We assume that i, j ∈ I(n, r) are such that Q(i) = Q(j); we must prove that there exists a sequence of words i = i(1), (2), . . . , i(s) = j with the properties listed in (D.2a). First make the (D.2c) Deﬁnition. Let i ∈ I(n, r). We deﬁne the size of i, denoted sz(i), by sz(i) := i1 + i2 + · · · + ir . This is a positive integer, and from the deﬁnitions of f˜c , e˜c it is clear that if f˜c (i) = ∞ then sz(f˜c (i)) = sz(i)+1, and if e˜c (i) = ∞ then sz(˜ ec (i)) = sz(i) − 1. We make the convention sz(∞) = 0. Let λ ∈ Λ+ (n, r) and Q ∈ Q(λ). Let w ∈ I(Q, ≈) (see (C.1e)), and let S(w) denote the set of all words of the form e˜c1 e˜c2 · · · e˜ct (w), where c1 , c2 , . . . , ct are arbitrary elements of {1, 2, . . . , n − 1}. We allow that t may be zero, in which case e˜c1 e˜c2 · · · e˜ct (w) = w. In general, e˜c1 e˜c2 · · · e˜ct (w) has size sz(w)−t. Let S be minimal amongst the sizes of the elements of S(w), and choose an element w := e˜c1 e˜c2 · · · e˜ct (w) of S(w) of size S (there may be many such w ). Then e˜c (w ) = ∞ for all c ∈ {1, 2, . . . , n − 1}, because if e˜c (w ) = ∞, then e˜c (w ) would be an element of S(w) of size S − 1.

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D Theorem A and some of its consequences

By Proposition (D.2b), all the elements of S(w) lie in I(Q, ≈). But then Theorem (D.1g) tells us that w = e˜c1 e˜c2 · · · e˜ct (w) = iQ . This implies that w = f˜c · · · f˜c f˜c (iQ ). In other words, any element w ∈ I(Q, ≈) can be t

2

1

f˜

joined to i by a ﬁnite sequence of steps of the form i(ν) −→ i(ν + 1); equivae˜ lently iQ can be joined to w by a sequence of steps of the form i(ν+1) −→ i(ν). So given i, j ∈ I(Q, ≈), we can join i to j by a sequence of the type described in the statement of Theorem A, by ﬁrst joining i to iQ and then joining iQ to j. This completes the proof of Theorem A. The arguments just given, together with Theorem (D.1g), provide valuable information on the ≈-classes. We summarize this in the Q

(D.2d) Proposition. Let λ ∈ Λ+ (n, r) and Q ∈ Q(λ). Then: (i) There is a unique word iQ in I(Q, ≈) lying in T, i.e. such that e˜c (i) = ∞ for all c ∈ {1, . . . , n − 1}. This word is speciﬁed by Sch(iQ ) = (λ, Tλ , Q). (ii) There is a unique word iQ in I(Q, ≈) lying in Υ, i.e. such that f˜c (i) = ∞ for all c ∈ {1, . . . , n − 1}. This word is speciﬁed by Sch(iQ ) = (λ, Zλ , Q). (iii) The following three conditions on a word i ∈ I(n, r) are equivalent (i.e. each condition implies the other two): (1) i ∈ I(Q, ≈). (2) There exist c1 , . . . , ct ∈ {1, . . . , n − 1} such that i = f˜c1 · · · f˜ct (iQ ). (3) There exist d1 , . . . , ds ∈ {1, . . . , n − 1} such that i = e˜d1 · · · e˜ds (iQ ). In (2) and (3), we allow t and s to be = 0, respectively. In these cases we interpret f˜c1 · · · f˜ct (iQ ) to be iQ and e˜d1 · · · e˜ds (iQ ) to be iQ , respectively. Proof. All the statements above can be deduced easily from Theorem (D.1g), Theorem (D.2a) (i.e. Theorem A) and the proof of Theorem (D.2a). Weights. Remember (see (A.3g)(3), or §3.1) that the weight wt(i) of a word i is the n-vector (w1 , . . . , wn ), where for each ν ∈ n, wν is the number of ρ ∈ r such that iρ = ν. Classical representation theory of GLn (C), which can be regarded as a sequel to classical invariant theory, uses weights extensively—they describe the (polynomial) representations Kλ of the diagonal subgroup Tn (C), see §3.2; then these are “induced” to give irreducible (polynomial) representations of GLn (C), see the end of Chapter 4. (D.2e) Remark. It is clear that wt(i) = wt(j), if i, j ∈ I(n, r) are such that j = iπ for some π in the symmetric group Sym(r) (the symmetric group is denoted G(r) in §2.1). In particular, wt(i) = wt(KP (i)) for any i ∈ I(n, r), because the entries in KP (i) are the same as the entries in i, apart from a place permutation π ∈ Sym(r). In the classical representation theory of GLn (C), which is essentially the representation theory of the Schur algebra S(n, r), the (isomorphism types of) simple modules are indexed by dominant weights, i.e. by the elements of Λ+ (n, r). We shall see in §D.4 that this holds also for the (isomorphism

D.3 Properties of the operator C

127

types of) simple modules for the Littelmann algebra L = L(n, r), although the argument is diﬀerent from that which applies to S(n, r). The weights of the elements of I(Q, ≈) have properties given in the next proposition. (D.2f ) Proposition. Let λ ∈ Λ+ (n, r) and Q ∈ Q(λ), then (i) wt(iQ ) = λ = (λ1 , . . . , λn ), (ii) wt(iQ ) = (λn , . . . , λ1 ), (iii) iQ (respectively, iQ ) is the only word in I(Q, ≈) having weight (λ1 , . . . , λn ) (respectively, (λn , . . . , λ1 )), and (iv) the weight ω of any word in I(Q, ≈) satisﬁes the inequalities (λ1 , . . . , λn ) ω (λn , . . . , λ1 ). (If ξ, η ∈ Λ(n, r), we write ξ η to mean that the diﬀerence ξ − η lies in the set U = α∈Σ Z+ α; see [33, page 3]). Proof. (i) From (D.1g)(i) we know that P (iQ ) = Tλ . Therefore wt(iQ ) is the same as the weight of KTλ (see Remark (D.2e)). It is very easy to see that wt(KTλ ) = (λ1 , . . . , λn ). (ii) In the same way, we deduce from (D.1g)(ii) that wt(iQ ) is the same as the weight (u1 , . . . , un ) of KZλ . So for each δ ∈ n, uδ is the number of pairs (s, t) ∈ [λ] such that n − βt + s = δ. For each t ∈ n, there is exactly one entry δ in column t of Zλ , if and only if 1 ≤ βt − (n − δ). Therefore uδ equals the number of columns of Zλ of lengths greater than or equal to n + 1 − δ. But this number is λn+1−δ . (iii) and (iv) Let i be any word in I(Q, ≈). By (D.2d) we know that there exist integers c1 , . . . , ct in {1, . . . , n − 1} such that i = f˜c1 · · · f˜ct (iQ ). From (A.3g)(3), we know that wt(i) = wt(iQ ) − αc1 ,c1 +1 − · · · − αct ,ct +1 . Therefore i iQ ; moreover the case wt(i) = wt(iQ ) = (λ1 , . . . , λn ) occurs only if t = 0 i.e. only if i = iQ . A similar argument shows that the weight of any word i ∈ I(Q, ≈) is wt(iQ ), with equality only if i = iQ .

D.3 Properties of the operator C First we want to understand how the action of C is related to the action of the Littelmann operators. Comparing the height functions of i and Ci. Fix i ∈ I(n, r), and consider the height function hic for some c ∈ {1, 2, . . . , n − 1}. This depends on the iν which are equal to c or c + 1. The operator C turns c and c + 1 into n − c + 1 and n − c, respectively. This suggests comparing hic with hCi n−c . Take some s ∈ {1, . . . , r}, then by deﬁnition (D.3a) hic (s) = #{ ν ≤ s : iν = c } − #{ ν ≤ s : iν = c + 1 }.

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D Theorem A and some of its consequences

Now write Ci = j1 . . . jr for a moment and consider (D.3b) hCi n−c (r − s) = #{ ρ ≤ r − s : jρ = n − c } −#{ ρ ≤ r − s : jρ = n − c + 1 }. We have jρ = n−ir−ρ+1 +1 for all ρ ∈ {1, . . . , r}. Furthermore, n−iν +1 = n−c if and only if iν = c + 1, and n − iν = n − c if and only if iν = c, for all ν ∈ {1, . . . , r}. So (D.3b) gives (D.3c) hCi n−c (r − s) = #{ ν ≥ s + 1 : iν = c + 1 } − #{ ν ≥ s + 1 : iν = c }. By using the notation: Πb := #{ ν ∈ Π : iν = b }, for every subset Π of {1, . . . , s}, and every element b of n, formula (D.3c) becomes (i) hCi n−c (r − s) = −{s + 1, . . . , r}c + {s + 1, . . . , r}c+1 . Also, by deﬁnition of the height function hic , we have (ii) hic (s) = {1, . . . , s}c − {1, . . . , s}c+1 . If we subtract (i) from (ii) we get (D.3d) If i ∈ I(n, r) and Y = hic (r), then hic (s) − hCi n−c (r − s) = Y for all s ∈ {0, . . . , r}. Example. Let n = 3, r = 5, c = 1, and consider i = 22111 ∈ I(n, r). Then Ci = 33322 and n − c = 2. The height functions hi1 and hCi 2 are hi1 (0), hi1 (1), hi1 (2), hi1 (3), hi1 (4), hi1 (5) = ( 0, −1, −2, −1, 0, 1),

Ci Ci Ci Ci Ci hCi (5), h (4), h (3), h (2), h (1), h (0) = (−1, −2, −3, −2, −1, 0). 2 2 2 2 2 2

Note that hi1 has the maximum at place r and hCi 2 has maximum value zero. (D.3e) Lemma. Let c ∈ {1, 2, . . . , n − 1}. Then, for each i ∈ I(n, r), we have C(˜ ec (i)) = f˜n−c (Ci) and C(f˜c (i)) = e˜n−c (Ci). Proof. Since C 2 is the identity, the second part follows from the ﬁrst. We prove the ﬁrst part. By (D.3d), we have a geometric description of how the height functions are ˜ = hCi , one reﬂects the graph of h in the related: given h = hic , then to ﬁnd h n−c ˜ vertical line x = r, and translates it in the “y-axis” direction so that h(0) = 0. i Explicitly, let s + t = r and Y = hc (r), then i hCi n−c (t) = hc (s) − Y.

D.4 The Littelmann algebra L(n, r)

129

˜ is a reﬂection about a vertical line, the last maximum of hi (at Since h c place q¯), becomes the ﬁrst maximum of hCi ¯). Furthermore, n−c (at place r − q ˜ is zero. This shows that if e˜c (i) = ∞ if q¯ = r then the maximum of h ˜ then fn−c (Ci) = ∞. Assume now that q¯ < r. By (A.3c) we know that iq¯+1 = c + 1, and e˜c (i) is obtained from i by replacing iq¯+1 = c + 1 by c. We get the word C(˜ ec (i)) = · · · (n − c + 1)(n − iq¯ + 1) · · · where the letters shown are at places r − q¯ and r − q¯ + 1. Now consider C(i) = · · · (n − c)(n − iq¯ + 1) · · · where the letters shown ˜ assumes its maximum at are at places r − q¯ and r − q¯ + 1. We know that h ˜ place r − q¯ for the ﬁrst time. So fn−c (Ci) replaces the letter n−c at place r − q¯ ec (i)). by n − c + 1. Hence f˜n−c (Ci) is equal to C(˜ From the deﬁnition of C we see immediately the following. (D.3f ) Lemma. If a word i ∈ I(n, r) has weight µ = (µ1 , . . . , µn ), then the word Ci has weight (µn , . . . , µ1 ). We want to show now: (D.3g) Lemma. The operator C preserves the shape. Proof. Let i ∈ I(n, r) and Q(i) = Q, and suppose Q has shape λ. Assume ﬁrst that i = iQ , then C(i) = iR for some standard tableau R. By (D.2f) we can identify the shapes of the words iQ and iR from their weights. The weight of iQ is λ, hence the weight of C(iQ ) is (nλ1 , (n − 1)λ2 , . . .). So iR also has shape λ. In general, by (D.2d)(iii), there are c1 , . . . , ct ∈ {1, 2, . . . , n − 1} such that i = f˜c1 · · · f˜ct (iQ ). From (D.3e) it follows that C(i) = e˜n−c1 · · · e˜n−ct (CiQ ). But we have already seen that CiQ has shape λ; now Proposition B implies that C(i) also has shape λ. (D.3h) Remark. The operator C does not preserve the Q-symbol in general. But it gives a pairing on the set of standard tableaux of the same shape.

D.4 The Littelmann algebra L(n, r) (D.4a) Let V be an n-dimensional vector space over a ﬁeld F with basis v1 , . . . , vn . Then the r-fold tensor product V ⊗r has basis {vi : i ∈ I(n, r)}. (In §§1–6, V ⊗r is called E ⊗r .) For each α ∈ Σ, where α = αc,c+1 (see §A.3), we let f˜c and e˜c act on the tensor space by linear maps, deﬁning

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D Theorem A and some of its consequences

f˜c vi := vf˜c i ,

e˜c vi := ve˜c i

and using linear extension. We set v∞ := 0. Let L = L(n, r) be the subalgebra of EndF (V ⊗r ) generated by these linear maps f˜c and e˜c , for c ∈ {1, 2, . . . , n − 1}. This algebra will be called the Littelmann algebra. (D.4b) As an F -space, the Littelmann algebra L is spanned the set of all monomials m = m1 m2 . . . mt of lengths t ≥ 1, where each mτ is either f˜c or e˜c (for some c ∈ {1, 2, . . . , n − 1}). We do not include the monomial m = 1EndF V ⊗r of length zero. But it may happen that L does contain this element (see Proposition (D.4e), below). (D.4c) An element H ∈ EndF (V ⊗r ) will often be described by its matrix (Hi,j )i,j∈I(n,r) , whose entries Hi,j ∈ F are deﬁned by the equations (D.4d) Hvj = i∈I(n,r) Hi,j vi , all j ∈ I(n, r). We often identify H with its matrix (Hi,j )i,j∈I(n,r) , and we often identify f˜c , e˜c with the elements of EndF (V ⊗r ) deﬁned by (D.4a). (D.4e) Proposition. L has an identity element, viz. DS , the diagonal matrix having (DS )i,i = 1, 0 according as i ∈ S or not; here S := I(n, r) \ (Υ ∩ T). Reminder: from §D.1 we have Υ = i ∈ I(n, r) : f˜c (i) = ∞ for all c ∈ {1, 2, . . . , n − 1} and T=

i ∈ I(n, r) : e˜c (i) = ∞ for all c ∈ {1, 2, . . . , n − 1} .

Proof of Proposition (D.4e). For any subset A of I(n, r), deﬁne DA to be the element of EndF (V ⊗r ) whose matrix with respect to the basis {vi : i ∈ I} is / A. The following diagonal, and (DA )ii = 1 or 0, according as i ∈ A or i ∈ facts are easily checked. DI(n,r) is the identity element of EndF (V ⊗r ). (ii) For any c, the matrix of f˜c e˜c is equal to DZ(c) , where Z(c) is the set of all i such that e˜c (i) = ∞. Similarly e˜c f˜c = DY (c) , where Y (c) is the set of all i such that f˜c (i) = ∞. (i)

(iii) If A, B ⊆ I(n, r), then DA DB = DA∩B and DA∪B = DA + DB − DA∩B . (iv) If A1 , . . . , Aw ⊆ I(n, r) such that DAt ∈ L for all t = 1, 2, . . . , w, then DA ∈ L where A = A1 ∪ A2 ∪ . . . ∪ Aw . Now check that I(n, r) \ T = c Z(c) and I(n, r) \ Υ = c Y (c), hence S = I(n, r) \ (Υ ∩ T) =

c

Z(c) ∪

c

Y (c).

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The partition I(n, r) = S ∪ (Υ ∩ T) of I(n, r) allows us to decompose each linear operator H ∈ EndF (V ⊗r ) in matrix form as H (1,1) H (1,2) H= , H (2,1) H (2,2) with H (1,1) ∈ EndF (S, S), H (1,2) ∈ HomF (Υ ∩ T, S), H (2,1) ∈ HomF (S, Υ ∩ T) and H (2,2) ∈ EndF (Υ ∩ T). For each c ∈ {1, 2, . . . , n − 1}, it is easy to verify the following facts. (v) If H = e˜c , or if H = f˜c , then H (1,2) , H (2,1) , H (2,2) are all zero matrices; also f˜c is the transpose of e˜c . (vi) If H ∈ L, then H (1,2) , H (2,1) , H (2,2) are all zero and H (1,1) 0 H= . 0 0 (vii) DS is the matrix shown, with H (1,1) the identity matrix. Proposition (D.4e) follows from these facts. (D.4f ) Example. If n = r = 2, then I(n, r) = {11, 12, 21, 22} = S ∪ (Υ ∩ T), where S = {11, 12, 22}, and Υ ∩ T = {21}. We have ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 0 1 0 0 0 0 0 0 1 0 0 0 ⎜0 0 1 0⎟ ⎜1 0 0 0⎟ ⎜0 1 0 0⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ e˜1 = ⎜ ⎟ , f˜1 = ⎜ ⎟ , DS = ⎜ ⎟. ⎝0 0 0 0⎠ ⎝0 1 0 0⎠ ⎝0 0 1 0⎠ 0 0 0 0 0 0 0 0 0 0 0 0

D.5 The modules MQ Let λ ∈ Λ+ (n, r). For each standard tableau Q in Q(λ) we deﬁne MQ to be the subspace of V ⊗r which has F -basis all vi such that Q = Q(i), that is, all i in Iλ (Q, ≈). By Proposition B, this is an L-submodule of V ⊗r . We get therefore a direct sum decomposition of the tensor space V ⊗r into L-submodules MQ . V ⊗r = λ∈Λ+ (n,r) Q∈Q(λ)

(D.5a) MQ = LviQ = LviQ . This follows from (D.2d). For z = i ξi vi ∈ V ⊗r , deﬁne the support of z to be supp(z) = { i ∈ I(n, r) : ξi = 0 }.

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D Theorem A and some of its consequences

(D.5b) Lemma. If z = i ξi vi and c ∈ {1, 2, . . . , n − 1} such that e˜c z = 0, then supp(z) lies in the set I(n, r) \ Z(c). Proof. Let U := { i : e˜c (i) = U := { i : e˜c (i) = ∞ }. ∞ } = Z(c) and Then z = z + z , where z = i∈U ξi vi and z = i∈U ξi vi . We have by assumption ξi ve˜c (i) = e˜c z = 0. (∗) i∈U

But for each i ∈ U , e˜c (i) = ∞, hence f˜c e˜c (i) = i. Applying f˜c to (∗), we get 0 = i∈U ξi vi . But the vi are linearly independent, so all the ξi (for i ∈ U ) are zero. Therefore z = 0 which shows that z = z has support in U = I(n, r) \ Z(c). (D.5c) Corollary. If z ∈ V ⊗r is annihilated byall e˜c , c ∈ {1, 2, . . . , n − 1}, then supp(z) lies in c I(n, r) \ Z(c) = I(n, r) \ c Z(c) = T. Similarly, if z ∈ V ⊗r isannihilated by all f˜c , c ∈ {1, 2, . . . , n − 1}, then supp(z) lies in I(n, r) \ c Y (c) = Υ. There may be a word i ∈ I(n, r) such that vi is annihilated by the algebra L. According to Proposition (D.2d), we must have iQ = i = iQ in this case, and the ≈-class of i consists of i alone. From (D.2f), the shape λ of Q has the property (λ1 , . . . , λn ) = (λn , . . . , λ1 ), hence λ = (k n ) (and r = nk). In this case MQ = F vi . There may be more than one such i ∈ Iλ (n, r). For example, I(2,2) (2, 4) contains i = (2211) and j = (2121) in Υ ∩ T. As a consequence, we have to allow L-modules which are not unital. An L-module M is then deﬁned to be an F -space on which L acts by linear transformations so that x(ym) = (xy)m, for all x, y ∈ L and m ∈ M . The L-module M is deﬁned to be simple (= irreducible) if either (1) LM = 0 and M is a simple F -space, that is, has F -dimension 1, or (2) M = 0, the element DS of L acts as the identity on M , and M has no L-submodules except M and {0}. We aim to show that the modules MQ are simple as L-modules. To do so, it is helpful to exploit a subalgebra of L. (D.5d) The involutory anti-automorphism J of L = L(n, r). At this point we ﬁnd a strong similarity with the involutory anti-automorphism J of S(n, r) deﬁned in §2.7. Deﬁne a symmetric bilinear map , on V ⊗r by the rule vi , vj = δi,j for all i, j ∈ I(n, r). Given H ∈ EndF (V ⊗r ), we deﬁned in (D.4c), (D.4d) its matrix (Hi,j ). Now deﬁne J(H) ∈ EndF (V ⊗r ) by the rule: the matrix of J(H) is the transpose of the matrix of H.

D.5 The modules MQ

133

By (D.4d), Hi,j = Hvj , vi for all i, j ∈ I(n, r). Replacing H by J(H), we get J(H)i,j = J(H)vj , vi . But by deﬁnition, J(H)i,j = Hj,i = Hvi , vj . Therefore Hvi , vj = J(H)vj , vi = vi , J(H)vj for all i, j ∈ I(n, r). Equivalently, (D.5e) Hv, w = v, J(H)w for all v, w ∈ V ⊗r . We may use (D.5e) as a deﬁnition of J(H) when H is given. From the elementary properties of transposed matrices we see that the linear map J : EndF V ⊗r → EndF V ⊗r is involutory (i.e. J 2 is the identity) and is an anti-automorphism (i.e. J(H1 H2 ) = J(H2 )J(H1 ) for all H1 , H2 ). But from our present viewpoint the important fact is (D.5f ) For all c ∈ {1, 2, . . . , n − 1} there holds J(f˜c ) = e˜c and J(˜ ec ) = f˜c . These facts follow from the properties stated in (A.3g)(4). We leave the details as an exercise for the reader. But from (D.5f) we see that J maps L into itself, so it gives a map J : L → L which is an involutory anti-automorphism of the algebra L = L(n, r). (D.5g) Take a (total) order ≤ on the set I(n, r) such that sz(i) ≤ sz(j) implies that i ≤ j for all words i, j in I(n, r). Using such an order, the matrix of e˜c is upper triangular, and the matrix of f˜c is lower triangular. We have therefore: (D.5h) Corollary. Let L+ be the subalgebra of L, generated by the elements e˜c , c ∈ {1, 2, . . . , n − 1}. Then L+ is nilpotent. (D.5i) Lemma. The module MQ is simple. Note that this holds for arbitrary ﬁelds F . Proof. This is clear if MQ is an 1-dimensional module which is annihilated by L, so we assume that this is not the case. We ﬁx i = iQ , then vi generates MQ as an L-module, by (D.2a). Let 0 = x ∈ MQ , it suﬃces to show that vi lies in the L-submodule generated by x. To do so, we consider the L+ -submodule of MQ generated by x. By (D.5h), this submodule contains some non-zero element z such that e˜c z = 0 for all c. By (D.5c), the support of z lies in T. But it also lies in I(Q, ≈) since z ∈ MQ . It follows that z is a scalar multiple of vi , by (D.2d)(i). We assumed z is non-zero, hence viQ lies in the submodule generated by x. (D.5j) It follows by a well-known theorem on ﬁnite dimensional algebras (or on rings with minimum condition; see e.g. [11, Theorem (25.2), page 164]), that L is semisimple.

134

D Theorem A and some of its consequences

Furthermore, every unital simple L-module M occurs as a submodule (and hence as a summand) of V ⊗r . Take any non-zero element x ∈ M . Then M = Lx, and the map θ : L → M which takes u → ux is an epimorphism of L-modules. But since L is semisimple, it follows that θ maps some simple submodule N of L isomorphically . And the simple sub onto M modules of L are submodules of V ⊗r = λ∈Λ+ (n,r) Q∈Q(λ) MQ (see the displayed formula, above (D.5a)). This shows that every unital simple L-module is isomorphic to MQ for some Q ∈ Q(λ), for some λ ∈ Λ+ (n, r). To classify the simple L-modules we must ﬁnd out when MQ and MR are isomorphic.

D.6 The λ-rectangle Fix λ ∈ Λ+ (n, r) and use the following notation: (D.6a) P(λ) = {P1 , P2 , . . . , Pdλ } is the set of all standard λ-tableaux whose entries all lie in n, and (D.6b) Q(λ) = {Q1 , Q2 , . . . , Qfλ } is the set of all standard λ-tableaux whose entries are {1, 2, . . . , r} in some order (see D.1). Deﬁnition. If P ∈ P(λ) and Q ∈ Q(λ), let P : Q denote2 the word i ∈ I(n, r) such that P (i) = P and Q(i) = Q. In the notation of B.7, (D.6c) P : Q = M(λ, P, Q) = Sch−1 (λ, P, Q). It is useful to display the set of words of shape λ in the following “λ-rectangle” P1 : Q1 P1 : Q2 P2 : Q1 P2 : Q2 (D.6d)

.. .

··· ···

.. .

Pdλ : Q1 Pdλ : Q2

P1 : Qfλ P2 : Qfλ .. .

...

Pdλ : Qfλ

This rectangle has the following properties: (D.6e) (i) Every element of Iλ (n, r) appears once and only once in (D.6d) (see (B.6a)). (ii) The hth row {Ph : Q1 , . . . , Ph : Qfλ } is the ∼-class Iλ (Ph , ∼), for each h ∈ {1, . . . , dλ } (see (C.1d)). (iii) The k th column {P1 : Qk , . . . , Pdλ : Qk } is the ≈-class Iλ (Qk , ≈), for each k ∈ {1, . . . , fλ } (see (C.1e)). 2

Not to be confused with the bideterminant (Ti : Tj ) deﬁned in (4.3a).

D.7 Canonical maps

135

(D.6f ) From now on we shall arrange the notation in the λ-rectangle (D.6d) so that P1 = Tλ and Q1 = Q(λ) . Recall from (C.2i) that Q(λ) ∈ Q(λ) has the property: an element i ∈ Iλ (n, r) has Q(i) = Q(λ) if and only if i = KP (i).

D.7 Canonical maps Fix λ ∈ Λ+ (n, r) again. The entries P : Q in (D.6d) are elements of I(n, r). From now on we shall make the (D.7a) Convention. When convenient, we shall regard each i ∈ I(n, r) as the element vi of V ⊗r . With this convention, the column of the rectangle (D.6d) corresponding to a given Q ∈ Q(λ) is a basis of the L-module MQ (see §D.5). (D.7b) Deﬁnition. If Q, R ∈ Q(λ), then the F -linear map γQ,R : MQ → MR which takes Ph : Q → Ph : R for each Ph ∈ P(λ), is the canonical map from MQ to MR . Since any two columns in (D.6d) have the same length, the canonical map is an F -linear isomorphism. It is clear that γQ,R γS,Q = γS,R and γQ,R = (γR,Q )−1 , for all Q, R, S ∈ Q(λ). Our ambition in this section is to prove that any canonical map is an isomorphism of L-modules (see (D.7f) and (D.7i)), and that any L-homomorphism MQ → MR is a scalar multiple of the canonical map γQ,R (see (D.7h)). (D.7c) Lemma. Let Q ∈ Q(λ) and P ∈ P(λ), and let i = P : Q. Then KP (i) = P : Q(λ) = γQ,Q(λ) (i). In other words, the operation i → KP (i) is achieved (for i ∈ Iλ (n, r)) by the canonical map γQ,Q(λ) . Proof. By (C.3p) one may make a sequence of basic moves joining i to KP (i). Since basic moves do not change P -symbols, we know that KP (i) = P : Q for some Q ∈ Q(λ). But KP (i) equals KP (KP (i)), hence its Q-symbol is Q(λ) (see (C.2i)). Therefore KP (i) = P : Q(λ) . Notice that this holds for any i in column Q of (D.6d), and in particular it holds for f˜c (i), for any c ∈ {1, . . . , n − 1}. By Proposition B (see C.6) we have f˜c (KP (i)) = KP (f˜c (i)), and in our case this gives (D.7d) f˜c (P : Q(λ) ) = KP (f˜c (i)) or, as a commutative diagram,

136

D Theorem A and some of its consequences

(D.7e)

KP (i) ⏐ ⏐ "

γ

←−−−−

i ⏐ ⏐ "

γ KP (f˜c(i) ) ←−−−− f˜c(i)

where γ = γQ,Q(λ) and the vertical arrows indicate action of f˜c . Thus the action of f˜c commutes with γ, when applied to any i in the Q-column of (D.6d). In the same way, one has a diagram like (D.7e), with e˜c replacing f˜c . Then we can replace f˜c in (D.7e) by any element of L and still have a commutative diagram, since the elements f˜c and e˜c , c ∈ {1, . . . , n − 1} generate L as F -algebra. So we get a (D.7f ) Corollary to (D.7c). For each pair Q, R of tableaux in Q(λ), the canonical map γQ,R : MQ → MR is an isomorphism of L-modules. Proof. The argument above shows that the corollary holds if R = Q(λ) , and −1 . this gives the general case, since γQ,R = γR,Q (λ) γ Q,Q(λ) (D.7g) Lemma. Suppose Q, R are both tableaux whose entries are 1, 2, . . . , r in some order (possibly of diﬀerent shapes). If ψ : MQ → MR is a homomorphism of L-modules, then ψ(viQ ) = αviR for some α ∈ F . Proof. Let z = ψ(viQ ), then e˜c (z) = ψ(˜ ec (viQ )) = 0 for all c ∈ {1, 2, . . . , n−1}. Therefore supp(z) ⊆ T, by Corollary (D.5c). But also supp(z) ⊆ I(R, ≈) since z ∈ MR , hence supp(z) ⊆ T ∩ I(R, ≈) = {iR } (see (D.2d)); this means that z = αviR for some α ∈ F . (D.7h) Corollary. Suppose Q, R are tableaux whose entries are 1, 2, . . . , r in some order. If Q, R have the same shape then HomL (MQ , MR ) = F γQ,R . Proof. If ψ ∈ HomL (MQ , MR ), i.e. if ψ : MQ → MR is an L-homomorphism, then ψ(viQ ) = αviR for some α ∈ F . But in the present case we know that γQ,R : MQ → MR also is an L-homomorphism, by (D.7f). Therefore we have L-homomorphisms ψ and αγQ,R which take viQ to the same element αviR . Since viQ is an L-generator of MQ , by (D.5a), the map ψ is equal to αγQ,R . The module MQ is simple, and MQ ∼ = MQ(λ) . For each λ, let Mλ = MQ(λ) . (D.7i) Lemma. Let λ, µ ∈ Λ+ (n, r), then Mλ ∼ = Mµ if and only if λ = µ. Proof. Suppose that there is an isomorphism ψ : Mλ → Mµ of L-modules. Then ψ(viλ ) = αviµ for some α ∈ F , by (D.7g). (Note that Mλ = MQ(λ) and, (λ) (µ) by (D.1i), iQ = iλ and iQ = iµ .) If we apply repeatedly f˜1 ’s to iλ then we replace each time the last 1 by a 2, and we can do this λ1 − λ2 times, and the next time we get zero. Similarly

D.8 The algebra structure of L(n, r)

137

if we apply f˜1 to iµ repeatedly, then we can do this µ1 −µ2 times before we get zero. The isomorphism shows now that λ1 −λ2 = µ1 −µ2 . The same argument with f˜2 shows that λ2 − λ3 = µ2 − µ3 , and so on. Both λ and µ have degree r which forces λ = µ. Exercise 1. Let λ = (5, 4, 2). Find f˜1 viλ and verify that f˜12 viλ = v∞ = 0. Find also f˜2t viλ for t = 1, 2, 3. Exercise 2. If λ = (k, . . . , k), where kn = r, we have 1 1 2 2 Tλ =

··· ···

.. .. . . k k

1 2 .. . .

···

k

Check by direct calculation that f˜c (KTλ ) = ∞ = e˜c (KTλ ) for all c.

D.8 The algebra structure of L(n, r) Each Mλ has endomorphism algebra F . It follows now that L is isomorphic to the direct sum of matrix algebras, Mdλ (F ), L∼ = λ

where the sum is taken over all λ ∈ Λ+ (n, r) with λ = (k n ), and where dλ denotes the dimension of Mλ . This follows from the Frobenius–Schur theorem, see [11, Theorem 27.8, page 183], but we shall give a direct proof that the representation L → EndF (Mλ ) aﬀorded by the simple module Mλ is surjective, for all λ ∈ Λ+ (n, r), λ = (k n ). The problem is to give elements of L which realize the “matrix units”. Fix λ ∈ Λ+ (n, r). The module Mλ is simple, it has F -basis { vi : i ∈ I(λ) } where we set I(λ) := I(Q(λ) , ≈) = { i ∈ Iλ : KP (i) = i } (see (C.2i)). An element Φ of EndF (Mλ ) is regarded as a matrix (Φij )i,j∈I(λ) in the usual way, Φ(vj ) = Φij vi , all j ∈ I(λ). i∈I(λ)

Monomials in f˜c , e˜c are often identiﬁed with the matrices of the linear transformations which they determine on Mλ .

138

D Theorem A and some of its consequences

(D.8a) Lemma. Suppose Φ is a monomial in the e˜c and f˜c , c ∈ {1, . . . , n−1}. Let (Φij )i,j∈I(λ) be the matrix of Φ restricted to Mλ . (i) Each row or column of the matrix has at most one non-zero entry (which is then equal to 1). (ii) If (Φij )i,j∈I(λ) has rank 1, then it is a matrix unit. Proof. For each i ∈ I(λ), Φ(vi ) is either zero, or a basis element. So all but at most one entries of each column are zero, and if there is a non-zero entry then it is equal to 1. This implies part (i), since if Φ = m1 · · · mt , then J(Φ) = J(mt ) · · · J(m1 ) is also a monomial. But (J(Φ)) is the transpose of (Φ) (see (D.5d)). Part (ii) follows. (D.8b) We want to show that every element of EndF (Mλ ) can be represented by some element of L, i.e. that the map L → EndF (Mλ ) is surjective. And we would like to do this by showing that each “matrix unit” Ei,j ∈ Mdλ (F ) can be represented by some polynomial in the f˜c , e˜c . It is enough to prove that Eiλ ,iλ can be represented by a monomial in L. Namely, if s, t ∈ I(λ) then by (D.2d) there are monomials p and q in f˜’s and e˜’s with p(viλ ) = vs and q(vt ) = viλ . The linear map pEiλ ,iλ q of V ⊗r has rank at most 1 (the rank of Eiλ ,iλ ), so by (D.8a) it is equal to Est , and this is then also represented by an element in L. The L-module Mλ = MQ(λ) has basis { vi : i ∈ I(λ) = I(Q(λ) , ≈) }. The dλ elements of I(λ) can be arranged (see (D.1g) and (D.6d)) as (λ)

i(1) = iQ

= iλ ,

i(2),

...,

i(dλ − 1),

i(dλ ) = iQ(λ) = iλ .

(D.8c) Proposition. (i) There are c(1), . . . , c(b) ∈ {1, . . . , n − 1} such that Φ := f˜c(b) . . . f˜c(1) maps iλ to iλ . (ii) The number b in (i) is given by sz(iλ ) + b = sz(iλ ). (iii) If d(1), . . . , d(s) ∈ {1, . . . , n − 1} are such that f˜d(s) . . . f˜d(1) iλ = ∞, then s ≤ b. (iv) Φ(vi(a) ) = 0, for all a ∈ {2, 3, . . . , dλ }. This shows that the matrix of Φ on Mλ is the matrix unit Eiλ ,iλ . Proof. (i) is a direct application of (D.2d)(iii)(2). We recall the proof, because it brings up useful information. The idea is to apply operators f˜c(1) , f˜c(2) , . . . in succession to the word iλ , in such a way that, for each t = 1, 2, . . . f˜c(t) · · · f˜c(1) iλ = ∞. The word f˜c(t) · · · f˜c(1) iλ has size sz(iλ ) + t, by (D.2c). The sizes of words in I(n, r) are bounded by rn. Hence, however we choose c(1), c(2), . . ., we

D.9 The character of Mλ

139

must reach b such that f˜c(b) · · · f˜c(1) iλ = ∞, but z = f˜c(b) · · · f˜c(1) iλ has the property f˜c (z) = ∞ for all c ∈ {1, . . . , n − 1}. This implies z ∈ Υ; however z ∈ I(λ) = I(Q(λ) , ≈), by (D.2b), hence z = iλ by (D.2d)(ii). This proves parts (i) and (ii) of (D.8c). To prove part (iii), note that the argument above (replace t by s) shows that, applying further operators f˜d(s+1) , f˜d(s+2) , . . . , f˜d(b ) to f˜d(s) . . . f˜d(1) iλ if necessary, we must reach b such that f˜d(b ) · · · f˜d(s+1) f˜d(s) · · · f˜d(1) iλ = iλ . Taking the size of each side of this equation, we get sz(iλ ) + b = sz(iλ ); this shows that b = b. Therefore s ≤ b = b. (iv) If a ∈ {2, 3, . . . , dλ }, then by (D.2d)(iii), i(a) = f˜d(1) . . . f˜d(u) iλ for some d(1), . . . , d(u) ∈ {1, . . . , n − 1}, and u ≥ 1. But this implies that Φ(i(a)) = f˜c(b) · · · f˜c(1) f˜d(u) · · · f˜d(1) iλ . If this were = ∞, it would contradict (iii), since b + u > b. Therefore Φ(i(a)) = ∞, hence Φ(vi(a) ) = 0. Remarks. (i) In (D.8c)(i), there may be several ways of choosing c(1), . . . , c(b) so that Φ = f˜c(b) . . . f˜c(1) maps iλ to iλ . But by (ii) the length b of any such sequence is always the same, namely b = sz(iλ ) − sz(iλ ). (ii) For any Φ ∈ L, the matrix of J(Φ) is the transpose of the matrix of Φ. This is true by deﬁnition if the matrices are deﬁned in terms of the natural basis { vi : i ∈ I(n, r) } of V ⊗r (see (D.5d)), hence it is true also for the matrices deﬁned in terms of the basis { vi : i ∈ I(λ) } of Mλ . Therefore, if Φ = f˜c(b) . . . f˜c(1) as in (D.8c), then the map J(Φ) = e˜c(1) . . . e˜c(b) has matrix Eiλ ,iλ . Example (see chapter E). . Take λ = (2, 1, 0) and Q(λ) = 1 3 . We can 2 take i(1) = 211,

i(2) = 311,

i(3) = 312,

i(4) = 322,

i(5) = 323.

i(3) = 213,

i(4) = 313,

i(5) = 323.

Another possibility is to take i(1) = 211,

i(2) = 212,

D.9 The character of Mλ The basis { vi : i ∈ I(Q, ≈) } for MQ consists of eigenvectors for the diagonal matrices in the general linear group GL(n, F ). Hence MQ has a formal character (as deﬁned in §3.4), also when the ﬁeld F is ﬁnite. Explicitly, let (MQ )α = ξα MQ (see §3), then the formal character of MQ is by deﬁnition

140

D Theorem A and some of its consequences

ΦMQ (X1 , . . . , Xn ) =

dim(MQ )α X1α1 · · · Xnαn .

α∈Λ(n,r)

The P -symbol preserves weights, hence dim(MQ )α = dim(Mλ )α , where λ is the shape of Q. Therefore ΦMQ = ΦMλ , that is, the formal character of MQ depends only on the shape of Q. Let Vλ be the “Weyl module” associated to λ; this is a module for the Schur algebra, see section 5.2. The following is due to P. Littelmann, in far more generality [35, Introduction]. (D.9a) Corollary. The modules Mλ and Vλ have the same formal character. Proof. In (5.4a) we saw that (Vλ )α has F -basis indexed by standard λ-tableaux of weight α. We also know from the characterisation of Mλ given above that (Mλ )α has basis vi labelled by standard λ-tableaux of weight α. Hence Mλ has the same formal character as Vλ . Note that it follows that ΦMλ = ΦMµ if and only if λ = µ. (This is also visible directly, by considering the “highest terms” of the formal characters.)

D.10 The Littlewood–Richardson Rule Suppose λ and µ are partitions with λ ∈ Λ+ (n, r), µ ∈ Λ+ (n, s). Then cνλ,µ ΦVν . ΦVλ · ΦVµ = ν

The coeﬃcients cνλ,µ are non-negative integers, and the Littlewood–Richardson rule is a combinatorial rule for computing these integers. As we have seen, the L-module Mλ has the same formal character as the Schur algebra module Vλ . Then we have ΦMλ · ΦMµ = cνλ,µ ΦMν . ν

Here the sum is taken over all ν ∈ Λ+ (n, r + s). This leads to the following combinatorial description of the coeﬃcients. (D.10a) Let W be the set of words i ∈ I(n, r + s) of the form i = jk with the following properties: (a) KP (j) = j and P (j) has shape λ, (b) k = iµ , and (c) the reverse B(i) of i is a lattice permutation of weight ν. Then cνλ,µ is equal to #W, the number of elements of W.

D.10 The Littlewood–Richardson Rule

141

A number of proofs of (D.10a) exist, and Littelmann gives a wide ranging generalization to cover any complex symmetrizable Kac-Moody Lie algebra [35, Introduction]. The proof we give is the special case which applies to gl(n) or to GLn . Proof. By deﬁnition Mλ is a direct summand of V ⊗r , and Mµ is a direct summand of V ⊗s . Then Mλ ⊗ Mµ is a direct summand of V ⊗(r+s) , as a vector space, since vi ⊗ vj = vij . It is invariant under the linear maps f˜c and e˜c . For example, f˜c (vij ) is either vf˜c (i)j or vif˜c (j) , or zero; and each of these belong again to Mλ ⊗ Mµ . Furthermore, Mλ ⊗ Mµ is the direct sum of L-modules MR for some standard tableaux R with shapes in Λ+ (n, r + s). Therefore cνλ,µ is precisely the number of such R such that MR occurs as a direct summand in Mλ ⊗ Mµ and MR ∼ = Mν . Each MR contains a unique “highest weight vector” vi such that e˜c (i) = ∞ for all c, namely the basis vector for i = iR . Hence cνλµ is equal to the number of words i = jk where (i) i belongs to T (see §D.1), and P (i) has shape ν; (ii) k = KP (k), and P (k) has shape µ; (iii) KP (j) = j and P (j) has shape λ. We know i ∈ T if and only if B(i) is a lattice permutation (see (D.1b)). For i ∈ T, the shape is the same as the weight (see (D.2f)). Furthermore, if B(i) is a lattice permutation then so is B(k), and since k = KP (k), we have k = iµ . This completes the proof of (D.10a). (D.10b) Very often, the Littlewood–Richardson rule is stated in a diﬀerent form. It says that the coeﬃcient cνλ,µ is equal to the size of the set C of standard (skew) tableaux T of shape ν \ µ and of weight λ such that the word w(T ) is a lattice permutation. Here the word w(T ) is obtained by reading T from right to left and from rows 1, 2, 3, . . .. We will now show directly that #C = #W, by means of a bijection from W onto C. Suppose i belongs to W, where i = jk as in (D.10a). Then always k = iµ , and we must consider the tableau P (j). Let rts be the number of times the letter s occurs in row t of P (j); since P (j) is standard, row t of P (j) starts with some letter ≥ t, and it has the form trtt (t + 1)rt,t+1 . . . nrtn , t ≥ 0 Write the multiplicities rst as an upper triangular matrix: ⎛ ⎞ r11 r12 ··· ⎜ r22 r23 · · ·⎟ ⎜ ⎟ . U =⎜ r33 · · ·⎟ ⎝ ⎠ .. .

142

D Theorem A and some of its consequences

By transposing this matrix, we can deﬁne a skew tableaux T = ψ(U ), depending on i = jk, as follows: The tth row of T starts at position (t, µt + 1) and has the multiplicities taken from the tth column of U , that is, row t is trtt (t − 1)rt−1,t . . . 1r1t . The associated word is then w(T ) = 1r11 (2r22 1r12 )(3r33 2r23 1r13 ) · · · We will show that T belongs to C, and that the map ψ : U → T is a bijection between W and C. (1) The word j has weight ν \ µ if and only if for each s, the sum of the entries in column s of the matrix U is equal to νs − µs . This means for the skew tableau T that the sum of the entries in row s is equal to νs − µs , for each s, that is T has shape ν \ µ. (2) The tableau P (j) has shape λ provided row t of P (j) has λt entries, for each t, that is rtv = λt . v≥t

This is equivalent with saying that the skew tableau w(T ) has weight λ. (3) The tableau P (j) is standard if and only if v+1

rs+1,y ≤

y=s+1

v

rs,y .

y=s

for all s ≥ 1 and all v ≥ s. This means for the word w(T ) that in each initial section the number of entries equal to s is ≥ the number of entries equal to s + 1, for each s ≥ 1. That is, P (j) is standard if and only if w(T ) is a lattice permutation. (4) The word B(i) is of the form (1µ1 2µ2 · · · )(· · · xr1x · · · 2r12 1r11 )(· · · xr2x · · · 2r22 )(· · · xr3x · · · 3r33 ) · · · This is a lattice permutation if and only if for each s ≥ 1 and each v µs +

v y=1

rys ≥ µs+1 +

v+1

ry,s+1 .

y=1

This is equivalent with T being standard. Combining (1) to (4), we see that if i = jk ∈ W and if U is the matrix encoding j, then the skew tableau T = ψ(U ) belongs to C. Conversely if we start with some T ∈ C, then T is the transpose of a matrix U , and this encodes a word i = jk in W. So ψ is a bijection.

D.11 Lascoux, Leclerc and Thibon

143

D.11 Lascoux, Leclerc and Thibon This is a brief summary of Chapter 6 of the collective work “Algebraic combinatorics on words” [38]. This chapter is called “The plactic monoid”, and its authors are A. Lascoux, B. Leclerc and J.-Y. Thibon. We refer to this chapter, and to its authors, as LLT. Our main purpose is to show that LLT prove facts which imply Theorem A and Proposition B (see (D.11h)). Reference numbers for sections, propositions, etc. in LLT are enclosed in square brackets (so that, for example, [6.1] stands for [38, 6.1]). (D.11a) The background of LLT is work of M. P. Sch¨ utzenberger, which expresses the combinatoric background of work by A. Young, G. de B. Robinson, D. E. Littlewood, etc. on the representation theory of the ﬁnite symmetric group. (D.11b) Words and tableaux. In LLT the set of all words on the alphabet A = {1, . . . , n} is denoted A∗ . So in our language, A∗ = r≥0 I(n, r). In LLT (page 3), a tableau3 is a word i in A∗ such that i = KP for some standard tableau P in the sense of section B.1. For example, i = 544135 is a 1 3 5 . If we know that i is a tableau, tableau, because i = KP for P = 4 4 5 the corresponding tableau P (which LLT call its planar representation) is uniquely deﬁned. The shape λ of i is, by deﬁnition, the shape of P . In the example above, the shape is λ = (3, 2, 1, 0, 0). (D.11c) In [6.1] the Schensted algorithm is described. It takes each word i to a tableau KP (i). We can take P (i) to be the tableau deﬁned in (B.4b), (B.4c). The equivalence ∼ on A∗ is deﬁned in [6.2, bottom of page 4]: if i, j are words, then i ∼ j means KP (i) = KP (j). The equivalence ≡ on A∗ is deﬁned on page 5 to be the equivalence on A∗ generated by basic moves (see (C.3c), (C.3d) and [6.2.3, 6.2.4]. LLT do not use the term “basic move”.) (D.11d) Knuth’s theorem (C.3a), [6.2.5] says that ≡ coincides with ∼. This is proved in [6.2], elegantly and economically, by a theorem of C. Greene [21]. Greene’s theorem itself is also proved in [6.2]. Now the main (D.11e) Deﬁnition (see [6.2.2]). The plactid monoid Pl(A) := A∗ /∼ is the quotient of A∗ by ∼. Elements of Pl(A) are the ∼-classes, or “plactic classes” in A∗ .

3

We write tableau (underlined) for a word which is a “tableau in the sense of LLT”. A tableau (not underlined) is a standard tableau in the sense of section B.1 of this Appendix. Later in LLT a tableau KP and its planar representation P are often identiﬁed.

144

D Theorem A and some of its consequences

Knuth shows that ∼ is compatible with the product of words: if u, u , v, v are words, then u ∼ u and v ∼ v implies uu ∼ vv (see [34, Corollary, page 724]). Product of words is by concatenation, so that uu = u | u ; see (A.3g)(6). Therefore Pl(A) is a monoid (i.e. a semigroup with identity): the product of the ∼-class of u with the ∼-class of u , is deﬁned to be the ∼-class of uu . If u is any word, then u ∼ KP (u) (see (C.3p), [6.2.3]). Every ∼-class contains exactly one tableau; see Theorem [6.2.5]. (D.11f ) A main theme in LLT is that it is often useful to “lift” a symmetric polynomial to Z[Pl(A)]. Suppose that M is any monoid. Then Z[M ], which is the free Z-module with M as Z-basis, is a ring. In case M = A∗ we can identify the ring Z[A∗ ] with the tensor ring T (V ) = Z ⊕ V ⊕ (V ⊗ V ) ⊕ · · · over the free Z-module V = Zν1 ⊕ · · · ⊕ Zνn , by identifying each word i = i1 · · · ir ∈ A∗ with the tensor product νi = νi1 ⊗ · · · ⊗ νir (compare with (D.4a)). Yet another interpretation of Z[A∗ ] is as the ring of all polynomials (over Z) in non-commuting variables ν1 , . . . , νn ; here one regards every tensor product νi = νi1 ⊗ · · · ⊗ νir as the monomial νi1 · · · νir . Now suppose that ξ1 , . . . , ξn are commuting variables. Then there is an epimorphism of rings κ : Z[A∗ ] → Z[ξ1 , . . . , ξn ] which takes νσ → ξσ for all σ ∈ {1, . . . , n}. And this map factors through the map π : Z[A∗ ] → Z[Pl(A)] induced by the natural epimorphism A∗ → Pl(A); this means that i ∼ j implies κ(i) = κ(j). (It is enough to check this in case i is connected to j by a basic move.) So there exists a ring epimorphism η : Z[Pl(A)] → Z[ξ1 , . . . , ξn ] such that κ = ηπ. In section [6.4] LLT deﬁne a “plactic Schur function” Sλ in Z[Pl(A)] which is mapped by η onto the classical Schur function in the variables ξ1 , . . . , ξn (see remark (iii) in section 3.5). Then they deduce the Littlewood–Richardson rule from an identity in Z[Pl(A)] (see Theorem [6.4.5]). (D.11g) Returning to section [6.3]; LLT deﬁne Schensted’s Q-symbol. So for any i ∈ A∗ , one deﬁnes the tableau Q(i) (or more correctly the tableau KQ(i)) which is a byproduct of the sequence of tableaux P (i1 ), P (i1 i2 ), . . . which is used to make P (i); see the example (B.4c), or the example in LLT (page 7). By its construction, Q(i) is what LLT call a “standard” tableau, i.e. if i ∈ I(n, r), then the entries of Q(i) are the numbers 1, 2, . . . , r in some order. The shape of Q(i) is the shape λ of P (i). LLT prove the Robinson– Schensted theorem [6.3.1], which says that the map ρ : i → (P (i), Q(i)) induces a bijection from the set Iλ (n, r) of all words i of given shape λ (see §C.1) to the set Tab(λ, A) × STab(λ). (In our notation, Tab(λ, A) = P(λ) and STab(λ) = Q(λ); see (D.6a) and (D.6b).) This is essentially the theorem (B.6a) which says the map Sch is bijective. It is proved in the same way, by constructing the inverse map ρ−1 . The rest of section [6.3] is devoted to applications to representations of the symmetric group S(n). A permutation σ of {1, . . . , n} is regarded as a word σ = σ1 · · · σn of length n. Then G. de B. Robinson discovered and Sch¨ utzenberger proved the theorem [6.3.3]: Q(σ) = P (σ −1 ). LLT give a short

D.11 Lascoux, Leclerc and Thibon

145

proof of this fact, and also generalize it to obtain, for any word i, a description of Q(i) as P (σ −1 ) for a certain permutation σ constructed from i (see [6.3.7]). Then a further generalization, gives them a generating function for the number dλ of plactic classes of given weight λ (see [6.3.10]). Notice that dλ appears in the “λ-rectangle” (D.6d). (D.11h) In section [6.5], the set of all i ∈ A∗ for which Q(i) is a given “standard” tableau Q is called a coplactic class. In our terminology (see section C.1) this is the ≈-class Iλ (Q, ≈), where ≈ is the equivalence relation on A∗ deﬁned in (A.4b): i ≈ j means Q(i) = Q(j). (LLT do not give a symbol for ≈.) In order to give “structure” to the coplactic classes, LLT introduce three operations on words (which then induce linear operations on Z[A∗ ]). For a given c ∈ {1, . . . , n − 1}, the LLT operators are called ec , fc , σc . We shall see in (D.11i) that ec , fc are just the Littelmann operators e˜c , f˜c deﬁned in section A.3. We do not have the operator σc in the Appendix, but it is used extensively in the latter part of LLT. Proof of Theorem A. Theorem [6.5.1(i)] says that if θ is either ec or fc , then Q(θi) = Q(i) for any word i such that θi = 0. This is Proposition (D.2b); it is the “if” part of Theorem A. The “only if” part follows from Proposition [6.5.2(i)]; one deﬁnes a graph Γ (called the Littelmann graph c in section E.2) to have for vertices all words i ∈ A∗ , with arrow i −→ j where fc i = j. If i, j are such that i ≈ j, i.e. if i, j are in the same coplactic class, then [6.5.2(i)] says that i, j are in the same connected component of Γ, which means that we connect i and j by a chain of links, each link being of c c the form either −→ or ←−. But this is “only if” for Theorem A. Proof of Proposition B. Theorem [6.5.1(ii)] says that LLT operators are compatible with the equivalence ≡. For example, if i, j ∈ A∗ and if i ≡ j, then for any c ∈ {1, . . . , n − 1} there holds fc (i) ≡ fc (j). (This includes the statement fc (i) = 0 if and only if fc (j) = 0.) But this is essentially the Lemma (C.6c). To deduce Proposition B, we combine (C.6c) with (C.3p), which says that for any i there holds i ≡ KP (i). However LLT have proved this in Proposition [6.2.3]. Therefore LLT have proved Proposition B. (D.11i) We sketch the proof that the LLT operators ec , fc are the same as the Littelmann operators e˜c , f˜c , respectively. Let c ∈ {1, . . . , n − 1}, and keep this ﬁxed. To calculate e˜c (i) and f˜c (i) for a given word i of length p, use the function hic (t) (see (A.3b)). This gives parameters M i , q i , q i , and these are suﬃcient to determine e˜c (i) and f˜c (i). Let us say that words i, j are isologous if M i , q i , q i are equal to M j , q j , q j , respectively. To calculate hic (t), one needs only the entries c, c + 1 in i. We say that letters other than c, c + 1 are neutral. In the example below i = 235342233 is a word in I(5, 9), and c = 2. Our ﬁrst move is to replace each neutral entry by the empty square, indicated by a “ · ” in the third line of table D.1 below. Now we look for an adjacent (c + 1, c) pair, i.e. entries ia , ib of i

146

D Theorem A and some of its consequences t

1 2 3 4 5 6 7 8 9

it

2 3 5 3 4 2 2 3 3

it

2 3

·

3

·

2 2 3 3

jt

2 3

·

·

·

·

2 3 3

kt

2

·

·

·

·

·

·

3 3

Table D.1. Successive construction of isologous words.

such that ia = c + 1, ib = c, a < b, and iz is neutral for all places z such that a < z < b, if there is any such place. In our example, (i4 , i6 ) is an adjacent (3, 2) pair. Now replace both entries ia , ib by neutral letters. It is (very) easy to see that the resulting word j is isologous to i. We next look for adjacent (c + 1, c) pairs in j; in the example, (j2 , j7 ) is such a pair. Then “neutralize” this pair, etc. After a ﬁnite number of steps we reach a word k that contains no adjacent (c + 1, c) pair. In this word, there may be r entries c, and they all occur before any of the s entries c + 1. (Either or both of r, s may be zero.) By construction k is isologous to i. But it is very simple to describe the function hkc (t): starting from the left, it ascends by the r steps c, moves horizontally if there are some neutral entries between the last c and the ﬁrst c + 1, then descends by the s entries c + 1. If r = 0 then M i = M k = 0, and if s = 0 we have hic (p) = hkc (p) = M k ; if r > 0 then q i = q k is the last place with entry c, and if s > 0 then q i = q k is the place immediately before the ﬁrst c + 1. In the example given in table D.1, we have exactly one c at place 1 in k, and two c + 1’s at places 8, 9, respectively; hence q i = 1 and q i = 7. We now have all that is needed to construct e˜c (i) and f˜c (i). We leave it to the reader to compare our construction of e˜c , f˜c with LLT’s construction of their operators ec , fc , and to show that the two constructions are identical.

E Tables

E.1 Schensted’s decomposition of I(3, 3) Let n = 3 and r = 3. For each i ∈ I(3, 3), the Q-symbol Q = Q(i) of i then contains each of the numbers 1, 2, 3 exactly once. We write I(Q) = I(Q, ≈) for all these tableaux Q. Then ∪˙

∪˙

I(111) ⎞ ⎛ 1 =I 1 2 3 ∪˙ I ⎝ 2 ⎠ . ∪˙ I 1 2 ∪˙ I 1 3 2 3 3

I(3, 3) =

I(300)

I(210)

Table E.1 below contains, for each λ ∈ Λ+ (3, 3), the tableaux ψ (λ) , Q(λ) (see (C.2g) and (C.2h)), the tableaux Tλ , Zλ , and the words iλ , iλ obtained from these (see (D.1d) and (D.1e)).

λ

ψ (λ)

Q(λ)

Tλ

iλ

Zλ

iλ

(3, 0, 0)

1 2 3

1 2 3

1 1 1

111

3 3 3

333

(2, 1, 0)

2 3 1

1 3 2

1 1 2

211

2 3 3

323

(1, 1, 1)

3 2 1

1 2 3

1 2 3

321

1 2 3

321

Table E.1. Various data associated with λ ∈ Λ+ (3, 3).

148

E Tables

The elements of the sets I(Q) with their P -symbols and Q-symbols are listed in table E.2.

λ

Q(i)

(3, 0, 0)

(2, 1, 0)

(1, 1, 1)

P (i)

i

P (i)

1 1 1

111

1 1 2

121

211

1 1 2

112

1 2 2

122

1 2 2

221

212

1 1 3

113

1 1 3

131

311

1 3 2

231

213

1 2 3

132

312

1 3 3

331

313

2 2 3

232

322

2 3 3

332

323

1 2 3

1 3 2

2 2 2

222

1 2 3

123

2 2 3

223

1 3 3

133

2 3 3

233

3 3 3

333

1 2 3

i

P (i)

i

1 2 3

321

1 2 3

Table E.2. P -symbols and Q-symbols of the words i ∈ I(3, 3).

E.2 The Littelmann graph I(3, 3) Let n, r be positive integers. Following Littelmann [35, §2], we deﬁne the structure of a graph on I(n, r) by saying that i, j ∈ I(n, r) are connected

E.2 The Littelmann graph I(3, 3)

149

by an edge if there exists an element c ∈ {1, . . . , n − 1} such that f˜c (i) = j or f˜c (j) = i. This graph is the Littelmann graph (it is the undirected form of the directed graph Γ in (D.11i)). The connected components of this graph are precisely the coplactic, or ≈-classes I(Q, ≈), where Q is a standard tableau with entries 1, . . . , r in some order. This follows from Theorem A (and (A.3g)(5), where we have seen that f˜c (i) = j if and only if e˜c (j) = i). Therefore we can use these tableaux to label the connected components of the Littelmann graph. For n = r = 3, the Littelmann graph is shown in table E.3. In this display, two words i, j are connected by a (directed) edge labelled by 1 or 2 according as f˜1 (i) = j or f˜2 (i) = j.

1 2 3

1 2 3

1 3 2

1 2 3

111

1 2 1; ;2 ; ;

2 1 1; ;2 ; ;

321

1

1

1 1 2;

1 2 2M M 1

1

;

;

2 2 1

2

113

M

2

M

1

M& 2 2 2 123 qq 1 qq q 2 qqqqq 2 x 2 2 3; 133 ; ; 2 ; 1 2 3 3 2

131

2

;

1

2 3 1 2

1

132

3 3 1;

1

232

;; ;; 2 1 ;; 332

2 1 2

311

2

2 1 3 2

1

312

3 1 3;

1

322

;; ;; 2 1 ;; 323

333

Table E.3. The four Littelmann graphs in I(3, 3).

Note that, if λ ∈ Λ+ (n, r) and Q = Q(λ) , then iλ is at the top and iλ is at the bottom of the corresponding connected component of the Littelmann graph (see table E.1 and (D.2d), (C.2c)).

Index of symbols

e

evaluation map

14

2

Y

= Ker e

14

Chapter 1 ◦ ∆

coproduct

3

E

natural module

17

ε

counit

3

Dr,K

rth symmetric power

19

F (K Γ )

ﬁnitary functions

3

V◦

contravariant dual

20

rab

coeﬃcient function

4

J

anti-auto of SK (n, r) ⊗r

20

4

,

4

MK (n, r)

modA (KΓ)

4

Chapter 3

com(A)

right A-comodules

5

Λ(n, r)

weights

23

= HomK (A, K)

6

W

= G(n), Weyl group

23

δab

Kronecker delta

modA (KΓ)

∗

A

canonical form on E

22

Λ+ (n, r) dominant weights

Chapter 2

21

23

11

ξα

= ξi,i where i ∈ α

23

ΓK

= GLn (K) (≥§2)

11

Vα

weight-space

24

cµν

coeﬃcient function

11

Tn (K)

diagonal subgroup

24

11

χα

character of Tn (K)

24

AK (n, r)

11

r

Λ E

exterior power

24

I(n, r)

11

ΦV

formal character

26

11

mλ

monomial symm fct

26

11

er

elementary symm fct

26 27

GLn (K)

AK (n)

G(r)

symmetric group

∼ ci,j

= ci1 j1 ci2 j2 · · · cir jr

11

eµ

= eµ1 · · · eµr

MK (n)

= modAK (n) (KΓ)

12

ϕV

natural character

27

12

Fλ,K

irreducible module

28

SK (n, r) Schur algebra

13

Φλ,K

irreducible character

28

ξi,j

13

Φλ,p

= Φλ,K if char K = p

29

MK (n, r) = modAK (n,r) (KΓ) basis element of SK

152

Index of symbols decomp numbers

29

u

= (1, 2, . . . , r)

s(w)

sign of w ∈ G(n)

30

f

Schur functor

Sλ

Schur function

30

V(e)

dλ,µ

Chapter 4 [λ] T

λ

R(T )

shape of λ

33

54 54 55

a

mod S → mod S

55

h

mod eSe → mod S

56

=a◦h

56

= G(r)

57

∗

basic λ-tableau

34

h

row stabilizer

34

G

+

C(T )

column stabilizer

34

Λ

= Λ (n, r)

57

Ti

= i ◦ Tλ

34

ST,K

Specht module

58

(Ti : Tj ) bideterminant

34

Sλ,K

= ST λ ,K

58

l

35

S T,K

dual Specht module

59

Dλ,K

35

S λ,K

= S T λ ,K

59

ϕK

⊗r EK → Dλ,K

36

Ωπ,π

60

Dr,K

= D(r,0,...,0),K

36

Ks

63

D(1r ),K

= D(1,...,1,0,...,0),K

36

d

MK (N, r) → MK (n, r)

65

λ

AK (n, r) right λ-weight-space

37

α∗

= (α, 0, . . . , 0)

65

β(i)

37

comp multiplicity

68

Chapter 5 NK

∗

Λ(n, r) nλ (V )

= Ker ϕK

Vλ,K

65

43

Chapter A

43

I(n, r)

words

Vr,K

= V(r,0,...,0),K

44

n

= {1, 2, . . . , n}

73

V(1r ),K

= V(1,...,1,0,...,0),K

45

Λ(n, r)

weights

73

[X]

=

π

45

Λ+ (n, r) dominant weights

73

{X}

=

s(π) π π∈X

45

λ(i)

shape of i

74

fl

= el {C(T )}

45

P (i)

P -symbol of i

74

bi

= ξi,l fl

π∈X

73

46

Q(i)

Q-symbol of i

74

46

αa,b

root

75

47

ω

Dλ,K

47

hic

height function

75

,

49

Mci

maximal height

75

50

qci

50

q ci

51

Littelmann operator

Chapter 6

e˜c f˜c

Littelmann operator

76

ω

= (1, 1, . . . , 1, 0, . . . , 0)

53

B

reversing operator

76

S

= SK (n, r)

53

eα

root operator

76

root operator

76

weight of i

76

Ω max

Vλ,K

min

(r, α) hr

=

r! α1 !···αn !

complete symm fct

Xλ,Z

S(α)

= ξα Sξα

53

fα

fα

MK (n, r) → mod S(α)

53

wt(i)

75

76 76 76

Index of symbols

153

i|j

concatenation of i, j

76

W

121

∼

P -equivalence

78

T

122

≈

Q-equivalence

78

C

122

KP

Knuth unwinding of P

79

Tλ iλ

Chapter B

122 = KTλ

122

[λ]

shape of λ

81

Zλ

T (n, r)

triples (λ, P, Q)

81

iλ

Sch

Schensted’s map

81

Q(λ)

123

U ← x1

insertion

82

iQ

123

122 = KZλ

122

(µ, U, V ) ← x1

82

iQ

k(a)

83

sz(i)

size of i

z

84

v∞

=0

130

εz

84

L(n, r)

Littelmann algebra

130

123 125

J

insertion map

89

DA

130

E

extrusion map

89

Z(c)

130

M

inverse of Sch

92

Y (c)

Es

94

MQ

irr. L-module

Js

94

supp(z)

support of z ∈ V ⊗r

131

J

anti-auto of L

132

Chapter C Iλ (n, r)

words of shape λ

95

P(λ)

Iλ

= Iλ (n, r)

95

dλ

130 131

134 = #P(λ)

134

Iλ (P, ∼) = {i ∈ Iλ : P (i) = P }

95

fλ

= #Q(λ)

134

Iλ (Q, ≈) = {i ∈ Iλ : Q(i) = Q}

95

P :Q

= M(λ, P, Q) ∈ I(n, r)

134

I(P, ∼)

= Iλ (P, ∼)

96

γQ,R

canonical map

135

I(Q, ≈)

= Iλ (Q, ≈)

96

Mλ

= MQ(λ)

136

(λ)

, ≈)

X[t]

98

I(λ)

= I(Q

ψ (λ)

100

Ei,j

matrix unit

Q(λ)

100

W

140

103

C

141

K K

ξ(a, t) f˜c (P )

basic move basic move = f˜c (KP )

138

103

A

words on A

143

104

≡

=∼

143

114

Pl(A) Z[M ]

Chapter D Υ

∗

137

121

∗

= A /∼

143 144

References

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Index

aﬃne group scheme, 8, 16, 18 aﬃne ring, 5 alphabet, 73 antisymmetric tensor, 45

decomposition number, 9, 17, 29, 69 deﬁned over Z, 7, 14, 19 diagonal subgroup, 24 dominant weight, 23 D´esarm´enien matrix, 45, 46, 48

basic move, 78, 103, 135 basis for Dλ,K , 36 basis for Vλ,K , 46 bideterminant, 34 bump, 85, 86

entry of a tableau, 81 equality rule, 13 equivalence ≈, 75, 78, 95 equivalence ∼, 78, 95 equivalent categories, 15 representations, 2 exterior power, 24, 36, 45 extrusion, 89 sequence, 90

canonical form, 21, 43 canonical map, 135 Carter-Lusztig basis, 45 Carter-Lusztig lemma, 38 Carter-Lusztig module, 43 character, 26, 139 formal, 26, 139 natural, 27 coalgebra, 3, 12 coeﬃcient function, 4 space, 4 column stabilizer, 34 column standard, 88 comodule, 5 completely reducible, 133 composition multiplicity, 68 concatenation, 76, 103 contravariant, 19 dual, 20 form, 20

ﬁnitary function, 3 Garnir relations, 38 Hecke ring, 54 height function, 75 hyperalgebra, 7 induced module, 42 insertion, 82 insertion parameters, 99, 104 insertion, sequence, 83 invariant matrix, 4, 17 involutory anti-automorphism, 132 James module, 40 James’s theorems, 63, 70

160

Index

KΓ-bimodule, 21, 31, 35 KΓ-isomorphism, 2 KΓ-map, 2 KΓ-module, 2 irreducible, 28 K-space, 2 Knuth unwinding, 79, 96 Knuth’s theorem, 103 Knuth, Donald, 78, 96 L-homomorphism, 135 ladder, 92, 93 λ-rectangle, 134 lattice permutation, 122, 140 letter, 73 Littelmann algebra, 130, 137 Littelmann, Peter, 73, 96, 140 Littlewood–Richardson rule, 140 Martins theorem, 69 matrix unit, 137, 138 modular reduction, 8, 16, 68 modular theory, 6, 16 module MQ , 131 Morita equivalent, 67 move basic, 78 multi-index, 11 operator B, 76, 122 operator C, 122, 127 operators e˜c , f˜c , 75 P -symbol, 74 partition, 23, 60 column p-regular, 60 path, 74 path model, 73 pivot, 105 place, 81 place permutation, 11 Proposition B, 78, 116 Q-symbol, 74 representation, 2 A-rational, 4 matrix, 4 polynomial, 5 Robinson–Schensted algorithm, 74

root, 75 simple, 75 root operator, 73, 76 row stabilizer, 34 row standard, 88 Schensted, 81 Schensted process, 74, 81 inverse, 89 Schensted’s map, 81 theorem, 89 Schur algebra, 7, 13 Schur function, 30 Schur functor, 53, 54, 57 semigroup, 2 semigroup-algebra, 2 semisimple, 133 shape of a tableau, 74, 81 of a weight λ, 81 of a word, 95, 129 size of a word, 125 Specht module, 58, 63 dual, 59, 62 standard, 81 column, 88 row, 88 standard tableau, 36, 74, 81 Steinberg’s tensor product theorem, 50 support, 131 symmetric function, 26 complete, 50 elementary, 26, 29 monomial, 26, 29 ring of, 27 symmetric group, 11, 53 symmetric power, 19, 36, 44 symmetric tensor, 44 tableau, 33, 74, 81 basic λ-, 34 standard, 36, 74, 81 Young, 74 Theorem A, 78, 121, 124 unit matrix, 137, 138

Index unital, 132, 134 unwinding, Knuth, 79, 96 weight, 23, 73, 126 dominant, 73 of a word, 76

weight space, 23 Weyl group, 23 Weyl module, 10, 33, 43, 140 word, 73 Z-form, 7, 8, 16, 50

161

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Recent Reprints and New Editions Vol. 1618: G. Pisier, Similarity Problems and Completely Bounded Maps. 1995 – 2nd exp. edition (2001) Vol. 1629: J.D. Moore, Lectures on Seiberg-Witten Invariants. 1997 – 2nd edition (2001) Vol. 1638: P. Vanhaecke, Integrable Systems in the realm of Algebraic Geometry. 1996 – 2nd edition (2001) Vol. 1702: J. Ma, J. Yong, Forward-Backward Stochastic Differential Equations and their Applications. 1999. – Corr. 3rd printing (2005) Vol. 830: J.A. Green, Polynomial Representations of GLn , with an Appendix on Schensted Correspondence and Littelmann Paths by K. Erdmann, J.A. Green and M. Schocker. 1980 – 2nd corr. and augmented edition (2007)

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