Representation Theory of the Symmetric Groups

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

682 G. D. James

The Representation Theory of the Symmetric Groups

Springer-Verlag Berlin Heidelberg New York 1978

Author G. D. James Sidney Sussex College C a m b r i d g e C B 2 3HU Great Britain

AMS Subject Classifications (1970): 20 C15, 20 C 20, 20 C30

ISBN 3-540-08948-9 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-08948-9 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

Preface

The r e p r e s e n t a t i o n by Frobenius by Young.

Althou~h

Day dividends, difficult

presented

this

are

These

identical

to a student

a proof

learn more

to the general

easily

at the expense to check

with

fill

of s u p n l y i n g This

theorems,

(see [16]

than with

and

leaving

for h i m s e l f

which

is e s p e c i a l l v

him to w r i t e

that the reader

from the p a r t i c u l a r

unpleasant

argument

details

known

the type-

of this book we have

on the ~ r i n c i p l e

since many w h o read e a r l y proofs in the details

line,

the complete

at

subject,

can often be best p r e s e n t e d

the sometimes

for himself.

one of the central

Rule,

example,

in the

Many of the results

and chalk

In many rlaces

by t r a n s l a t i n g

However,

quicker

to any of

III course

theorems

unpublished.

the c o r r e c t

than by readinn

perhaps

has n e v e r

some of the t e c h n i q u e s

for a Part

arguments

if he wishes.

by a w o r k e d

for a full proof.

~iven

a blackboard

combinatorial

Drool

for one,

undoubtedly just how

will be found here

that

all the basic

previously

with

bv i n d i c a t i n q

out a complete nreceded

on those

and include

since

The author,

realize

studied

of papers

to his.

as some m a t e r i a l

word,

first

work w o u l d

this will

it is p r o b a b l e

are based

to explain

groups was

in a lon~ series

study of Younq's attempted

and so no reference

although

in 1977,

are easier

will

task,

notes

Cambridge

written

a detailed

anyone who has

proofs,

as well

of the symmetric

and then d e v e l o p e d

it is to read his papers.

undertaken Youn~'s

theory

and Schur,

notation

required

is alwa~Is included, the reader m i g h t important

find

when d e a l i n g

as the L i t t l e w o o d - R i c h a r d s o n of this Rule

for a d e s c r i p t i o n

find

it d i f f i c u l t

of the p r o b l e m s

to

encount-

ered). The

approach

adopted

is c h a r a c t e r i s t i c - f r e e ,

places,

such as the c o n s t r u c t i o n

grouns,

where

reader who

the results

themselves

is not f a m i l i a r w i t h

fields must not be d e t e r r e d ordinary

renresentation

ally at the more thought

that t e c h n i c a l

knowledge

theory,

ies w h i c h make

it p o s s i b l e

The most e c o n o m i c a l general I0-Ii

theorems

since

wav

should

is r e q u i r e d

for this book

to learn

(notinq the remarks

Nor

except tables

in those of s y m m e t r i c

the ~round

field.

The

theory over a r b i t r a r y in fact,

to u n d e r s t a n d

the s y m m e t r i c

that

the

by looking

initi-

he be put off by the

for c h a r a c t e r i s t i c - f r e e

groups

enjoy

special

propert-

to be

largely

self-contained.

the i m p o r t a n t

results

without

from r e p r e s e n t a t i o n

Many of the theorems

upon

we believe,

is e a s i e r

situation.

representation

denend

representation

by this;

theory

general

of the c h a r a c t e r

following

theory Example

rely on a certain

is to read 17.17), bilinear

sections

then form,

using 1-5,

15-21. and towards

any

IV

the end we show that this b i l i n e a r by using

it in a new c o n s t r u c t i o n

remarkable symmetric

that its s i q n i f i c a n c e qrouDs

I wish and p a t i e n t

was only

to express

Orthoqonal

in the r e p r e s e n t a t i o n

recently

my thanks

form m u s t have been of Y o u n q ' s

known Form.

to Young, It is

theory of the

recoqnized.

to Mrs.

Robyn B r i n q a n s

for her careful

tvDin~ of my m a n u s c r i p t .

G, D. J a m e s

Contents

1.

Background

from

2.

The

3.

Diagrams,

symmetric

4.

Specht

5,

Examples

6.

The

character

7.

The

Garnir

8.

The

standard

9.

The

Branching

representation

group

tableaux

modules

p-reqular

Ii.

The

12.

Composition

table

of G n

basis

of

15

Sequences

. . . . . . . . . . . . . . . . . . .

27 29

. . . . . . . . . . . . . . . . . . .

34

of

~

36

. . . . . . . . . . .

39 42

. . . . . . . . . . . . . . . . .

44

. . . . . . . . . . . . . . . . . . . . . . .

51

. . . . . . . . . . . . . . . . . . . . . . . . .

54

The

18

Hooks

19

The

Determinantal

2O

The

Hook

21

The

Murnaghan-Nakayama

22

Binomial

23

Some

24

On

25

Young's

26

Representations

Littlewood-Richardson series

for

Mu

skew-hooks

for

Orthogonal of

. . . . . . . . . . . . . . . . . . .

73

. . . . . . . .

Rule

74

. . . . . . . . . . . . .

77

. . . . . . . . . . . . . . . .

modules

matrices Form the

2 and

. . . . . . . . . .

79

. . . . . . . . . . . . . . . .

decomposition

primes

60 65

Snecht

decomposition

. . . . . . . . . . . . . .

dimensions

coefficients

irreducible

Rule

. . . . . . . . . . . . . . . . . .

Form

Formula

The

. . . . . . . . . .

n . . . . . . . . . . . . . . . . . . . .

A

and

module

. . . . . . . . . . . . . . . . . . . .

16

Specht

Specht

homomorphisms

Rule

18

the

representations

17

Index

8 13

22

Theorem

factors

Semistandard

the

5

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

partitions

Young's

References

tabloids

. . . . . . . . . . . . . . . . . . . . . .

irreducible

13

for

1

. . . . . . . . . . . . . . . . . . . and

relations

14

Appendix.

. . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

iO.

the

theory

87

. . . . . . . . . . . . . . of ~

89

. . . . . . . . . . . .

98

n . . . . . . . . . . . . . . . . . .

general

matrices

linear

of

group

the

114

. . . . . . . .

symmetric

groups

3 with

n m< 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

~n 136 153 155

i.

BACKGROUND We

the

shall

group

FROM REPRESENTATION

assume

the m o s t

elementary

possible

to p r o v e

theory

that

algebra,

FG,

factor

properties

all

the

If M is

of the

Proof:

group

Let m be

module

o f M,

isomorphism

The because

first we

concept F,

right-)FG-modules. in the

only

following:

the

of

and with It is

representation

F G - m o d u l e t t h e n M is a c o m p o s i t i o n

a l ~ e b r a r FG.

a non-zero

element

o f M.

M is i r r e d u c i b l e ,

Then

mFG

M = mFG.

is

a non-zero

sub-

The map

(r c FG) f r o m FG o n t o M.

By

the

first

@ ~ M

composition

factor

isomorphism

shall work

isomorphic

theorem

over

use case

1.2 M A S C H K E ' S

will

appear

on many

an a r b i t r a r y

field,

when

certain

G-invariant

of Maschke's

occasions,

an F G - m o d u l e

can

THEOREM

bilinear

If G is a f i n i t e

L e t e l , . . . , e m be bilinear

forms,

as in the p r o o f

Theorem:

field of real numbers r then every

a unique

to M.

but not decomposable.

We often

Proof:

the

a field

theorems

to be a n F G - h o m o m o r p l l i s m

a top

of a special

is

using

with

G over

theorem,

be r e d u c i b l e

the

familiar

(unital

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

FG/ker so F G h a s

of

group

and since

seen

is

group

important

0: r + m r is e a s i l y

reader

of a f i n i t e

of the s y m m e t r i c

i.i T H E O R E M

the

THEORY

an F - b a s i s

group

for o u r

f o r m ~ on M s u c h

a n d F is

FG-module

a subfield

is c o m p l e t e l y

FG-module

M.

of

reducible.

Then

there

that

(ei,ej) # = 1 i f i = j, a n d O i f i ~ j. Now,a

new bilinear

This

f o r m is

f o r m can be d e f i n e d

= [ ( u g , v g) geG

G-invariant,

Given

a submodule

= O for e v e r y

using which

the

fact

is the

real numbers,

then

for all

i f u ~ U,



that g in G.

then

required ~ O,

so U n U ± = O.

u g-lcu.

U ± is

f o r U i to b e since

We

by definition,

that



Thus

= O,

f o r m is G - i n v a r i a n t .



and therefore

in the s e n s e



But =

our

condition

If u ~ O,

= d i m M,

that

by

f o r all u , v i n M.

U o f M, v E U i m e a n s ,

u in U.

=

#

shall

This

shows

a submodule

F is a s u b f i e l d prove

an F G - m o d u l e

below

t h a t v g ~ U l, o f M.

o f the

field

complementing

U in M as

required. We now

remind

the

reader

of some elementary

of

that dim U + dim U A

algebra

involving

bilinear

M

is

by

forms.

Let

M be

the

vector

M*.

Let

a finite-dimensional space

of

el,...,e k be M.

O if

By

considering

M*

can

~ j.

element

~ of

el, .... ,em, is

a basis

thus: of

el~

=

...=

M

Suppose

now

is

m'

M with

in

that

a linear ker

is

® =

M and

M*.

every

s ubspace

1.3

dim

V ~

that

@ are

linear,

9v

ker0

1.4

i =

we

see

that

combination

j,

and

any of

el,...,E m

m

of

V,

spans

if

and

only

if

V ° and

every

bilinear

form,

non-zero

m

in

< M

, >,

on

M

there

is

an

where

( x E M).
is

since x ~ M,




<m,x>

= dila M

is

in

the

linear

second in

the

= O}=

O,

since

@ is

an

isomorphism

, so

V ± corresponds

place, first

the~]Jnear

and place.

form

between

to V °.

Thus,

for

M

equation

given

x + V± ÷

shows

dual

to

1 if

between

dimensions

gives

V.

+ V ±,

and

U ±±

for

m + ~m

M

V + dual

This

0=

a basis

extend =

of

M.

identification,

this

=

~v:

Since

denoted

subspaces

O

c U

c V

= M,

we

have

V l c U ±,

define

x + V ± = x'

V/ker

is,

all

V ± = dim

V ±±,

9:

If

be

Therefore,

annihilator

ek+l,...,e

since

dim

this

generally,

may

a linear m.

dual

V,

V + dim

More

and

e i ej

The will

Define by

I for But

V ±l

we

as

a symmetric

transformation, {m £ M

the

<m,x>

~m e M*,

Under

Since

have

~ O).

x +

non-singular.

and

we

M + M

~m:

0 is

and

e l , . . . , e m,

...+(em~)e

V ° = dim

(That

<m,m'>

Now,

on

V,

by

F.

F,

M*

Therefore,

non-singular

see

over

into

ej ~ M*

uniquely

to V °,

V + dim that

9:

We

M

a subspace

action

(el~)E 1 +

= dim

= O. dim

which

be w r i t t e n

# belongs

ek~

of

define

the

space

from

and

dim Further,

1 ~ j ~m,

~ =

M*

maps

a basis

el,...,e m of i

For

vector

linear

= U ~ V, V/U

v~hen O of

M / V ±.

If

M is

into g U

then

of

U±/V l b y v + %v, w h e r e



(xE U±).

x - x'e

V±,

is w e l l - d e f i n e d . but =

~ V

In



the

-

same



as b e f o r e ,

= O.

9v

now

{v ~ V l f o r

ker the

and

all

0 = U.

We

dual

of

c M~

V/U~

an F G - m o d u l e

for

x ~ U±



therefore

U ± / V ±. dual

the

,

have

Again, of

group

=

=

can

V n

U ±±

a monomorphisra

dimensions

U I / V ±.

G, w e

O}

In

particular~

turn

from

give:

the

dual

V~

space

M*

into

an F G - m o d u l e

by

m(~g) Notice This not

that

means

that

in g e n e r a l

representing

This

of the

M

(which we to :I.

respect

assume

of M w h e n that

homomorphism.

But =

For e v e r y

form

this,

(x + V l ) ~ v g

pair

be d e d u c e d

and V ±, we

also

from

find

that

this

book,

the

the

the

= <x,vg>

=

T' (g -I)

el,...,em conjugate

complex

numbers.

~v g,

definitions.

1.4

0: v + ~ =

If U are

is

FGa G-

(xg -[ + V ± ) ~ v

=

as r e q u i r e d .

U and V of M,

next

then

basis

in

= <xg-l,v>

(~g)h.

complex

isomorphisms that

=

of M ) is

< , > is G i n v a r i a n t .

the

U± + V± =

~(g~

is the m a t r i x

dual

is the

over

w e faust show

of s u b s p a c e s

that

the dual

if T(g)

to the

of M*

are w o r k i n g

then

call

e l , . . . , e m of M,

respect

(x + V ±) (~v g) ' and ~vg

can e a s i l y

Throughout

we

of M,

To v e r i f y

Indeed,

character

the b i l i n e a r

and V are FG-subraodules isomorphisms.

g with

the

g ~ G).

to e n s u r e

shall

to the b a s i s

representing that

~ E M*,

of g a p p e a r s

FG-isomorphic

means

(x + V ~ ) g - l ~ v

(meM,

inverse

the m o d u l e

character

~ow

(rag-~

the

g with

is the m a t r i x of M*.

=

letting

(U + V) ±

Replacing

=

U± n V±

as

,

U and V by U ±

(U n V) ±. picture

will

be useful:

M

I V+

V±

\

/ V~vnv±

I

O The s e c o n d i s o m o r p h i s m t h e o r e m g i v e s

V/(V nV ±) ~ (V+ V ~ ) / V ± .

But

(V + V±)/V ± ~ d u a l o f V/(V + V±) ±, by 1 . 4 = d u a l o f V / ( V n V±), so 1.5

F o r e v e r ~ F G - s u b m o d u l e V o f N r V/(V n V±) i s

a self-dual

FG-

module. Every up in this

irreducible

It is v e r y submodule a basis

~mportant

V of M.

of V?

of c a l c u l a t i o n ined with

representation

of the s y m m e t r i c

group

will

turn

fashion.

The

How

can we

answer

if V has

respect

to n o t i c e

to a b a s i s

entry

of A be

<ei,ej>.

1.6

THEOREM

The

compute

is s i m p l e

large

dimension

that the

V n V ± can be n o n - z e r o dimension

in t h e o r y ,

dimension.

The

e l , . . . , e k of V by

of V / ( V n

but will

of V / ( V n V ±) e q u a l s

to the

V±),

require

Gram matrix, letting

for a

the

A,

given

a lot

is def-

(i,j)th

rank

of the

Gram matrix Proof:

with

respect

As usual,

map

to a g i v e n

V + dual

0: v ÷ ~v L e t e l , . . . , e k be basis

of V*.

Since ~e i

Thus matrix

the

of V* .

the

rank

But,

of the

The

only

use w i t h o u t

and

the

with

we

and

respect

el,...,

e k be

"

el,...,e k coincides

to the b a s e s

with

e l , . . . , e k of V and

0 = V n V ± , so d i m V / ( V n V ±)

ker

the d u a l

have

the

el,...

= dim

Im @ =

Gram matrix.

results

from general

are

those

irreducible

group

over

43.18

and E x e r c i s e

1.7

Let

the n u m b e r

,

for the b a s i s

visibly,

following C,

(u E V)

of V,

= < e i ' e l > e l + ' ' ' + < e i ' e k > gk

proof

and p - m o d u l a r

basis

= <ei,ej>

Gram matrix

of @ t a k e n

£k

U @ v =

given

ej~e{

of V.

of V by

where

the

basis

field

us h o w m a n y

representations

well-known

the

representation

telling

result

of c o m p l e x

theory

a finite

about

which

inequivalent group

(cf.

Curtis

shall

possesses,

representations

numbers

we

ordinary

of a f i n i t e

and

Reiner

~ ]

43.6).

S be an i r r e d u c i b l e of c o m p o s i t i o n

C G - m o d u l e t and M be any ~ G - m o d u l e .

factors

of M i s o m o r p h i c

that

results

Then

to S e q u a l s

dim HOm~G(S,M). In fact, approach, foolish

it turns

and T h e o r e m

to p o s t p o n e

Readers Frobenius characters,

out

i.i g i v e s

proofs

interested

Reciprocity so we

these

Theorem

in c h a r a c t e r

Theorem

assume

everything

until

these

and

are

redundant

we w a n t , i.i

values

when

it w o u l d

be

can be a p p l i e d . will

be

the o r t h o g o n a l i t y

results

but

in o u r

familiar relations

discussing

with for

characters.

the

2.

THE

SYMMETRIC

The proofs

of the

any e l e m e n t a r y

book

A function of n n u m b e r s , with

degree

n, w h i b h and

will

~n

{l,2,...,n},

we

as in the

onto

itself

of f u n c t i o n s ,

be d e n o t e d

shall

common

to see

write

by

~X

We

numbers

is

~n"

(where

can be

is c a l l e d

found

in

for the

a permutation

of n n u m b e r s , the

Note

O~

practice

that

to w r i t e

symmetric

that

= i).

~n

together

group

of

is d e f i n e d

If X is a s u b s e t

subqroup

of

~n w h i c h

i~

2~

3~

the

orbits

for

of

fixes

every

usually

as f o l l o w s :

of the

group

generated

as a p r o d u c t

by n

, it is

of d i s j o i n t

cycles,

:

suppress

the

if ~ i n t e r c h a n g e s fixed,

~

n~

~ can be w r i t t e n

example

a permutation

( 1 2 3 4 5 6 7 8 9 ) 3 5 1 9 6 8 7 2

example,

section

X.

By c o n s i d e r i n g simple

in this

set of all p e r m u t a t i o n s

has n~ e l e m e n t s

outside

It is

stated

theory.

{l,2,...,n}

the

composition

n

number

results

on g r o u p

from

and

the u s u a l

~ O,

GROUP

then

4

1-cycles

=

when

the d i f f e r e n t

~ is c a l l e d

(2568)(13) writing

numbers

a transposition

( 4 9 ) (7)

a permutation.

a,b

and

leaves

and is w r i t t e n

For the o t h e r

as ~ =

(a b). All

our m a p s w i l l

(i 2) ( 2 3 )

=

(i 3 2 ) .

mathematicians Since

would

on the right;

This

must

point

interpret

(i I i 2 . . . i k)

any p e r m u t a t i o n ,

be w r i t t e n

=

in this way,

be n o t e d

the p r o d u c t

as

carefully,

as a p r o d u c t

as some

(i 2 3 ) .

(i I i2) (i I i3)... (i I ik),

can be w r i t t e n

we h a v e

any

cycle,

and h e n c e

of t r a n s p o s i t i o n s .

Better

still, 2.1

The This

transpositions is b e c a u s e ,

(b-3,b-2)... If n product Hence

(a,a+l)

2.2

when

= ~i ~ 2 " ' ' a j

there

that

sgn

~ =

DEFINITION

are n o n - n e g a t i v e

1 < x sn g e n e r a t e can

conjugate

(b-l,b)

then

are

two w a y s

of w r i t i n g

it can be p r o v e d

by

(b-2,b-l)

that

~ as a

j - k is even.

function

~ {±i}

(-i) ] if ~ is a p r o d u c t I =

~n"

(a b).

= T1 Y 2 " ' ' T k

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

with

a < b, we

to o b t a i n

of t r a n s p o s i t i o n s ,

sgn: such

(x-l~x)

(Ii,12,~3,...)

integers,

with

of j t r a n s p o s i t i o n s .

is a p a r t i t i o n

of ~ if ~ i , 1 2 , ~ 3 , . . .

11 _ > 12 al 3 ~ . . . and

[ I i = n. i=l

The permutation the g r o u p (4 9)(7)

~ is s a i d

generated has

following

by ~ h a v e

cycle-type

will

to have lengths

is, we

usually

repeated

often

parts

Since

be

2.3

The

titions

have

We

should

2.5

EXAMPLE

this

same

the

(4,2,2,1)

zeros

are

=

at the

conjugate

cycle

as the

(4,2a,i).

end of

l, and i n d i c a t e

in

~n

if and o n l y

classes

of

~n

equals

if the

the n u m b e r

group

of

inequi~alent

number

the n u m b e r

of n.

ordinary

us

look

of p a r -

~n

permutes

F of d i m e n s i o n

at an e a s y

representation

basis

~n

which

1,2,...,n

elements

of M(n-l'l) ; the

not hard

to

guesswork.

spot

find

another

If F = ©,

Maschke's

Theorem

on M (n-l'l)

and

then

an

~

n

we

=

but

~n

be

~ n ).

acts

suppose

of r a t i o n a l

construct

an

an i n v a r i a n t

We

inner

product

shall

space

we w i s h

n complement

Then

an

certainly

Notice

though,

whatever

is a c o m p l e m e n t

that

the

(,) gives

field.

an

S (n-l'l)

to U if and o n l y

U spanned It is

to e l i m i n a t e

inner

of product

to U. (*)

=

Then O}

S (n-l'l)

of U ±, and it is e a s y to see that we have e q u a l i t y . M (n-l'l) = S (n-l'l) @ U w h e n F = ~.

M (n-l'l)

is a subThus

~

- i n v a r i a n t b i l i n e a r f o r m on n is a l w a F s a s u b m o d u l e , too (It

if c h a r

F ~

n.)

S (n-l'l)

is a S p e c h t

module. Are

there

any o t h e r

easy ways

of

,

denote

the p r o o f

-invariant

on M (n-l'l)

a I +...+

E ~

(~ - [ ) F ~ n .

a vector

trivially.

numbers, ~

for

directly

1,2,...,n

= 1 if i = j and O if i z j

{[ a i [ I ai

=

S (n-l'l)

field

U ± will

-invariant U ±

Let

on w h i c h

submodule,

the

suggests

defines

a submodule

~ is a s u b m o d u l e

~n

arises

; take

called

We

can e a s i l y

of

example:

= i-~ (~ e

by ~ + ~ + . . . +

representations

a representation

the n u m b e r s

n, w i t h

irreducible

of G, so

of n.

first

is a n a t u r a l

classes

irreducible

a i m to c o n s t r u c t

Let

of i n e q u i v a l e n t

of c o n j u g a c y

of p a r t i t i o n s

therefore

that

G,

act on the s p a c e by [ z (n-l,l) r e p r e s e n t a t i o n by M

module

(2 5 6 8) (1 3) such

type,

to the n u m b e r

There

fact

over

let

finite

the

partition

and

=

the

of c o n j u g a c y

number

each

from

the

is e q u a l

equals

space

suppress

for any

The ~n

Thus,

Abbreviations

by an index.

number

Now,

of

12 ~...

of

of n.

~G-modules 2.4

I if the o r b i t s

adopted:

two p e r m u t a t i o n s

permutations

11 ~

(4,2,2,1,0,O,...).

(4,2,2,1,0,O,.~.) That

cycle-type

constructing

representation

modules

for

~n ?

by u n o r d e r e d an F ~ n - m o d u l e difficult

Consider

pairs

iT

if we

define

to h a n d l e ,

but

< j s n } is a t r i v i a l moment,

but

the v e c t o r

(i ~ j). ~

submodule.

observe

generally,

we

M (n-2'2),

has

= i~,j~.

it is n o t

simply

space

M (n-2'2)

This

irreducible, We

do n o t

over

dimension

F spanned

(3) , and b e c o m e s

space

should

since

[ { ~

go i n t o

details

that M (n-2'2) s u p p l i e s m o r e

not

be

Ii ~ i for the

scope

for i n v e s -

tigation. More

by u n o r d e r e d this

space

there

cible)

loss

F~n-module

Flushed can do. shall

Let

denote

be

the

ij

followed

with

space

vectors

unless

by u n o r d e r e d

a m.

parts

space

we

This have

M (n-m'm)

spanned

j = k).

Since

(n-m)-tuples,

means

that

for e v e r y

a corresponding

(redu-

at o u r d i s p o s a l . this

success, be the

(i ~ j).

spanned

we

The

be d e n o t e d

by

~

go on and see w h a t

spanned action

&n

no two

, but

--

by o r d e r e d

is i[ ~ = ~j~ _

consisting

k,__where

and have

should

space

by v e c t o r s

by a l - t u p l e

may

notation

that n-m

vector

i 3 ~ ik

spanned

two n o n - z e r o

M (n-2'12) by ~[

the

(where

to t h a t

in a s s u m i n g

of n w i t h

with

i I. • .i m

is i s o m o r p h i c

is no

partition

m-tuples

can w o r k

.

it seems

as a b a s i s

we

which

2-tuple

and k are equal.

that we vector

we

L e t M (n-3'2'I)

of an u n o r d e r e d

of i,j

else

pairs,

should

These

change

of M (n-3'2'1)

our in

i I ....... in_ 3 in- 2 in_ 1 i n place

of in- 2 in- 1 i n By now,

each partition introduced contains char

it s h o u l d

in the n e x t a Specht

F = O.

be

I of n.

clear

The

section•

module

how

to c o n s t r u c t

notation

we n e e d

M 1 is r e d u c i b l e

S I, w h i c h

it turns

an F G

n

to do this

out,

(unless

-module formally I =

M 1 for is

(n)),

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

if

but

3.

DIAGRAMS,

3.1

DEFINITIONS.

{(i,j) I i,j If

TABLEAUX

• •

(i,j) • [I],

pectively,

We

then

shall

There

axis

right

brackets

[4,22,1],

not

3.3

of

[I].

of t h o s e

following

[I]

is

set o f i n t e g e r s ) .

T h e k th r o w nodes

whose

(res-

first

example:

about which

giving

work with

one

upwards:

examples

way

round

diagrams

their

first

coordinate

It is c u s t o m a r y

of d i a g r a m s ,

to d r o p

so w e w r i t e

[(4,22,1)].

If

and write

of n is p a r t i a l l y

I and ~ are p a r t i t i o n s

I ~ ~, p r o v i d e d j,

lli

->

l I z U, w e w r i t e

EXAMPLE.

is s h o w n

the d i a g r a m

is k.

convention

a n d the s e c o n d

DEFINITION.

I ~- U a n d

then

Z is the

x x x x Ill = x x x x x

(4,22,1)

for all If

a node

consists

as in the

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

~,

called

coordinate

when

of n, (Here,

Some mathematicians

the i n n e r

3.2

is

is n o u n i v e r s a l

to the

inates

(i,j)

of a diagram

draw diagrams

be shown.

The

1 ~ j ~ I i}

second)

I =

should

If I is a p a r t i t i o n

1 ~ i

column)

(respectively,

AND TABLOIDS

by the

The

ordered

by

of n, w e

say

that

I dom-

that

[ Zi i=l

I ~ U.

dominance

relation

on the

set of partitions

of

6

tree:

(6) (5111

/(4!21\ (3,3)

(4,12 )

/ /\ (3,2,11\ (3,13 )

(23 )

~(22,121'/ (2!14 )

i (16 ) The

dominance

partitions, the s e t 3.4 only

but

order

of p a r t i t i o n s .

DEFINITION if the

is c e r t a i n l y

it is s o m e t i m e s

least

If

The I and

useful

one w e

use

the

"correct"

to h a v e is g i v e n

~ are p a r t i t i o n s

j for which

lj ~ ~j

order

a total

for

>, on

by

o f n, w r i t e

satisfies

to use

order,

lj

> ~j.

I > ~ if a n d (Note t h a t

some

authors

nary

order

write

It is s i m p l e order

this

to v e r i f y

~, in the s e n s e

lication

relation

as

I < ~).

This

is

called

the d i c t i o -

> contains

the p a r t i a l

on p a r t i t i o n s .

is

false

that

that

the

total

I m ~ implies

order

I > ~.

But

the

reverse

imp-

since

(6)>(5,1)>(4,2)>(4,12)>(32)>(3,2,1)>(3,1~)>(2~)>(22,12)>(2,1~)>(16). 3.5

DEFINITION

obtained tition

only

use

to t a k e

the

be m o r e

than one

are b o t h way

Ill

of n conjugate

The which

If

the

rows

of the rows

total

of the

order

next

partitions

ordering

a bijection 3.6

from

partitions,

to d e f i n e

[I]

DEFINITION by

allowing

no repeats.

to

replacing

example,

each

1245

permutation to t h e wins

and

the

t and

f o r m of the n e x t

3.7

for every columns Proof: t 2 in must

such at

the

Since

order

there

in

may

(4,2,12 ) a n d

(32,2)

is n o

"syn~letrical"

order

is r e v e r s e d

by

that

~ l'. This

c a n be d e f i n e d

but we prefer

Ill b y o n e

4573

are

l-tableaux

result,

sends

the

arrays

o f the

less

as

formal

of integers

integers

1,2,...,n,

(4,3,1)-tableaux.

in t h e n a t u r a l

the

first

definition

the n e w

o f the

which

relates

way;

~,

thus

tableaux

of a tableau

tableau

representation

COMBINATORIAL

that

Imagine If]

~n.

though,

t and a permutation

~ gives to the

t I is

i the numbers

o f t I.

have

is

the

the

above

as a f u n c t i o n compositions

of

t~).

theory

of

~

the d o m i n a n c e

depends upon a n order on partitions

of t a b l e a u x .

THE BASIC

and suppose

the

in

the

a tableau

approach

to a p r o p e r t y

there

to see,

say,

(e.g.

is o n e o f the n~

(253)

(Of c o u r s e ,

Given

Every

[I'3

l' is the p a r -

6 set of

(i 4 7 8 6 )

functions

diagram

Ill.

218

on t h e

second.

here.

of

of n

so t h a t

if ~'

node

8 acts

table

8),

{l,2,...,n},

367

~n

in

is a h - t a b l e a u .

A h-tableau

obtained

For

of

It is i n t e r e s t i n g

thing

conjugate

> is to s p e c i f y ,

partition

I ~ ~ if a n d o n l y The

the

columns

character

self-conjugate

conjugates.

and

to I.

self-conjugate

of totally

~aking

is a d i a g r a m ,

by interchanging

Then

LEMMA

a l-tableau f r o m the

and

I a n d ~ be p a r t i t i o n s

t 2 is a ~ - t a b l e a u .

ith r o w o f

t2 belong

o f n,

Su~ose

that

to d i f f e r e n t

I ~ ~.

that we

can place

that no

two n u m b e r s

least

Let

~i c o l u m n s ;

the

that

~I n u m b e r s

are is

f r o m the

in the s a m e l I a ~i"

first

column.

Next

insert

row of

Then the

Ill ~2

10 numbers

f r o m the s e c o n d

to so this, we r e q u i r e have

row of t 2 in d i f f e r e n t ll+ 12 >- ~i + ~2"

columns.

Continuing

To have

space

in this way,

we

I ~- ~.

3.8

DEFINITIONS

subgroup

of

i.e.

~n

If t is a t a b l e a u ,

keeping

its r o w - s t a b i l i z e r ,

R t = {7 E ~ n I for all i, i and iT b e l o n g

The c o l u m n For e x a m p l e ,

stabilizer

when

R t, is the

the rows of t f i x e d setwise.

Ct, of t is d e f i n e d

t = 1245 367

,

Rt =

~{i

to the same row of t}

similarly.

245}

x

~{367}

x

~{8}

8 and

[Rtl = 4' 3' i' Note

3.9

t h a t Rtw = ~ - * R t ~

DEFINITION

Define

and

an e q u i v a l e n c e

t a b l e a u x by t I ~ t 2 if and o n l y tabloid

{t} c o n t a i n i n g

equivalence

Ctw = z - I C t ~ . relation

on the set of l-

if tl~ = t 2 for some zE Rtl

t is the e q u i v a l e n c e

The

class of t u n d e r this

relation. ,!

It is b e s t to r e g a r d entries". the rows 345 12

In e x a m p l e s , of t.

245 13

Then z-l~

235 14

135 24

¢ ~-IRtl~

(i)

When

and

the

132 54

are m a n y o t h e r

on p a r t i t i o n s ,

row

lines b e t w e e n

so {tl~}

134 25 = 123 45

sensible for m o s t

= {tla~}

DEFINITION

(3,2)-tabloids

Given

tabloid

= {t2~}.

if for some i

ordering

{t 2}

{t2}.

in this order,

of h - t a b l o i d s ,

any t a b l e a u

less than or e q u a l

action

by

of o u r p u r p o s e s .

the b e s t

This

t 2 = tlO for some o in Rtl.

r o w of {t I} than

orderings

123 45

.

j > i, j is in the same row of {t I} and

the

124 35

by {t}z = {tz}.

implies

h-tabloids

i is in a h i g h e r

is s u f f i c i e n t

234 15

{t I} < {t 2} if and o n l y

We have w r i t t e n

of e n t r i e s

{tl} = {t2}

= Rtl~,

order

DEFINITION

(ii)

unordered

{t} by d r a w i n g

125 34

(3,2)-tabloids,

since

We t o t a l l y

3.11

denote

acts on the set of h - t a b l o i d s

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

3.10

we s h a l l

as a t a b l e a u w i t h

Thus 145 23

are the d i f f e r e n t

~n

a tabloid

As w i t h

above.

There

but the c h o s e n m e t h o d the d o m i n a n c e

is a p a r t i a l

t, let mir(t)

order

one:

denote

the n u m b e r

to i in the first r rows of t. T h e n

write {t I} ~ {t 2} if and o n l y if for all i and r

m i r ( t I) ~

mir(t2).

11

This compare

orders

only

the

By c o n s i d e r i n g m i r ( t I) 3.12 3.13

then

of all s h a p e s

associated

the

For

~-tabloids

EXAMPLES

the

first

(mirtl))

Therefore, The

i,

then

and

sizes,

the

largest

b u t we

r, such

{t2} ~ {t 1} ~ {t 2} i m p l i e s

If t I = 1 3 6

7 rows

and

shall

the same p a r t i t i o n . that

that

{t l} a n d

(i)

with

largest

< m i r ( t 2) , it f o l l o w s

(mir(t2))

(ii)

tabloids

tabloids

and

< {t2L.

t 2 = 124

257

356

4

7

3 columns

{tl~

of the m a t r i c e s

(mir(tl))

and

are

=

1

1

1

1

1

1

1

2

2

2

2

2

2

3

3

2

3

3

2

3

4

3

4

4

2

4

5

3

5

5

3

5

6

3

6

6

3

6

7

3

6

7

(mir(t2))

=

{t I} ~ {t2}. tree

below

shows

the

~ relation

on the

(3,2)-tabloids:

345 12

i 245

145

235

2 3

Y"T--

\234 24

15

/

\

/

125 34

134 25

35

t 123 45

Suppose row of t. 3.14

that w

Then

mir(t(wx))

< x and w is in the

the d e f i n i t i o n - mir(t)

of m

= /i

~-l

ir

(t)

ath row

and x is in the bth

gives

if b -< r < a

and

w < i < x

if a < r < b

and

w ~i <x

12

O otherwise. Therefore 3.15

{t}

~ {t(wx)}

W h e n we p r o v e the

tabloids

(or are

the

{t} same

if w

Young's

and

{t(x-l,x)}

3.16

LEMMA

If x-i

is no

l-tableau

First

mi,r(t*) first

is

lower

are

than

t I with

for any

= the n u m b e r

of t* =

By

{0

first

to k n o w

adjacent

in the

that 4 order

t is a l - t a b l e a u f

tableau

t* w i t h

of n u m b e r s

then

.

i* in the equal

r*th

row,

to i* in the

is

lower

than

x in t, and

{t}

~ {t I}

3.14,

mir(t) the

need

if r > r*

= mir(t(x-l,x))

m i r ( t I) = mir(t)

and

shall

if r < r*

that x-i

mir(t) Therefore

x in t.

~ {t I} ~ { t ( x - l , x ) }

(t*)

{t(x-l,x) } .

we

x in tf and

{t}

that

suppose

than

immediately

note

1

By

Form,

- mi,_l,r

r rows

Now

Othogonal

lower

tabloid):

there Proof:

< x and w is

- mi_l,r(t)

paragraph

and x a p p e a r

in the

l-tableaux.

Therefore,

same

of

place

if i ~ x-l.

if i z x-i = m i r ( t I) - m i _ l , r ( t I) if i ~ x-i the proof, in t and

{t I} = {t} or

all

t I.

the n u m b e r s But

{t(x-l,x)}

t and

except

x-i

t I are b o t h

as r e q u i r e d .

or x.

13 4.

SPECHT With n

by

MODULES

each

partition

~ of n, we

G~

=

The module

~ { 1 , 2 , .... ~i }x

study

M ~ of

on

~

.

The

the b a s e

field

different

Specht

modules,

the o r d i n a r y

The = {t~}

~ 4.2

over action

of

as

~ varies

subgroup

~

of

elements

and b e c a u s e

one

~n

with

S ~ is

of M ~,

numbers),

partitions

of

of n,

the give

all

n

field,

are

the p e r m u t a t i o n

a submodule

of r a t i o n a l

an a r b i t r a r y

basis

~ { ~ i + ~ 2 + i , . ., ~ i + ~ 2 + ~ 3 }x

starts

over

on t a b l o i d s has n E x t e n d i n g this a c t i o n

(~ £ ~ n ) .

and

let M ~ be

the v a r i o u s

already to be

been

~-tabloids.

defined,

linear

is t r a n s i t i v e

the vec-

by

{t}~

on M ~ turns

M~

on t a b l o i d s ,

with

tabloid,

M ~ is the p e r m u t a t i o n

a cyclic

field

~

an F G n - m o d u l e , stabilizing

(the

~n

module

representations

L e t F be

F whose

of

Specht

is ~

irreducible

DEFINITION

tor s p a c e

into

a Young

~ { ~ i + i , .... ~i+~2 }x

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

and w h e n

4.1

associate

taking

F Gn-module,

module

generated

of

~ on the s u b g r o u p ~ . M ~ is n an[ one t a b l o i d f and d i m M ~ = n~ /

by

(~i~_~2 ~ .... ) . 4.3

DEFINITIONS

sum,

Kt'

is the e l e m e n t

the e l e m e n t s to e a c h

Suppose

in the

of the

column

permutation.

that

t is a tableau. group

algebra

stabilizer

Then

F ~n

the s i ~ n e d

obtained

of t, a t t a c h i n g

the

column

by suma~ing signature

In short,

is

characteristic when

defined


0

= S(4' l)

I

4 (see E x a m p l e

5.1)

S(3,1) ,(3,2) n Ker ~i >_ O

I S(3,2) I

But dim M (3'2) In p a r t i c u l a r ,

~ S (3'2)

>- 5

= lO, so we have e q u a l i t y

dim S (3'2)

= 5 and S (3'2)

in all p o s s i b l e

= Ker ¢ o n

places.

Ker ~i' as we w i s h e d

to prove. 5.4

EXAMPLE I

S (2'2)

is s p a n n e d

2,

by the graphs

1

1

2.

4_

t

2.

-f

-I

-I

4~

3

Clearly, When

-I

4.

the first two

4

t

3

form a basis.

char F = 2, S (2'2)

that any p o l y t a b l o i d

~

-I

contains

c S (2'2)± none

of edges:

\/

The reason

or both

edges

underlying

this

of the f o l l o w i n g

is

pairs

22

6.

THE

CHARACTER

There cters

of

table

are m a n y ~n"

Theorem

~n

the o t h e r

hard all

9)

n

of e v a l u a t i n g

character

is v e r y

useful,

this w a y w e h a v e

hand, Rule

to use

if just (section

the o r d i n a r y of

but

to c a l c u l a t e

out

all

a few e n t r i e s

are

required,

is the m o s t

on this

irreducible

O n _ 1 is k n o w n ,

to w o r k

21)

a computer

table

formula.

the

character

the e a r l i e r

efficient

tables.

On

the M u r n a g h a n -

method,

The m e t h o d

chara-

the B r a n c h i n g

but

given

it is

here

finds

the e n t r i e s in the c h a r a c t e r t a b l e of ~ simultaneously. It is n to R . F . F o x , w i t h some s i m p l i f i c a t i o n s by G . M u l l i n e u x . l Let X d e n o t e the o r d i n a r y i r r e d u c i b l e c h a r a c t e r of ~n c o r r e s -

due

ponding S@

to the p a r t i t i o n Let

~X by

OF

ways

If the

(section

of

Nakayama

TABLE

1G denote

is a Y o u n g 4.2

~ - that

the

trivial

subgroup,

(The n o t a t i o n

is,

the

character

character

and that

+ ~n

1 ~X

+G m e a n s " i n d u c e d

of the ~ G

of a g r o u p is the

up to G" and

n Recall

G.

character

%G m e a n s

module that

of M ~ 1

"restricted

to G" .) All

the m a t r i c e s

by p a r t i t i o n s

of n,

a composition

factor

tions

~ with

6.1

%~he m a t r i x

c._!.l~ ~ +_ ~ n ' (see B =

p > ~

given

once,

m =

Imx~) lower

for ~ 5 '

Let partition

will

order

have

rows

(3.4).

and c o l u m n s

Since

factors

M

has

correspond

indexed

S~

as

to p a r t i -

4.13), ~iven

by ml~

triangular below).

= the

with

character

l's d o w n

It f o l l o w s

inner

product

the d i a ~ o n a l .

at o n c e

that

the m a t r i x

by

bx~ = l ~ I is u p p e r

section

and the o t h e r

(Theorem

X ~ ) is

the e x a m p l e (bxp)

in this

in d i c t i o n a r y

(xX,l~ +G n)

triangular. ~

denote

p, a n d

the

conjugacy

let A =

(alp)

be

class

of

~n

t/~e m a t r i x

corresponding

given

to the

by

al~ = IS x n 6~1 The is known,

matrix the

A is n o t h a r d

character

straightforward [ clp P Therefore,

matrix agp

[ bpl

I

B = CA', bp~

=

C =

(clp)

manipulations.

=

But,

to c a l c u l a t e ,

table

(×x, ~ , l where

A'

I G x lI

of First

@V)

is the (1 ~ X +

and we ~n

claim

that

once

can be c a l c u l a t e d

note

it

by

that

= blv. transpose

~n'

1 ~

of A.

+ G n)

~) =

l

;. P

(x

evaluated

on

an e l e m e n t

of type

23

14). I S~ n ~141 = [ (n'. / 1(~141)

I~x

14

n ~1411~,~

~ ~141

= X In: / 1~'1411 axu a14 14

If A is k n o w n ,

we

can s o l v e

these e q u a t i o n s

top left h a n d c o r n e r of D, w o r k i n g ceeding

to the n e x t

c o l u m n on the right.

t h e r e is o n l y one u n k n o w n be

found,

6.2

B =

If the m a t r i x A =

then we can

(b~14) s a t i s f [ i n g

6.3

EXA~LE

(5)

24

(4,1) ~/ A

=

/

t a b l e C of

Suppose (5)

~n

(a~14), w h e r e

a~

non-negative

=

I~

n

(4,1)

(3,2)

is ~ i v e n b~ C = BA'

-i

.

(3,12 )

(22,1)

(2,13 )

20

20

15

i0

1

6

0

8

3

6

1

2

2

3

4

1

2

0

3

1

1

2

1

(2,13 )

1

(i s )

(3,2) B

=

(3,12 ) (22,1) (2,13 )

(15 )

(15 )

30

(22,1)

1 1

(3,2)

(3,12 )

can

6!4 I is

upper triangular

Then

(3,12 )

(4,1)

and this

Therefore

I ~141)a114 a 14

n = 5.

(3,2)

(5)

and pro-

the e ~ u a t i o n s

[ b141 b14~ = [(n' and the c h a r a c t e r

at the

in turn,

at e a c h stage,

entries.

find the u n i q u e

by s t a r t i n g

column

Since B is u p p e r t r i a n g u l a r ,

to be c a l c u l a t e d

s i n c e B has n o n - n e g a t i v e

THEOREM

k n o w n,

d o w n each

(22,1)

(2,13 )

(15 )

(5)

(4,1)

120

24

12

6

4

2

1

24

12

12

8

6

4

12

6

8

6

5

6

4

6

6

4

4

5

2

4 1

matrix

24 (5)

(4,1)

(3,2)

(3,12 )

1

1

1

1

1

1

1

(4,1)

-i

0

-i

1

0

2

4

(3,2)

0

-i

1

-i

1

1

5

(3,12 )

1

0

0

0

-2

0

6

(5)

C

=

(22,1) (2,1 ~ ) (i s ) The usual

6.4

(1 s)

O

1

-i

-i

1

-i

5

0

1

1

0

-2

4

1

-i

-I

1

1

-i

1

columns

down

tionary

(2,13)

-i

of the

character

one - in p a r t i c u l a r ,

appear

(22,1)

table

the d e g r e e s

are in the r e v e r s e of the i r r e d u c i b l e

the last c o l u m n - b e c a u s e we have

o r d e r on b o t h

NOTATION

the rows

Equations

interpreted~ Aas s a y i n g to S ( ~ ) , S(~'I)

to the

to take the dic-

and the columns.

like

[3][2]

t h a t --M~3'2)

and S(3'2)~ •

chosen

order

characters

= [5] + [4,1]

has

composition

In g e n e r a l

+ [3,2]

factors

are to be

isomorphic

if I is a p a r t i t i o n

[11][12][13]... means

t h a t __M~ has

is the m a t r i x

defined

By d i v i d i n g of that c o l u m n obtained.

each

column

(which e q u a l s

of the m a t r i x I ~pl),

B by the n u m b e r

and t r a n s p o s i n g ,

[4,1]

[3,2]

[3,12 ]

[22,1]

[5]

1 1

[3][2] = [3][1] 2

1

1

1

2

1

1

[21211]

1

2

2

1

[2][1] 3

1

3

3

3

2

[i] s

1

4

5

6

5

Notice

Theorem

14 s h o w s

6.2 has

COROLLARY

Proof:

(mlp)

at the top

the m a t r i x m is

[2,13 ]

[i s ]

1

[4][1]

1

= [5] + [4,1]

are in a g r e e m e n t w i t h E x a m p l e s

Rule in s e c t i o n

product

(m =

1

t h a t the r e s u l t s

+ [3,2]

6.5

.

in 6.1).

[4][1]

[4,1]

ml

In the a b o v e e x a m p l e ,

[5]

m

S~_

= ~ mlp [~] P as a f a c t o r w i t h m u l t i p l i c i t y

of n,

how to e v a l u a t e

and

[3][2]

5.1 and 5.2.

= [5] + Young's

the m a t r i x m d i r e c t l y .

the i n t e r e s t i n g

The d e t e r m i n a n t

of all the p a r t s

of the c h a r a c t e r

of all the p a r t i t i o n s

t a b l e of

of n.

all = ~ (I i - i) : and bll = I ~ iI = ~ lit i i S i n c e A and B are u p p e r t r i a n g u l a r and B = CA', we h a v e

~n

is the

25

det C = ~ H A i , as claimed. A i Recall that the p a r t i t i o n l' conjugate A on its side"

(see d e f i n i t i o n

3.5).

to I is o b t a i n e d by "turning

The c h a r a c t e r table of

~5

in

Example 6.3 exhibits the property: l' l (in) 6.6 X = X ® X We prove this in general by showing 6.7

THEOREM

Remark

~

~ ~ ~(in)

Since S ~

is i s o m o r p h i c to the dual of S ~

is self-dual, we may omit the words

"the dual of"

from the s t a t e m e n t of the Theorem, but we shall later prove the analogous T h e o r e m

over an arbitrary

field, w h e r e the d i s t i n c t i o n b e t w e e n

S A' and its dual m u s t be made. Proof:

Let t be a ~iven A-tableau,

and let t' be the c o r r e s p o n d i n g i'

tableau. e.g.

if t = 1 2 3

then t'= 1 4

45

25 3

Let Pt' = ~{~I~ E R t, } and - d i m ~ n ~ /~Q Similarly,

A'

A"

shortens

~ dim S ~

A

) = dim S ~

Therefore, (*) . l

= dim S~.

At

Therefore, dim S~ = dim S ~ Xx and we have e q u a l i t y in (,). Thus, Ker @ = S ~ . The t h e o r e m is now A' proved, since we have c o n s t r u c t e d an i s o m o r p h i s m b e t w e e n M ~ /S~ '± (~ A' I ® _(i n) dual of S ~ , by 1.4) and S ~ ~ e Remark

dim S ~

A t

Thus

C o r o l l a r y 8.5 will give dim S l = dim S A', trivially, but this the proof by only one line.

There is one n o n - t r i v i a l e v a l u a t e d quickly,

c h a r a c t e r of

namely x(n-l'l) :

~n

w h i c h can always be

26

LEMMA

6.9

the number Proof:

The value

The

is clearly

trace of

7, acting

on the p e r m u t a t i o n

of fixed points

M(n-l,l)

_(n)

5.1),

the

We can thus w r i t e (= X (n-l'l)

the c h a r a c t e r characters

on a p e r m u t a t i o n

table

of

from these,

less than

of 7.

module

M (n-l'l) ,

Since

~ s(n-l,l)

result down

follows

at once. (in) X (n) ,X (n'l'l) , X and at once. The best way of finding

four characters,

® X (In)) of ~n

z is one

of z.

the n u m b e r

(cf. E x a m p l e

X (2'In-~)

of X (n-l'l)

of fixed points

%

for small

using

n is to deduce

the column

the r e m a i n i n g

orthogonality

relations.

27 7.

THE G A R N I R

RELATIONS

For this s e c t i o n , elements

of the g r o u p

let t be a 9 i v e n a l g e b r a of

~n

w-tableau.

which

We w a n t

annihilate

to find

the q l v e n pol V-

tabloid e t . L e t X be a s u b s e t of the ith c o l u m n of t, and Y be a s u b s e t (i + l) tn c o l u m n

of the

of t.

W'i+ 1

!

Let ~i'''''

Ok be c o s e t r e p r e s e n t a t i v e s

for

~X

x

~y

in

~XuY'


~i

Write and

. then e t G x , y ~X

~Y

= 0

(for an.~ I base

for [ { ( s g n

~XuY

~)~I~ ~ ~ X

for [ { ( s g n o ) o I ~ E

field).

× ~Y}

~Xu

y}

!

Since

IXu YI

p a i r of n u m b e r s way,

{tT} ~ X u Y Now,

~X

> ~i'

for e v e r y

T in the c o l u m n

in X u Y are in the same = O. ~Y

Therefore,

stabilizer

row of tT.

Hence,

of t, some

in the u s u a l

{ t } < t ~ XuY = O.

is a f a c t o r of ~t'

and

~XuY

= ~X~Y

GX,y"

28 Therefore O = {t}K t ~ X u Y Thus,

{t}K t GX, Y = O w h e n

tabloid 7.3

coefficients

EXAMPLE

here

Referring

=

IXl~IYl~{t}Kt

the b a s e

GX,y

f i e l d is ~, and since

are i n t e g e r s , to E x a m p l e

the same h o l d s

7.1, we have

O = e t GX, Y = etl - et2 + et3 + et4 _ et5 + et6 so

et

et2

et3

et4

et 5

et 6

all the

o v e r any field.

29

8.

THE

8.1 the

STanDARD

BASIS

DEFINITIONS rows

polytabloid

annoying

5.2,

the

of t.

{t}

if the n u m b e r s

is

equivalence

a standard

class

{t}.

increase

tabloid

along

if t h e r e

e t is a s t a n d a r d

tabloid

5.2,

et5

that

contains

a unique

along

rows

defined

over

~-tabloids

involve

4 5

standard

any

and

the

corresponding

standard

of

more

and

tableau,

a standard

since

the

tableau.

It is

standard

tabloid

than

one

form

a basis

2 4).

polytabloids

for the S p e c h t

field.

have

independence

the

may

involves

the

(3,2)-tableaux

listed.

a polytabloid

We p r o v e

The

tableau

5 standard

are

to i n c r e a s e

that

(In E x a m p l e

linear

in the

polytabloids

have

module,

columns

MODULE

if t is s t a n d a r d .

A standard numbers

the

tableau

In E x a m p l e standard

SPECHT

t is a s t a n d a r d

and d o w n

is a s t a n d a r d

OF THE

been

of the

totally

standard

ordered

by d e f i n i t i o n

polytabloids

follows

3.10.

from

The

the

tri-

vial 8.2

LEMMA

Suppose

i t i} is the

last

different r then Proof: = O ved

We may (a i ~ F)

in vj It is

assume

that

aj+ 1 = ...=

that

is s t a n d a r d ,

and

this

tabloids

linearly

using

the p a r t i a l

8.3

LEMMA

{t}

order

lower

than

is the

on

in e t s a t i s f y

x.

Thus,

in et,

tabloid to d e d u c e

by

3.15,

of t' {t'}

shows

all

If alv I + . . . + since

{tj}

amV m

is i n v o l -

a I = ...= a m = O. involved that

in e t w h e n

the

standard

go for a s t r o n g e r

down

{t'}

~ a non-iden~ty

induction

{tm}.

aj = O,

increasing

column

are

t poly-

result,

tabloids:

numbers

in some

integer

entries,

and by row

reducing

a s s u m e t h a t the f i r s t k rows of N (which Ii of S Q ) are l i n e a r l y i n d e p e n d e n t m o d u l o p.

in N m o d u l o

2--M~'

l (MF' ~ ) "

we m a y

to the b a s i s

tabloids

an e l e m e n t

that 0 c HornF ~ n

= < fi,{tj}

assume

rows,

coefficients

p, we

fl,...,fk

nij

@EHom~ ~n_~M©,_MQ)

of p e l e m e n t s .

a basis

is

v involves

technical

inteqer

trivial

of S Q to o b t a i n

first ~

that

modulo

field

It is

different

rather

coefficients

But

Therefore + 3ft 5

inteqers

Take

5.3).

- et4

F is the l± ~ ~ SF .

Proof:

5

(cf.

- et3

where Ker

3.4

zero

- et2

LEMMA

involved

5,

the s u m of the edge

valency

v = -etl

Next we want 8.14

has

-I

p, w e

obtain

a s e t of v e c t o r s

the

corres~R e d u c i n g

in M~,

the

%

last m - k which S F1 •

are

of w h i c h linearly

are

the

standard

independent

basis

and o r t h o g o n a l

and the

to the

f i r s t k of

standard

basis

of

Since d i m S FIi = d i m MFI - d i m S F1 =

we h a v e when

of S~,

constructed

the

tabloid

Now,

any one

combination all i n t e g e r s to zero,

a basis

are

of S 11 ~ whose

coefficients

are

of o u r b a s i s

of l - t a b l o i d s , reduced

as r e q u i r e d .

k ,

reduced

elements

and is s e n t

modulo

elements modulo

of S ~ ± is to

zero by

p, 0 c e r t a i n l y

give

a basis

l± of S F

p. an i n t e g r a l 0.

sends

linear

Therefore, the b a s i s

when l± of S F

33 We can now complement T h e o r e m 6.7 by p r o v i n g 8.15

I'

THEOREM

Over any field r S 1 ® S (In) is isomorphic

to the dual

of S Proof:

It is sufficient

to consider

is F, the field of p elements,

the case where

the ground

field

since we have p r o v e d the result when

F=~. In the proof of T h e o r e m 6.7, we gave a ~ G - h o m o m o r p h i s m 8 from I' 1 ~in~ ~,,n M~ into M ~ 8 S~ " and proved that Ker 0 = S ~ ~. Using the Lemma above, 0, defined by ~: {t'n} +

(sgn 7)

{t~}Ktn ® u

is an F ~ n - h o m o m o r p h i s m onto S F 8 S~ In) whose kernel contains By dimensions, Ker 0 = ~F _I'± , and the result follows.

S~'±.

34 g.

THE B R A N C H I N G

THEOREM

The B r a n c h i n g

Theorem

ducible

representation

symbols

~ ~n-i

Using notation 9.1

EXAMPLE

Proof:

+ [4,2,1 ~] + [4,22 ]

cases

9.3

of

+ ~n+l

~ @ {SI~ I[I] is a d i a g r a m

obtained

by a d d i n ~

a

fiX] is a d i a g r a m

obtained

by t a k i n ~

a

}.

Theorem.

THEOREM

When

of the T h e o r e m Part

a series with each

The

factors

factor

occurring are t h o s e

T h e o r e m I and S li o c c u r s (See P e e l

that a node (e.g. w h e n the d i a g r a m

(ii)

are e q u i v a l e n t ,

follows

S ~ is d e f i n e d

has

Proof:

Let

q i v e n by p a r t

f r o m the rith row of r l , r 2 , r 3 = i, 3, 4).

by r e m o v i n g

L {~} is {t}, w i t h When 9.4

n-i for ~ n-l"

of the B r a n c h i n ~

if I i ~ xJ.

a node

[~] to leave Suppose

that

such a diagram [li]

is

from the end of the r.th row of l

0

if n % rith row of {t}

{~} if n c r th row of l

{t}

n removed.

t is s t a n d a r d ,

0i: e t + ~ e ~

Lo

if n e rith row of t if n e r l t h , r 2 t h , . . . , o r

ri_ith

row of t.

Let V i be the s p a c e s p a n n e d by those p o l y t a b l o i d s a standard

~-tableau

Then

Vi_ 1 ~ K e r 0 i

since

S~%~

(M ~ , M li) by

0ic H O m F ~ n _ l {t} + I

field/

r I < r 2 = < het,

{t}

> = < et.

= < et*,{t}< t > = < et*,e t > . The which

last

line

is c o p r i m e

of the p r o o f

of L e m m a i O . 4

to p if and o n l y

shows

if ~ is p - r e g u l a r .

that h = j~l(Zj:)J,

39

ll.

THE

IRREDUCIBLE

The

ordinary

REPRESENTATIONS

irreducible

OF

representations

at the e n d of s e c t i o n

4.

We n o w

characteristic

the

characteristic

one,

by

ii.i

p,

allowing

THEOREM Then

Proof:

S ~ _= S ~l

tha£

S~/(S ~ n S ~)

polytabloids

of ~

that our O case

S U is d e f i n e d

is n o n - z e r o

if and o n l y

e t and at,

integer

assume

were

ground

constructed field

has

can be s u b s u m e d

in this

p = ~. Suppose

tic p.

the

and

~n

g~ d e f i n e d

if

in S ~.

< at,at, But

in 10.3,

over

a field

if and o n l y

this

of c h a r a c t e r i s -

if ~ is p - r e g u l a r .

> = 0 for e v e r y is e q u i v a l e n t

and C o r o l l a r y

10.5

pair

of

to p d i v i d i n g

gives

the

desired

result. Shortly, are

given

by

we

DEFINITION

=

and As

endent

tion

that

we

shall

that 4.10,

LEMMA Let

the c h a r a c t e r i s t i c

the

suffix

two D~'s

unless that

are

said

F when

of F is p

our

isomorphic,

that

(prime

or

results

are i n d e p -

we

S 1 is s e n t

need

to zero

a generalizaby e v e r y

element

k >- ~. I and

p are p a r t i t i o n s

of M U and S U p R o s e

MU/U.

The

submodule part

Then

U is i n s i g n i f i c a n t

of the L e m m a

element

says

of H O m F ~

13.17).

Proof:

of n t and

that

k is p - r e g -

8 is a n o n - z e r o

I ~ p a n d if

I = ~q

then

(See P e e l

[20]).

Let

het@

Since

in t to o b t a i n

= et,Kte

h ~ O and

I = ~,

for

of this

I p-regular,

unless

result.

S 1 is sent

I ~ ~ . (cf.

Coro-

the

a l-tableau tableau

and

t*.

reverse

the o r d e r

By C o r o l l a r y

10.6,

h ~ O.

= at, e< t

8 is n o n - z e r o ,

at, e< t ~ U.

By L e m m a

4.6,

I >- ~,

then at8

The

(SI,M ~)

t be

at,< t = h e t w h e r e But

in the p r o o f

that,

n

row e n t r i e s

and if

F ~n-modules

L e t D~ = S F / (~ S ~ n S~ ±) .

from S 1 into

to zero b y e v e r y

the

drop

which

Suppose

essential

llary

irreducible

(S ~ + U)/U.

Remark The

that

U be a s u b m o d u l e

F_~n-homomorphism Im 0 =

no

(M l, M ~)

11.3

the

field.

of L e m m a

ular.

all

D~ w h e r e

Suppose

To p r o v e

n

that

U is p - r e g u l a r .

usual, of the

of H o m F G

prove

the m o d u l e s

11.2 ~)

shall

result

= h-let,%< t = a multiple

follows,

because

of e t + U ~ (S ~ +

S l is g e n e r a t e d

by e t.

U)/U.

of

40

11.4

COROLLARY

regular. F ~n

Let

Suppose

U be

nomomorphism

Proof:

~Je c a n

that

I and

a submodule

~ are p a r t i t i o n s

f r o m D 1 i n t o MZ/U.

lift

of n,

o f M ~ a n d supp__Qse_t_~t @ ~s Then

@ to a n o n - z e r o

1%

~ and

add

I is p-

a non-zerg_

I ~ U if U ~ __S ~.

of H o m F ~ n (SI,M~/U)

element

as fol-

lows: S~

~ S ~ / ( S ~ n S ~±) canon•

Therefore, submodule 11•5

of

teristic

Each inq

i ~ ~, b y t h e L e m m a .

(S ~ + U ) / U ,

THEOREM p

D ~ varies D ~ is field

Proof:

(James

(prime over

= D ~ ~ M~/U 8

so U d o e s

[73)

self-dual for

not

Suppose

or = ~).

a complete

As

If

contain

that our

Z varies

s e t of

I = ~ then S~

ground

field F has

over p-regular

inequivalent

and absolutely

I m 8 is a n o n - z e r o

partitions

irreducible

irreducible.

charac-

Every

of n,

F~n-raodules.

field

is s p l i t -

n

Theorems

4.9

and

ii.i

show

t h a t D ~ is s e l f - d u a l

and absolutely

irreducible. Suppose

t h a t D 1 ~ D ~.

Then we have

a non-zero

F ~

-homomorphism n

f r o m D 1 i n t o M I / ( S Z n S~±),

and by Corollary

11.4,

I ~ ~.

Similarly,

~ I, so I = ~. Having question: section

shown

Why have we

got

17 w e

prove

representation gives

our

every

83.7:

83.5:

absolutely

4.12

More

two

left with

factor

over

o f the

D ~, a n d t h e n T h e o r e m artificial

results

and Reiner

approach,

the F?

regular i.i the

from representation

[2]: field

for a g r o u p

G,

then

for G,

the n u m -

f o r G.

If F is a s p l i t t i n g

shows

irreducible), 82.6:

ucible

FG-modules

is

FG-modules

field equals

then

the nur~ber of p-

~ is a s p l i t t i n g

(to m a k e

is to comJoine C u r t i s The number

less

field,

Lemma

use o f o u r k n o w l e d g e and Reiner

of inequivalent

than or equal

10.2

83.5 w i t h

absolutely

to the n u m b e r

now

t h a t D ~ is

irred-

of p-regular

o f G.

Theorem THEOREM

n over

field

subtle,

and Reiner

11.6

this

If ~ is a s p l i t t i n g

irreducible

Curtis

classes

follow

to a c c e p t

are

representations

composition

to s o m e

we

of G•

Theorem

us h o m e .

irreducible every

from Curtis

is a s p l i t t i n g

classes

Since sees

than

prefer

we quote

and Reiner

the

that

are i s o m o r p h i c ,

F is i s o m o r p h i c

of i n e q u i v a l e n t

regular

all

Rather

and Reiner field

Curtis ber

over

probably

which

Curtis

shall

result.

reader will theory

that no two D~'s

1.6

a field

the p - r a n k

gives

The dimension

o f the i r r e d u c i b l e

of characteristic

o f the G r a m m a t r i x

with

p can be respect

representation

calculated

D ~ of

by evaluating

to the s t a n d a r d

basis

o f S ~.

41

11.7 11.6 we

EXAMPLE

We h a v e

in E x a m p l e

obtain

is

5.2.

(cf. E x a m p l e

The

p-rank

2 if char

11.8

THEOREM

of the

of this

If

The of

column

bilinear

f o r m has

that

Theorem

every

The

~n

Gram matrix

or 2 if p = 2,

3 or

= 0 if char

>3,

respectively.

F = 2, and d i m

D (2'2)

=

of e v e r y

non-trivial

2-modular

irreducible

is even. t is a ~ - t a b l e a u ,

char even

of t,

is even.

F = 2, and rank,

homomorphism

case

self-dual,

of a g r o u p

has

then

< e t , e t >, b e i n g

Hence

even

11.6

gives

that the

of a g e n e r a l

absolutely

the o r d e r

< , > is an a l t e r n a t i n g

it is w e l l - k n o w n

so T h e o r e m

1 1 . 8 is a s p e c i a l

non-trivial,

representation

of T h e o r e m

The

respectively.

dimension

stabilizer

form when

Remark

>3,

~ ~ (n) a n d

bilinear

(2,2).

5.4):

is O , i

F = 3 or

representation

an a p p l i c a t i o n

the p a r t i t i o n

S ( 2 ' 2 ) / ( S (2'2) n S t2'2)±)"

1 or

Proof:

illustrated now

[42 214

A =

Therefore,

already

Consider

an a l t e r n a t i n g

result.

result

irreducible

which

states

2-modular

dimension.

~ in the p r o o f

of T h e o r e m

8.15

sends

{t'} ~f e x c e p t

All the

(see section

of S ~ w h e n

(All p u b l i s h e d matrices

of T h e o r e m

composition

if ~ is p-re~ularf

Consider

Proof:

open.

of

14).

the field is

algorithms

for arbitrary

for

symmetric

answers.)

a general±sat±on

THEOREM

factors

decomposition

factors

field has c h a r a c t e r i s t i c

of M ~ are k n o w n

composition

is still

calculating

the g r o u n d

factors

the

characteristic

When

the c o m p o s i t i o n

the

following

4.13:

factors

of M ~ have

w h e n D ~ occurs

the

precisely

form D 1 with once.

picture:

Mu

I S ~ + S ~±

I

0

By C o r o l l a r y form D ~ w i t h so has

the same

and recall

Since

all the c o m p o s i t i o n

But S ~l is i s o m o r p h i c

composition

that every

S~/(S ~ n S ~±) D ~.

11.4,

X ~ ~.

factors,

irreducible

is n o n - z e r o

factors

in the o p p o s i t e

F ~n-module

if and only

0 ~ S ~ n S ~± ~ S ~ ~ M ~

of M~/S ~ have

the

to the dual of M~/S ~, and order.

(See 1.4,

is self-dual.)

if ~ is p-regular,

is a series

for M ~,

Now,

when

it equals

the T h e o r e m is

proved. 12.2

COROLLARY

If ~ is p-re~ular~

factor D ~ = S~/(S ~ n S~±). then D ~ D 1 for some factors

of S ~ have

Proof:

This

the

The d e c o m p o s i t i o n

p of the o r d i n a r y 12.2 give

a unique

If D is a c o m p o s i t i o n

I m ~.

matrix

irreducible irreducible

top c o m p o s i t i o n

factor of S ~ n S ~±

If ~ is p - s i n ~ u l a r ~

form D l w i t h

is an i m m e d i a t e

of the p - m o d u l a r

S ~ has

all the c o m p o s i t i o n

I m ~.

corollary

of T h e o r e m s

of a group

records

representations representations.

4.9 and 12.1.

the m u l t i p l i c i t i e s

in the r e d u c t i o n s Corollaries

modulo

8.11 and

43 12.3

The decomposition

COROLLARY

matrix

of

~n

for the prime p has

the form: D ~ (~ p-regular) I

S~(~ p-regular) 1 !

when the p-regular the p-singular 12.4

are placed

Consider

representation.

order before

n = 3, S (3) = D (3) is the trivial

S (I~) is the alternating

S (3) if and only if p = 2.

tion matrices

in dictionary

all

partitions.

EXAMPLE

S (13) ~

partitions

of

are :

D(3)

D(2,1)

S(3)

1

5.1,

when p = 2,

S(I 3 )

S (3)

1

S(2,1)

1

S(3)

D(2,1)

1 when p > 3

S(I 3 )

(By convention,

omitted matrix entries

are always

1 1

D(I 3)

S(2,1)

D(2, l)

when p =

S(I 3 )

D(3)

and

the decomposi-

D(3)

1

S(2,1)

representation,

Using Example

p-modular

zero.)

3

44 13

SEMISTANDARD Carter

of the

basis

for H O m F ~

of t h e i r has

and Lusztig

standard

basis

HOMOMORPHISMS Ill o b s e r v e d

( S I , M ~) w h e n

argumen~

is g i v e n

characteristic

2 are

tion

of t a b l e a u x

letters i,

to i n t r o d u c e

the n u m b e r

such

and

notation

in the

A slightly

some

for

copy

cases where

~i t i m e s

the m o d u l e s

of M ~.

repeated

tableaux.

i occurs

ideas

construction

can be modified

c h a r F ~ 2.

here,

a new

T having

to d e n o t e

the

module

to g i v e

a

simplified

form

the

field

ground

included.

We keep our previous convenient

that

o f the S p e c h t

This

entries,

and we

A tableau in T.

S l a n d M l, b u t

requires

T has

shall type

it is

the i n t r o d u c use

capital

~ if f o r e v e r y

For example

2 2 1 1 1 is a

(4,1)-tableau

13.1

DEFINITION

Remark:

We

s u m is n.

of

type ~(l,~)

allow

~ to b e

For example,

~i ~ ~2

For type

the

~

....

remainder

any

is a l - t a b l e a u

sequence

module

of non-negative

~ can be

of

~

of type

(4,5,O,1).

on a Young

~}.

integers, The

subgroup

whose

definition does

not

a n d M (4'5'O'I)n=_ M (5'4'I) of section

13f

let t be

a given

l-tableau

(of

(in)). If T E ~ ( I , ~ ) ,

position

let

as i o c c u r s (i) (Tz)

The

= {TIT

if n = iO,

of M ~ as the p e r m u t a t i o n require

(3,2).

=

(i)T b e

i n t.

the entry

Let

(iw'l)T

~n

in T w h i c h

act on

~(l,~)

(i ~ i ~ n, T ~ ~ ( l , ~ ) , n

action

forced

occurs

o f ~ is t h e r e f o r e t h a t o f a p l a c e -I to t a k e ~ in the d e f i n i t i o n to m a k e

same

~ ~n ) .

permutation, the

in the

by

and we

~-action

are

well-

defined. 13.2

EXAMPLE

If t = 1 3 4 5

and T = 2 2 1 1

2 T(I

2) = 1 2 1 1

and

T(I

2 3) = 2 1 1 1 .

2 Since ment

~n

is a Y o u n g

then

1

2 is t r a n s i t i v e subgroup

~

on

~(l,~),

, we may

a n d the s t a b i l i z e r

t a k e M ~ to b e

o f an e l e -

the v e c t o r

space

45

over F spanned have

defined

by

the

tableaux

If T 1 a n d T 2 b e l o n g (respectively, in the r o w 13.3 eT

in

M ~ in a w a y w h i c h

column)

to

~(l,~) .

depends

~(l,U),

equivalent

we

for

o f the

If T E ~ ( l , U } ,

define

to v e r i f y

that

the m a p

to T } S

eT b e l o n g s

why we

t h a t T 1 a n d T 2 are r o w

stabilizer

: {t}S + ~ { T I [ T 1 is r o w e q u i v a l e n t It is e a s y

say

soon emerge

I a n d U.

if T 2 = T I ~

(respectively,column)

DEFINITION

It w i l l

on both

some permutation given

h-tableau

t.

0T b y

(S~ F ~ n ).

t o HomF ~

(Mt,MU). n

13.4

EXAMPLE

If t = 1 3 4 5

and T = 2 2 1 1

2

{t}@T

then

1

= 2 2 1 1 + 2 1 2 1 + 2 1 1 2 + 1 2 2 1 + 1 2 1 2 + 1 1 2 2 1

1

1

1

1

and

1

{t}(123)@ T = 2 1 1 1 + 1 1 2 1 + 1 1 1 2 + 2 1 2 1 + 2 1 1 2 + 1 1 2 2 2

Notice different ponding

2

that

the w a y

tableaux

to w r i t e

whose

rows

1

down

contain

i

{t}@ T is s i m p l y the

same numbers

1

to s u m a l l as the

the

corres-

r o w o f T.

It is c l e a r 13.5

2

that

T = < {t}0,T~ =

Hence

and

since

for some


= < {t}0~-I,T

>

>

{t}@ = ~ < { t } @ , T i > { t } 0 T i i=l M 1 is a c y c l i c

as r e q u i r e d

module,

0 is a l i n e a r

: k @ = ~ < {t}@,T i > i=l @Ti

combination

o f 0T. 's l

51 14

YOUNG'S RULE It is now p o s s i b l e to d e s c r i b e the c o m p o s i t i o n factors of M ~

explicity. 14.1 ~

YO~L~G'S RULE

l The m u l t i p l i c i t y of S Q as a c o m p o s i t i o n

equals the number of s e m i s t a n d a r d

Proof:

l - t a b l e a u x of type ~.

Since @ is a s p l i t t i n g field for

, the n u m b e r we seek is n But this is equal to the n u m b e r of semi-

n (Sl,M~), by 1.7.

dim H o m ~

~

s t a n d a r d l - t a b l e a u x of type ~, by C o r o l l a r y Remark:

factor of

An i n d e p e n d e n t proof of Young's

13.14.

Rule appears in section 17.

Young's Rule shows that the c o m p o s i t i o n

factors of M ~ are o b t a i n e d

by w r i t i n g down all the s e m i s t a n d a r d t a b l e a u x of type ~ w h i c h have the shape of a p a r t i t i o n diagram. 14.2

EXAF~LE

We calculate

the factors of M (3'2'2)

The s e m i s t a n d a r d

t a b l e a u x of type ~ are: ii12233

111223

11122

3

33

111233

11123

11123

2

23

2 3

1112

ll

233

23

12

1113

3

22

3 1113

ii13

lll

223

22

223

3

3

ill 22 3 3 T h e r e f o r e in the n o t a t i o n of 6.4, [3][2][2]

= [7] + 216,1] + 3[5,2] + 2[4,3]

+ [5,12] + 214,2,1] + [32,1] + [3,22 ]

Remark:

Young's Rule gives the same answer w h i c h e v e r way we choose to

52 order the integers in the d e f i n i t i o n of "semistandard",

and does not

require ~ to be a p r o p e r partition: 14.3

EXAMPLE by

The factors of M (3'2)

1112

or by

2

2 2 1 1 1

Therefore, 14.4

EXAMPLE

are given by

1112

1 1 1

2

2 2

2 2 1 1

2 2 1

1

1 1

[3][2] = [5] + [4,1] + [3,2]

(cf. E x a m p l e 5.2).

If m s n/2 then

[n-m][m]

= In] + [n-l,l]

Since dim M (n-m'm) =

+ [n-2,2]+

... +[n-m,m].

(~), we deduce that

dim s(n-m'm)

= (~) _

(m~l) .

Notice that Young's Rule gives S ~ as a c o m p o s i t i o n factor of M ~ w i t h m u l t i p l i c i t y one, and the other S p e c h t modules S ~ we get satisfy m ~

in a g r e e m e n t w i t h T h e o r e m 4.13.

that the m a t r i x m =

(ml~)

~ e m e m b e r i n g that this shows

r e c o r d i n g factors of M ~ _ as ~ varies

(see 6.1)

is lower t r i a n g u l a r w i t h l's down the diagonal, we can use Young's Rule to w r i t e a given [~] as a linear c o m b i n a t i o n of terms of the form [~i][~2]...[i i]

(The m e t h o d of doing this e x p l i c i t l y i s given by the

D e t e r m i n a n t a l F o r m - see section 19). like

[~][Ul]...[u k]

Ul,...,Uk.

S~ @ S ~ U l ) ~ . . . ~ s ~ k ) +

More generally,

[~][u]( = S duct

(=

IIence we can calculate terms ~ n) for integers

Young's Rule enables us to e v a l u a t e

@ S ~ ~ ~ n ) for any pair of p a r t i t i o n s

~ and ~ . The pro-

[~][u] is the subject of the L i t t l e w o o d - R i c h a r d s o n Rule

(section

16), and the a r g u m e n t we have just given shows that the L i t t l e w o o d R i c h a r d s o n Rule is a purely c o m b i n a t o r i a l g e n e r a l i s a t i o n of Young's Rule. 14.5

EXAMPLE

Young's Rule.

We calculate By Example

[3,2][2]

= S (3'2)~ ® S ~ 2)+

14.4,

[3,2] = [3][2] - [4][1] To find [4][1][2], we use Young's Rule: 1111233

111123

11112

3

33

~7

using only

53

ll

1 1 3 3

ll

2

1 1 3

1 1 1 1 3

23

2 3

llll

llll

233

23 3

[3,2][2]

=

[3][2][2]

[3,2][2]

=

[7]

+

[32,1]

=

[5,2]

+ +

[3,22 ] [4,3]

+

-

[4][1][2]

+ 216,1] [7]

-

[4,2,1]

+

216,1] +

, and

3[5,2]

- 2[5,23

[32,1]

+

using

+ 2[4,3] -

[3,22].

+

Exan~le

14.2,

we

have

[5,12 ] + 214,2,1]

[4,3] (cf.

-

[5,12 ] ExamDle

[4,2,1]

16.6).

54

15

SEQUENCES In o r d e r

section,

we must

A sequence the

to s t a t e

the L i t t l e w o o d - R i c h a r d s o n

discuss

is s a i d

properties

to h a v e

type

of

~ if,

finite

Rule

in the n e x t

sequences

for e a c h

of i n t e g e r s .

i, i o c c u r s

~ i times

in

sequence.

15.1

EXAMPLE 2 2 11

The

1

sequences 2 12

x J J J J

1212

12

1

15.2

(i)

i's

15.3

An

the

EXAMPLES

sequences

l's

x / J ,l J

x ,; J J J

J V x / J

112

112

1112

12

VFVV/

the q u a l i t y

2

JVJJJ

of e a c h

is e i t h e r

if and o n l y

than

good

above.

if the n u m b e r

the n u m b e r

indicated

(3,2)

2 1

2 11

t e r m is d e t e r or bad).

are good.

greater

We h a v e

of type

12

in a s e q u e n c e

i + 1 is g o o d

is s t r i c t l y

2 1112

a sequence,

(each t e r m

All

are

1

VJJ/V

Given

as f o l l o w s

(ii) good

112

JJVJJ

DEFINITION

mined

(3,2)

2 112

x x J V J

VJJJJ

of type

11

of p r e v i o u s

the q u a l i t y

Here

of the

is a n o t h e r

of p r e v i o u s good

terms

(i+l)'s.

in the

example:

3 1 1 2 3 3 2 3 2 1 2

× / J J Jx It f o l l o w s and o n l y ious 15.4

immediately

if the n u m b e r

good

(i+l)'s.

15.5

c's

whose

non-negative

~,

Remark: on

contains

Let

integers

~i

a pair

As here,

shall

a partition

~, b u t w i l l

If the

condition

of n.

So,

definition

we

that

S ~i

~ is c a l l e d

let

which

(c-l)'s

an i+l is b a d

the n u m b e r

will

be n e e d e d

in s u c c e s s i o n ,

if

of p r e v later:

then

the

all good.

~

be

=

a sequence

(~,

for all

~,...)

be

of n o n - n e g a t i v e a sequence

of

i,

" of p a r t i t i o n s

frequently still

we

for e x a m p l e

, ~ # is a p r o p e r

15.5.

that

a Specht

for n.

drop

refer

~i ~ ~2 a "'" h o l d s

Note

that

i's e q u a l s

a result

(~i,~2,...)

and

such

good

m good

are

~ =

s u m is n,

~i+l ~ Then

we h a v e

in the s e q u e n c e

DEFINITION

integers

f r o m the d e f i n i t i o n

of p r e v i o u s

Hence

If a s e q u e n c e

next m

J V x V/

to

the ~

condition

shall

call

partition

module

~i a ~2

as a p a r t i t i o n ~ a proper of some

S ~ is d e f i n e d

n'

a "'"

of n. partition s n

only

for

in

55

a proper

partition,

but the m o d u l e M ~ s p a n n e d

by ~ - t a b l o i d s

may have

improper. 15.6

DEFINITION

Given

a pair

be the set of sequences

of type

good i's is at least

.

We w r i t e sequences

~.

Since

the n u m b e r

of O,

the n u m b e r

of good i's

~,

~ for n,

for each

i, the n u m b e r

so that s(o,~) of good

(i+l)'s

there has been

let s ( ~ , ~ )

consists

of

of all

in any sequence

no loss in a s s u m i D g


the n u m b e r

the p r o o f

~

bad

~c

s,

the n u m b e r

in

s -

(~c

--

for e v e r y

of

(c-l)'s

SF

)"

of previous

15.18, since

of previous ~c

c-i in s

assume

good

Therefore,

to s ( ~

holds

~ )th b l a c k c-i in s. ~c If the t e r m x in s • is a c-i a p p e a r i n g c-i i n s,

then x was white

in s* = the n u m b e r

of white

Now,

s contains

in s.

Also,

of previous

for a c-i

before

the n u m b e r

(c-l) 's b e f o r e

at m o s t

~ R c) , s o f o r a n y

(c-2) 's > the n u m b e r 15.18

of previous

(c-l) 's in s*.

c > 2.

s belongs

, the n u m b e r

the of

after

(~c -

the

c-i

in

(c-l) 's (~c

~c# )th b l a c k

(c-l)'s b e f o r e

x in s < the n u m b e r

x

of good

59 (c-l) 's b e f o r e bad

x in s by

15.17

c-i in s, by a p p l y i n g

(c-2) 's b e f o r e and 15.18 From

x

15.17

to the next

(the i n e q u a l i t y

is p r o v e d 15.18,

(the i n e q u a l i t y

in this

15.16

Theorem

15.14.

15.19

EXAMPLE

between

s((3),(3,2))

case

follows

Referring

being

to E x a m p l e

+

term)

strict

strict

if x is a

< the n u m b e r

if x is a good

of good

c-i in s),

too.

at once,

\ s((3,1),(3,2))

22111

being

and this

15.8, and

the

completes

the p r o o f

of

i-i c o r r e s p o n d e n c e

s((5),(5))

is o b t a i n e d

by:

s((3,1) , (3,2)) \ s((3,2) , (3,2))

and

11111

x x,/// The

1-1 c o r r e s p o n d e n c e

s((4,1),(4,1))

between

is given by 21211

11211

x / / / / 21121

11121

x//// 21112

llll2

x / J / / 12211

/ / x / /

12111

60

16

THE L I T T L E W O O D - R I C H A R D S O N

RULE

The L i t t l e w o o d - R i c h a r d s o n I is a p r o p e r

Rule

[I][~]

where

of r.

Remember

nodes,

and the i n t e r p r e t a t i o n

~I ~ S ~ + [~], ~ a

that

of G x H,

for groups

to

~

for c a l c u l a t i n g

and ~ is a p r o p e r p a r t i t i o n

combination

is that w h e n

result

G-module

Littlewood-Richardson factors

of n-r

is a linear

a

has S ~ as a c o m p o s i t i o n

~n

some i r r e d u c i b l e

partition

[i][~]

It is a w e l l - k n o w n

sentation

is an a l g o r i t h m

of d i a g r a m s

is the c o e f f i c i e n t

irreducible

G and H is e q u i v a l e n t

Rule enables

irreducible

of a Y o u n g

repre-

to S 1 × S 2, for

H-module

us to c a l c u l a t e

of any o r d i n a ~ l r e p r e s e n t a t i o n

of

factor w i t h m u l t i p l i c i t y

that every o r d i n a r y

S 1 and some

with n

the

$2, so the

composition

subgroup,

induced

up

.

n For the moment,

the additive integer}.

group

Given

we define

forget

any i n t e n d e d

generated

by

interpretation,

{[I]II

is a p r o p e r

any p a i r of p a r t i t i o n s

a group e n d o m o r p h i s m

~#

and c o n s i d e r

partition

of some

,~ as in d e f i n i t i o n

[~ ~,~]" of this

additive

= ~ a [~] w h e r e

a

15.5,

group

as foll~

ows : 16.1

DEFINITION

for every

of r e p l a c i n g (i)

[I] [ ~

'~]

i, and if I l• ~ ~i the nodes

The numbers

of

for every

[~]\[I]

(ii) The nur~bers are s t r i c t l y

and

(iii)When

reading

from right

I i ~ ~i

such

of ways

that

along rows

increasing

down

columns

to left in s u c c e s s i v e

rows,

we have

a

= ~, w h e n

~ is a fortiori

a proper partition,

we w r i t e

[~]"

[U, ~ The o p e r a t o r s

16.6

a~d 16.7.

16.2

LEMMA

are i l l u s t r a t e d

If N =

Proof:

When

that we have describes

[~

of type

then

[O] [O'~]

[O] [~]"

(iii) ~.

factors

= [~l]_~[~2]...[~k!.

= IN].

of d e f i n i t i o n But

and by E x a m p l e s

16.1 m e r e l y

says

[ ~ l ] [ ~ 2 ] . . . [ ~ k ], by d e f i n i t i o n ,

of M ~

, and the

first r e s u l t

follows

Rule. be a d i a g r a m by

(i) to

jth row w i t h

happens.

then

condition

a sequence

can be r e p l a c e d conditions

= O,

the c o m p o s i t i o n

from Young's Let

~

by the next L e m m a

(~I,~2,...,~{,

If ~ is a p r o p e r partition,

the

= O unless is the n u m b e r

in s(u ~ ,~).

If ~ for

by integers

are n o n - d e c r e a s i n g

and

sequence

i, then a

There

appearing

~i l's, (iii)

~2 2's,

in

[O] [u]

and so on,

of 16.1 hold.

Suppose

j < i, and let i be the are no

(i-l)'s

higher

Then

a way

in [~]

that

that some i appears

least n u m b e r

than this

the nodes

in such

for w h i c h

in

this

i, by the m i n i m a l i t y

of

61

of

i;

by

condition

from

nor

can

right

condition tion

(i).

(iii).

Proof:

Replace have

[~

,~]" that

of n - r

each

this

a sequence

by

a configuration

if we

(i) and Suppose

tions are

after that

a bad there

the

(ii)

which same

c must

(ii),

there

j+m-l) th p l a c e s .

all,

by

This

this,

This

a configuration the

a good

16.1

from

right

(c-l)'s place

same

we h a v e

in the

(ii)

in the

only

(i-l)th

immediately

The

to w o r r y

to the

of

left

good

left

it in the

16.1

of

immediately

(i-l,j) th

there

by

are

c's

conditions

m u s t be

c's

in

16.1 (i-l,

belonging

good,after

c occurs

to

(c-l) 's

see

cannot

(i,j)th that

in the

of the row

We

must

that

by

c in the

(i-l,j)th

row.

c is

Reading of

(good)

(i-l,j)th But

place

(since t h e r e

above 15.4.

a bad

place.

the n u m b e r

ith

to the

immediately occur

dis-

satisfy

immediately

c lies

of o u r b a d

(c-l) 's we

(ii).

This

the p o s s i b i l i t y

we

to

(i) and

with.

problem

c's

ith

c

in a s e q u e n c e

the S a d

c is in the rows,

a

There zight

can be c h a n g e d

or a b a d

first

to the

the n u m b e r

a bad

about

to the

in the

that

place

c's

started

row,

and a g o o d

below

we

15.14.

condi-

integers.

be

(ii)

satisfying

(ii).

changing

unless

in s u c c e s s i v e

row

and

conditions

same

column.

(i-l) th row

least

after

to

is the p o s s i b i l i t y

Then

good

(i,j)th

the b a d

of i n t e g e r s

place

to left

in the

is at

c

c-i

(i-l,j)th

all

c's

(i) and

(i-l,j)th,(i-l,j+l)th,...,

(c-l) 's are

16.1(i)

that

(i) and

(good)

Therefore,

that

of

a c-i

such

to left

from Theorem

because

places.

c in the

satisfying

configuration

c in the

in the

shows

suppose

cuss

of a

the

conditions

have

conditions

let m be m a x i m a l

all

the b a d

(c-l) 's and

c might

and

that we

right

16.1

follow c~'s to

happen,

(c-l) 's in the

Since

affecting

Conversely,

left

are

all

of i n t e g e r s

then

place

(i,j+m-l)th

= [~].

i.

such

from

~1ore c o m p l i c a t e d

(i,j)th

[9]

for e a c h so on,

configuration

cannot

~, > i, b v c o n d i -

and

I and 9 are .proper

satisfying

A bad

be bad,

s(~ # Ac,~) , our

15.4.

without

for o u r

and

changing

the b a d

occur.

c in the

To d e a l w i t h

to s(~ ~ ,~)\

c-i

row.

(i,j)th,(i,j+l)th,...,

(i) and

good

hold

that

reading

a configuration

might

itself

is a b a d

when

of i n t e g e r s

changed

row,

Ii < 9 i

~2 2's

that

reading

contradicting

row w i t h

ith

and

with

the Lenuna w i l l

not yet

@th

row,

r~Rc ]'_

of r,

~I l's,

prove

with

since

(i) and

c in the

place.

start

(ii),

two probleras

a good

We m u s t

we h a v e

16.1

in the

(i-l) 's w h e n

i is bad,

i is in the

in s(~ # ,~) \ s(~ # Ac,~)

rows.

the

+ [p~

of it in the s a m e

by no

and

respectively,

[9]\[I]

(c-l) 's gives

16.1

appear

every

= [~Ac,~]"

in s u c c e s s i v e

if and o n l y

rows,

~ is a p a r t i t i o n

in

right

i is p r e c e d e d

i can that

and n,

node

(i-l) 's to the

in s u c c e s s i v e

proves

Assume

partitions

any

B u t no

This

LEMPirA

be

Thus,

to left

(ii).

16.3

there

must

every have

is a g o o d

a

c in

62 ~e

(i,j)th place,

tions

16.1

and we end up w i t h a c o n f i g u r a t i o n s a t i s f y i n g condi-

(i) and

good c in the

(ii)).

This contradicts

(i,j)th place,

and completes

THE LITTLEWOOD-RICHARDSON

16.4

IX] [~]. Proof:

the fact that there is a the proof of the Lemma.

RULE

[X][~3

(James [10]9

If ~ is a p r o p e r p a r t i t i o n of n, then a p p l y i n g

operators A c and R c repeatedly to O, ~ we reach a c o l l e c t i o n of pairs of p a r t i t i o n s

~,~.

By Lemma 16.3, we may write

[O,v]" = ~ a [ ~ " w h e r e each a m in an integer,

a

= 1 and a m = 0 unless

Hence there are integers b

[~] ~ Iv].

and c B such that

If]" = ~ b [O,m]'and [~]" = Z cB[O,8]" By Lemma 16.2

[~][~]"

=

[0][~]'[~]

"

= [O]~ b~[O,~]

~ cB[O,8]

= ~Z bm [el]...[aj]

~ C 8 [81]-..[B k]

= [O] ~ b ~ [ O ' ~ ] ' [ O ] ~

16.5

COROLLARY

Proof:

=

[0]

=

[l][u]

For all [l],

[0] [~]

[~]

[v]'[p]"

cB[O'8]"

= [ ~ ] ' [ ~ ] ' = ([p][~])"

[l][~]

[~]

= [l][~][~]

= [l][~][~]

= [x]EP]'[~]" The C o r o l l a r y

is

extremely

hard

to prove

= [x]([~][~])"

directly.

More g e n e r a l l y ,

it follows from the L i t t l e w o o d - R i c h a r d s o n Rule that for every e q u a t i o n like

[3][2] = [5] + [4,1] + [3,2]

equation

[3] [2]

Of course,

- [5]

+ [4,1]

there is a c o r r e s p o n d i n g o p e r a t o r + [3,2]

the B r a n c h i n g T h e o r e m

the L i t t l e w o o d - R i c h a r d s o n

(part

is a special case of

Rule.

When applying the L i t t l e w o o d - R i c h a r d s o n the d i a g r a m El], then add ~i that at each stage

(5))

l's, then ~2

Rule, it is best to draw 2's and so on, m a k i n g sure

If], together w i t h the numbers w h i c h have been added,

form a p r o p e r d i a g r a m shape and no two i d e n t i c a l numbers same column.

appear in the

Then reject the result unless reading from right to left

83

in

successive

condition 16.6

rows

EXAS~LE =

following XX

[5,2]

i is p r e c e d e d and

[3,2][2] +

[4,3]

X 1 1

X XX

+

[4,2,1] (cf.

1

XXXI

1

XX22

XXXl

+

=

(i-l)'s

every

[32,1]

Example XXX

[3,2][2][2]

for

term

+ [3,22],

1

XXX

X X

X X 1

X X

1

1

1 1

[3,2] [2]

i

XXX

[2] 1 1

XXX

XX

XX

22

2

X X X

X X X

XXI

XX

1

12

12

ii

2

2

2

22

XXX

12

X X X l 2

XXI2

XXI

2

XXX XX

12 2

1

XXI 22

X X X 1

112

2

XXX

12

XXX

12

X XX

XX

X X

XX

12

1

1

1

X XX

22

XX

1 2

12

2

X X X 2

X X X 2

X X X

X X X 2

X X X

X X 1

X X 1

X X

X X

X X 2

12

1

122

11

ii

2

2

1

2

XXX

1122

XX

at t h e

XXXI

12

XX

XX

looking

1

XX2

XXX

(This

good.)

=

2

12

by

X XX

X X X 1

XXXI

i's.

14.5).

XX2

XX2

than to b e

[2] •

[3,2]

X X 1

EXAMPLE

by more

sufficient

=

configurations:

X X

16.7

each

is n e c e s s a r y

XXXI XXI

X X X 2 2

XXX2

XX

XX2

ii

ii

22

X X X I 2 2

X X X 2 2

XX

XXI

1

1

64

We h a v e

arranged

in s u c c e s s i v e line)

give

rows,

the d i a g r a m s

sequences

[3,2][2,2]

The

= [3,2] [2,2].

diagrams

reader

ond batch

may gives

in the

in s ( ( 2 , 2 ) , ( 2 , 2 ) ) ,

+ [4,3,23

The

so that,

the d i a g r a m s

the

to check

the

same

second that

+ [5,3,1]

answer

line

right

(before

to left

the

first

+ [5,22 ] + [42,1]

+ [32,2,13

give

changing

as

from

so

+ [4,3,12 ] + [4,22,13

before

care

= [5,4]

reading

first batch

+ [3,23 ]

[3,2] [ ( 2 ' 1 ) ' ( 2 ' 2 ) ]

a bad

2 to a 1 in the

[3,2] [3'1]" , in a g r e e m e n t

with

secLemma

16.3. [3,2][3,13

= [3,2] [3'13"

+ [5,22 ] + [5,2,12]

= [6,3]

+ [42,1]

+ [6,2,1]

+ 2[4,3,2]

+ [5,4]

+ 215,3,1]

+ [4,3,12 ] + [333

+ [4,22,1]

[32,2,1]. The bad,

last b a t c h

contains

and by c h a n g i n g [3,2][4]

+ [4,3,2] which

the

~ [3,2] [4].

2's

all

the

to l's,

= [7,2]

to v e r i f y

Lemma

+ [6,3]

,

is s i m p l e

configurations

directly.

16.3

where

both

2's

are

gives

+ [6,2,1]

+ [5,3,1]

+ [5,22 ]

65

17.

A SPECHT A better

What

Since

take

into

next

example

17.1

in this

to a S p e c h t

series.

account

f r o m the

top.

S (n)

reading

uences,

Thus

and d e d u c e module.

as the

intersection

arbitrary Let that

At

~~

the

with

Rule

time,

of c e r t a i n

we

F ~

n

called

in a S p e c h t does

with

factors

series.

standard

Throughout

The

M (n-l'l).

factors

D (n) ,D (n-l'l)

factors

15.14

basis

the

S (n-l'l) , factors

on seq-

of the

Specht

defined this

reading

5.1 has

Theorem

-homomorphisms

partition.

we must

matter:

with

characterize

iso-

a Specht

fields,

in E x a m p l e

use o n l y

a n d the

field.

factor

D (n-l'l) ,D (n),

series

series

shall

each

some

> 2, a n d c o n s i d e r

Specht

we

be

factors

has no Specht

Young's same

I is a p r o p e r

module

S~

on M ~, in the

section

F is an

field.

must

be

of p a r t i t i o n s

a proper

We w a n t

construct

n

an a r b i t r a r y

with

over

is u n i s e r i a l ,

The

section,

~ ~ ,~ be a p a i r

proper.

factors of the

is u n i s e r i a l

both

will

reducible

of the

F divide

M (n-l'l)

a series

a series

the o r d e r

char

important

Specht

case w h e r e

such

t h a t M (n-l'l)

S (n-l'l)

can be derived o v e r

completely

f r o m the top. (n) (n-l,l) S ,S

in the o r d e r In this

module;

that

Let

5.1 s h o w s that

Rule

is t h a t M ~ has

the o r d e r

shows

D (n) a n d

case

M ~ is n o t

EXAMPLE

Example

FOR M ~

f o r m of Y o u n ~ s

happens

morphic

SERIES

partition,

to d e f i n e

an o b j e c t

et

for n, while

a submodule which

we

S~

is

as d e f i n e d do n o t

in

15.5.

require

Recall

~ to be

'~ o f M ~, and to do this we

"between"

a tabloid

and a poly-

tabloid. 17.2

DEFINITION et

17.3

Suppose

that

t is a ~ - t a b l e a u .

'~ = Z {sgn n { t } z l n ~ C t and n fixes EXAMPLE

If t =

~

and

~

Let

the n u m b e r s =

(3,2,0),

outside

~ =

[~ ~ ]}

(3,4,2)

9 86 (part of t is b o x e d - i n et

'~ =

to s h o w w h i c h

235

135 2 74

9

-

86 17.4

only

17

175 49

86

DEFINITION

S~

numbers

'~ is the

-

2 34 86 subspace

will

be m o v e d ) ,

then

275 9

+

1349 86

of M ~ s p a n n e d

by e~ ~

'~

's

as t v a r i e s . Of

course,

S~

'~ is an F ~ n - s u b m o d u l e

of M ~, s i n c e

et

'~

= et~

66

If ~ ~

= O,

then

S~

17.5 If 11 = ~i we can a b s o r b the Sequences 17.6

'~ = M ~ and

come

CONSTRUCTION

ponding

~-tableau

t e r m is a g o o d If the

jth

into

play

Given t as

by w a y

a sequence

follows.

i, p u t

term

= ~ , then

S~

'~ = S ~.

1 ~i = ~i~ for i > i, then S I ~ '~ = S ~ ~ '~ p a r t of U i n t o ~ ~ (cf. 15.7).

and first

now

if U ~

j as

is a b a d

far l e f t

i, put

type

along

in the

j as

so

of of

Work

,

~,

the ith

far r i g h t

construct

sequence. row

a corresIf the

jth

of t as p o s s i b l e .

in the

ith

row

as p o s s -

ible. 17.7

3 1 1 2 3 3 2 3 2 1 2 1

EXAMPLE

x /V/Vx and

corresponds

to

//×

¢ s((4,3,2),(4,4,4))

///

12 3 iO 121

1

Different to d i f f e r e n t 17.8

The

M

tabloids,

in s(O,~)

gives

already

construction which

side

increase

17.9

We

where S~ ~c-i

'~

along

see

soon

the o p e r a t o r s '~ .

~ )' s i n c e w e

Otherwise,

given

the s u b g r o u p

rows

the

which

belong

between

s(O,~)

and

concept

t, w e m a y

of C t f i x i n g

is

the 8.2

take

the n u m b e r s

17.7).

involved

in-

But, in

in s(~ ~ ,~) b y

17.6}

a basis

'~ here.

of S ~

t h a t S ~ ~ '~/S ~ ~ A c ' ~ in 15.10.

true

convention coset

Example

tabloid

to lie

gives

have

defined

trivially

the

cf.

last

which

is

'~

to p r o v e

A c a n d R c are

of

T of t y p e

the n u m b e r s

colw~s-

actually

is

a basis

in s(~ ~ , ~ ) c o r r e s p O n d s

(i.e.

{t}is

of S ~ #

t h a t we

adopt

[~ ~ ]

to a s e q u e n c e

subset

This

the t a b l e a u

a sequence

and so L e m m a

though,

13,

of v i e w i n g

(2)T, .... , (n)T.

and d o w n

It c o r r e s p o n d s

objective,

A c ' ~ _c S ~

that inside

[u~,then

17.3),

independent

shall

Our main

ensures

inside

(cf. E x a m p l e { et

(1)T,

is s t a n d a r d

if t is s t a n d a r d

a linearly

correspondence

for in s e c t i o n

to the s e q u e n c e

a tableau

e~ W'~

a l-1

encountered

as a s e t of s e q u e n c e s ,

corresponds

[~#]

to t a b l e a u x

~ tabloids.

We h a v e

The

correspond

so

construction

the set of Remark

sequences

if ~

that

Ac, ~ = O , O

So, o

representatives outside

First,

is the

~ S~ note

'~Rc, that

(i.e. zero

Ol,...,Sk

if

module. for

[~ ~ ] in the s u b g r o u p

of

67

C t fixing the numbers outside -~ ~t

'~

Ace ~

=

~ (sgn Oi)O i . i=l

NOW we want an F ~ n - h o m o m o r p h i s m 17.10

e ~t

[p ~ Ac], w h e r e u p o n

DEFINITION

mapping S ~ ~ '~ onto S ~

Let ~ = (~i,~2,...)

'~Rc "

and

u = (~l,~2,...,~i_l,~i + ~i+l - v'v'ui+2'''') . Then ~i,v b e l o n g i n g to Ho*~ ~n(M~,M~) is defined by {t}~i, v = Z {{tl}I{tl} agrees with {t} on all except the ith and

(i+l) th rows,

subset of size v in the Remark

and th~

(i+l) th row of {t I} is a

(i+l) th row of {t}}.

It is slightly simpler to visualize

basis of M ~ viewed as sequences. all sequences

the action of ~i,v

~i,v sends a sequence

obtained by changing

all but v (i+l)'s to i's.

way you look at it, ~i,v is obviously

an F ~ n - h o m o m o r p h i s m .

tabloid involved in {t}~i,v has c o e f f i c i e n t

on the

to the sum of Whichever Every

i, so ~i,v is " i n d e p e n d e n t

of the ground field." 17.11

EXAMPLES

(i) When ~ = (3,2), ~i,o and ~i,i ~i appearing in Example 5.2. (ii)

~2,1

If ~ = (4,32,2),

:

1

2

5

3

4

9

6

7

8

are the h o m o m o r p h i s m s

~o and

then

iO ÷

1

2

5

lO

3

4

9

7

8

+

6

ii 12

2

5

I0

3

4

9

6

7

ii 12

+

1

Ii 12

i2SlO 34967 8 ii 12

(iii)

If n > 6 and ~ = (n-3,3)

v = 1 2 3 + 1 2 4 + 1 3 4 second row only), we have

+

and 2 3 4

(replacing each tabloid by its

v ~I,o = 4 ~ F v ~i,i

--i+~ +~+i+~+~+i+~ =

3(i

+~+~+~+~

+ ~ + ~)

v ~i,2 = 2(~-~ + ~'-~ + ~--~ + 2 3 + 2 4 + 3--4). Therefore, and

v(Ker~l,o v ( Ker ~i,i

n Ker ~ i , 2

if and only if char F = 2 if and only if char F = 3.

68 (iv)

Taking

n = 6 in e x a ~ p l e

(iii),

=~g+g-i-g-~=~-i

(4 5 6 - 1 5 6)~1, 1

(4 5 6 - 1 5 6 - 4 2 6 + 1 2 6 ) ~ i , i That

is,

for

t =

e~t

~

, p~

'~ @ l , 1

=

=

(3,1)

= O.

and p =

e~R2'~R2

where

(3,3),

tR 2 =

we ha~e

1 2 3 5 6 4

and

e~ Compare

17.12

A?,~

the

and

Let

dim s ~ A c ' ~ +

dim S ~ ' ~ R c

> IS(~Ac,~)I = is(]/~,~) I Everything (i),

are

[0] [ ~ ' ~ ] "

(ii) and

falls out~ (iii)

When ~ = ~ ,

+ Is(~]#,]/Rc)I by 17.9 by T h e o r e m

15.14

We must have equality everywhere,

so results

follow.

S~'~

= S ~, and so has a Specht series whose

are given by [0][~]" (see Lemma 16.2). ively that S ~ Ac'~ and S ~#'~Rc given by [O] [ ~ A c ' ~ ] " (i), and [ ~ , ~ ] "

by Lemma 17.12

= [~Ac,~]"

+ [~,~Rc]"

factors

we may assume induct-

have Specht series whose

and [O] [ ~ ' ~ R c ] "

a Specht series whose

Therefore,

factors

factors

are

Since we have proved conclusion (see Lemma 16.3),

S~'~

has

are given by [O] [ ~ ' ~ ] "

All we have used in the above proof are the purely

combinatorial

results 15.14 and 16.3 (In fact, it is much easier to show that [ 0 3 [ ~ , ~ ] " = [O][~*Ac,~]" + [~,~Rc]" than to prove Lemma 16.3 in its full form.)

We have therefore

polytabloids

form a basis

(Jii)),

given alternative

for the Specht module

and of Young's Rule

proofs (take ~

(take ~ # = O in ~art

that the standard = ~ in Dart

(iv)).

17.14 COROLLARY M ~ has a Specht series. More generally, S 1 @ S(~I)@...@ s(~k)+ ~ n has a Specht series. The factors order of appearance

are independent

of the ground

field,

and their

and can be

70 calculated by applvinq the operators A c and R c repeatedly

to [0,~] and

[l~ (l,~ir...,~k) ]J respectively. The factors of S l @ S(~I)@. .@ s(~k)+ are ~iven by [I] 'E'~l]"[~2]''''[~k]'. " (By (I,~I,...,Z k) we mean the partition where lj is the last non-zero part of I). Proof:

(ll,...,lj,

It is simple to see that Sl' (l'~l'''''~k) ~ Sl @ s(~l)@'''@ s(~k)+

n

~l,...,~k) ,

~n

and we just apply Theorem 17.13(ii) to obtain a Specht series., The last sentence is true because [O][l' (l'~l'''''~k)]" [l] [~l]''''[~k]" ~s can be easily verified. Remark

James and Peel have recently constructed a Specht series for

the module S ~ @ SI+ ~n " Here again, the factors and their order of appearance are independent of the ground field. The Specht factors are given by the Littlewood-Richardson Rule. 17.15 EXAMPLE We construct a Specht series for M (3'2'I) In the tree below, we always absorb the first part of ~ into ~ (e.g. M (3'2'I) = S O , (3,2,1) = S(3), (3,2,1) ; cf. 17.5). Above each picture we give the dimension of the corresponding module. iX x x x x I + R 3

1.6 X X X X] ~R2

l1 X X X X X I

xx

5

A2

IX X X X A2 5

0,0

Xl

W

24 R3

14 ~X X X

5 X]

x ~

A2

3O

A3

xxx l

14

I A2

i xx xl xl

ixxxxl xl 9

X

- +A3

9

10

~ ~2

r x~XXxxl

71

5 Therefore, M (3'2'1) has a Specht series with factors S (6) , S (5'1) , S(5,1) , S(4,2) , S (4,12 ) , S (4,2) , ~~(32) , S (3,2,1) , readinq from the top. This holds regardless of the ~round field 17.16

EXAMPLE

Consider

S(4'22'i)%

= S(4,22,1),(4,22, 12)

~iO

xx] Ixxxxl Ixxxxl xxx] R5

X~

R4R 3 >

~

R2 >

xxx Hence,

top,

S(4'22'I)%

isomorphic

Examples

2~,

to S (5'2

has a series with factors, reading from the ~) ' S(4,3,2,1) , S(4,23) , S (4,2 ~ '12) (ef.

9.1 and 9.5).

17.17 EXAMPLE Following our algorithm, we find that when m < n-m, M(n-m'm)has a Specht series with factors S (n) S (n-l'l) .... S (~-m'm) reading

from the top

(cf. Example

14.4).

There is a point to beware of here. M(n-m-l'm+l)/ S (n-m-l'm+l) is isomorphic modules

have Specht

is sometimes tion factors Appendix.)

series with

factors

It seems plausible that to M ~ - m ' m ) ; after all, both as listed above.

this

For instance, when char F = 2, S (6"2) has composiand D (7'I) (see the decomposition matrices in the Since D (6'2) is at the top of S (6'2) D (7'I) ~ S (6'2)

n S (6'2)± ~ M(6'2)/

(S (6'2)

Therefore, M(6'2)/ S (6'2) has a top factor isomorphic M (7'I) does not (see Example 5.1). Theorem

However,

false. D (6'2)

17.13 provides

the irreducible

an alternative

representations

of

~n

+ S (6'2)±) .

to D (7'I), while

method of showing

appear

that all

as a D v, thereby avoiding

72 the quotes S ~± has come

from Curtis

the same

from D ~

shows

factors

I ~ ~

w h e n M ~ is the regular irreducible Theorem 17.18

then

The C o r o l l a r y since

properties

representation

has

of M ~ 17.13

of F

to Sl's w i t h

composition

~n'

factor of M ~

case w h e r e

Theorem

~ =

I.i shows

(in),

that

to some D 9.

partition

of n, with k n o n - z e r o

5-1 ker ~ i - l , v v=O the m o s t

S ~ as a subset

(cf. E x a m p l e

in the section

factors

fact to the

is i s o m o r p h i c

Since

the useful

is perhaps

it c h a r a c t e r i z e s

certain length

~ i=2

this

11.5.

But T h e o r e m

isomorphic

S l c M 1 , every

If ~ is a p r o p e r

k S~ =

factors

Applying

F ~n-module

17.13(i)

COROLLARY

parts,

since

to some D 9.

composition

and from M~/S ~.

a series w i t h

By induction,

every

in the proof of T h e o r e m

as M~/S ~, all the

(if ~ is p-regular),

that M~/S ~ has

is i s o m o r p h i c

and Reiner

5.2).

d e a l i n g with

important

result of this

of M ~ c o n s i s t i n g

It will be d i s c u s s e d

decomposition

section,

of vectors

matrices

of

having

at g r e a t e r ~

n

.

73 18

HOOKS AND S K E W - H O O K S Hooks play an i m p o r t a n t part in the r e p r e s e n t a t i o n

theory of

~n'

but it is not clear in terms of modules w h y they have a r$1e at all~ For example,

it w o u l d be nice to have a direct p r o o f of the Hook for-

m u l a for d i m e n s i o n s

(section 20), w i t h o u t doing all the w o r k r e q u i r e d

for the s t a n d a r d basis of the S p e c h t module. The F

(i,j)-hook may be r e g a r d e d as the i n t e r s e c t i o n of an infinite

shape

18.1

(having the

EXAMPLE

(i,j)-node at its corner) w i t h the diagram.

X X X X X XX

The

(2,2)-hook is

XXXX X

X

XXX

~

X~X

and the hook graph is

6 5 4 2 5431 321

18.2

DEFINITIONS (i)

The

(irj)-hook

of [p] consists of the

the ~i- j nodes to the right of it

(i,j)-node along w i t h

(called the arm of the hook)

and the

!

~ j - i nodes b e l o w it

(called the le~ of the hook). !

(ii) (iii)

The length

of the

If we replace the

(i,j)-hook is hij = Pi + ~j + 1 - i - j (i,j)-node of [p] by the n u m b e r hij for

each node, we obtain the hook graph. (iv)

A skew-hook is a c o n n e c t e d part of the rim of [~] w h i c h can

be removed to leave a p r o p e r diagram. 18.3

EXAMPLE

X X X X

X X X ~ X X

X

X X

skew 4-hooks in [42,3]. skew 5-hooks,

and

X ~

show the only two

The d i a g r a m also has one skew 6-hook,

two skew 3-hooks,

two skew 2-hooks,

two

and two skew 1-hooks.

C o m p a r i n g this w i t h the hook graph, we have illustrated: 18.4

LEMMA

There is a n a t u r a l i-i c o r r e s p o n d e n c e b e t w e e n the hooks

of [~] and the skew-hooks of [~]. Proof:

The skew hook

F

1

j th column corresponds

to the

(i,j)-hook.

X ~ ith row

74

19

THE

DETERMINANTAL

We h a v e

seen

FO~M

that when

11 a 12a

[11][12][13].. and

the m a t r i x

(see 6.4

and

m =

(m1~)

4.13). [i]

is l o w e r

triangular

19 .i

Inverting

[~]

triangular

with

l's d o w n

the d i a g o n a l

that

(m -1) I~

a n d m -I is l o w e r EXAMPLE

= ~ ml~

It f o l l o w s = Z

.... ,

[ ~ i ] [ ~ 2 ] [ ~ 3 ]. "'"

with

l's d o w n

the d i a g o n a l .

the m a t r i x

m for

[3][2]

[3][1] 2

~5

given

in s e c t i o n

6, we

find [5] [5]

m

[4,1]

-i

[3,2]

0

-i

1

1

-i

-i

-i = [ 3 , 1 2 ] [22,1] [2,1 ~ ] [15 ] The

we

w e go

-i

-i

1

2

-i

-2

1

1

-2

-2

3

3

-4

in the m a t r i x found

FORM i -

[m]

= O if m < O. down

term

and

to the n e x t

(which b e h a v e s

Rule,

and

the

partition

of nf

then

i+j]l

the d e t e r m i n a n t

diagonal,

by Y o u n g ' s

1

by

If i is a p r o p e r

=I[i

the

m are g i v e n

directly

[i]

to w r i t e

f r o m one

then

for

[I]

is to p u t

let the n u m b e r s

in e a c h

row.

as a m u l t i p l i c a t i v e

Beware

[11],[12]..

increase

by 1 as

of the d i s t i n c t i o n

identity)

and O

(0 x any-

EXAMPLES [3]

I J 19.4

1 1

DETERMINANTAL

down

b e t w e e n [0] t h i n g = 0). 19.3

[i] s

1

O

define

The way in o r d e r

[2][1] ~

-i

in m -I can be THE

where

[21211]

1

coefficients

entries 19.2

[4][1]

1

[4] I [i]

= [3][1]

- [4]

[0]

= [3,1]

[3]

[4] I

=

- [4][1]

[i]

[2] I

+ [4] - [4]

=

[3,1]

I

EXAMPLE

[3][2]

= [3,2]

+ [4,1]

+ [5]

- [4,1]

- [5]

= [3,2] Suppose

we h a v e

proved

the d e t e r m l n a n t a l

form

for 2-

75 part partitions. column,

Then expanding

the f o l l o w i n g

determinant

we h a v e

[3]

[4]

[5]

[i]

[2]

[3]

[O]

[1]

[2]

I

:

I[[3]I] [4] [2] I

[2]

-

I[3][O] [4] I[1]

[3]

+ L[I] [2] I [o] Ill which, =

up the last

by i n d u c t i o n ,

is

[3,2][2]

- [ 3 , 1][3]

[5]

+ [12][5]

[3,22 ] + [32,1]

+ [4,2,1]

+ [4,3]

+ [5,2]

-([32,1]

+ [4,2,1]

+ [4,3]

+ [5,2])

- ([6,1] + [6,1]

+ [5,12]) + [5,12 ] = [3,22 ]

Diagrams

Diagrams

Diagrams

containing

containing

containing

X X~

X X~

X~

X~

X X

X~ P r o o f of the D e t e r m i n a n t a l in the case w h e r e

I =

the end of i do n o t c h a n g e has no n o n - z e r o having

part,

Form:

column hook

with

of [I]",

the r e s u l t

zero p a r t s

The r e s u l t

that w e h a v e p r o v e d

at

is true w h e n

the r e s u l t

for

parts.

in the l a s t c o l u m n of

lengths

to p r o v e

I k > O, since

the d e t e r m i n a n t .

so a s s u m e

f e w e r tha n k n Q n - z e r o

The n u m b e r s

It is s u f f i c i e n t

(ll,12,...,Ik)

([li - i+j])

hll,h21,...,hkl,

are the

"first

since

• nil = li + ii' + 1 - i - 1 = li - i + k. Let s Example

be the s k e w h o o k of [I] c o r r e s p o n d i n g to the (i,l)-hook l 19.4, s3,s 2 and s I are X X X X X X X ~ XX

Omitting with diagonal

the l a s t c o l u m n

X~

and ith row of

X~

([I i - i+j])

(In

).

gives

a matrix

terms

[ l l ] , [ 1 2 ] , . . . , [ l i _ l ] , [ l i + 1 - i ] , . . . , [ I k - i] and t h e s e

are p r e c i s e l y

of e x p a n d i n g induction

the p a r t s

the d e t e r m i n a n t

of

[i \ s i]

I[I i - i+j]I

.

Therefore,

up the

the r e s u l t

last c o l ~ a n and u s i n g

is [ikSk][hkl ] - [~Sk_l][hk_l,1]+...±[IkSl][hll]

(*)

76 NOW consider to

[i ks i]

column

[I\ si][hil].

This is e v a l u a t e d by adding hil nodes

in all ways such that no two added nodes are in the s~ae

(by the L i t t l e w o o d - R i c h a r d s o n Rule, or C o r o l l a r y

17.14).

[I \ s i] c e r t a i n l y contains the last node of the ist, 2 n d , . . . , ( i - l ) t h rows of [I], so we deduce that all the diagrams in [I\ si][hil] (i) contain the last nodes of and

the ist,2nd,..., (i-l)th rows of [I],

(ii) do not contain the last nodes of the

(i+l)th,

(i+2)th,...,kth

rows of [I]. Split the diagrams in [i \ si][hil]

into 2 set, a c c o r d i n g to w h e t h e r

or not the last node of the ith row of [I] is in the diagram.

It is

clear that [I] is the only d i a g r a m we get c o n t a i n i n g the last nodes of all the rows of [I], and a little thought shows that in c a n c e l l i n g in pairs to leave 19.5

Proof:

COROLLARY

[I].

dim S1 = n~ I (I i ~ i+j) ~ I where

[~i][~2]...

(*) we get sets

This proves the D e t e r m i n a n t a l Form.

[~k ] has d i m e n s i o n

n~ ~l~...~k :

(see 4.2), and the C o r o l l a r y is now immediate.

~1

= 0 if

r < O

77 20 20.1

THE HO0~ FORMULA FOR DIMENSIONS THEOREM

(Frame, Robinson

The dimension

and Thrall

[4])

of the Specht module S l is given by (hil - hkl)

dim S l = n'

i O,

a a V p ( b _ l ) = V p ( a _ b + l ) < Zp(a)

of the L e m m a

each

all of

)"

gives

our

result.

the g i v e n

89 23

SOME I R R E D U C I B L E SPECHT MODULES The Specht module S ~ is i r r e d u c i b l e over fields of c h a r a c t e r i s t i c

zero, and since every field is a s p l i t t i n g field for

~n' S~ is irre-

ducible over field of prime c h a r a c t e r i s t i c p if and only if it is i r r e d u c i b l e w h e n the ground field has p elements.

This then, is the

case we shall i n v e s t i g a t e and, except w h e r e o t h e r w i s e stated, F is the field of o r d e r p in this section.

The complete c l a s s i f i c a t i o n of irre-

ducible Specht modules is still an open problem,

but we tackle special

cases below. 23.1

LEMMA

Suppose that HornF ~ n ( S U t S ~) -- F.

Then S H is i r r e d u c i b l e

if and only if S H is self dual. Proof:

If S H is irreducible,

then it is certainly self-dual

(since its

raodular c h a r a c t e r is real.) Let U be an irreducible submodule of S ~. there is a submodule V of S H w i t h S ~ / V -~ U. S~

If S H is self-dual,

then

Since

~ S~/V ÷ U canon iso

gives a n o n - z e r o e l e m e n t of HornF ~ n ( S ~ , S ~) , we m u s t have U = S ~, so S ~ is irreducible. The h y p o t h e s i s H o m F ~ n ( S ~ , S ~) ~ F c a n n o t be o m i t t e d from this L e m m a (see E x a m p l e 23.1c/ill) below), h y p o t h e s i s holds

but C o r o l l a r y 13.17 shows that the

for m o s t Specht modules.

Before applying the Lemma, we w a n t a result about the integer g~' defined in 10.3 as the g r e a t e s t common d i v i s o r of the integers < et,et,

>

where e t and et, are p o l y t a b l o i d s

tition conjugate

R e m e m b e r that i,

y

> O

irreand

x + y =n. 1 Let

(y+2) . .. (y+x)

2

t =

(y+l) and

let

Kt

= Z

{sgn

o)s

Kt = For

the

moment,

I ~ e

(i work

~ { 2 , 3 ..... y + l } }"

(12) over

-

(13)

~.

-

...

-

Then

(l,y+l))< t

.

Then

{t}KtPtK t =

{t}Kt~tP t = y:{t}<tp

y:{t}Ktgt(l

-

t

Therefore,

= H(hook = But

g~'

=

the

Len~a

10.4,

we

if p

~ n,

23.8

If

are Im p ~

now

-

in

.

-

over

and

(12) of

(12)

working

8 = S~ n,

-

homomorphism {t}(1

where

...

is

t

...-

definition

...-(l,y+l))8 the

field

self-dual.

(l,y+l))

= by

{t}KtPtK t 23.2

.

so

-

Therefore,

S (x'ly)

-

[ ~ J ) { t } < t,

(x - i) ~ y : ( x + y ) { t }
i, all

{t} = {t*} {t}(l

-

is the

(12)

the

unique

proves

in et, involved

- ...-(l,y+l)), < {t~l

Therefore,

tabloids tabloid

{t}(l

-

-

(12)

Finally,

we p r o v e

y > 1 a n d p = 2.

that

case,

et,

Ker

row.

Hence

and

8.15,

> = i.

8 \ S ~l,

where

and T h e o r e m

23.4

p I n.

S ~ is r e d u c i b l e

By T h e o r e m

first

et,

so

...-(1,y+l))¢

in this

1 in the

in b o t h

(12)-...-(l,y+l)), -

S ~ ~s r e d u c i b l e

and

have

when

we may

~ =

assume

(x,1 y) w i t h

that

x a y.

x > i,

Observe

that [x][y3 and by

[x][l y]

2-modular

as a 2 - m o d u l a r

so X (x'ly)

Remark:

The

But when

and

Whence,

is c e r t a i n l y part

~n

a reducible

of the p r o o f

n are and

in the

(n-l,l)

n is odd, since

21.12 23.7

will

of

same

2- m o d u l a r

shows

~n'

are

help

since

2-block

that us

matrices

unfortunately,

X

(i y)

are

+ X (x,y)

all

in o u r

this

~n"

2-block,

the

2-part

5.1 p r o v e s Example

first

are

the

that

21.10).

in the n e x t

partitions,

case

of of

(n) When

2-blocks,

in d i f f e r e n t

result

For hook

2-block

partitions

in two d i f f e r e n t

(n-l,l)

is n o t

same

in the same

Example

of n lie

of

...

character.

in the are

(see also,

(n) and

+

that

n is even,

partitions

shows

the d e c o m p o s i t i o n

to c a l c u l a t e ;

When

2-block

are in the

the 2 - p a r t

Lemma

Theorem on

same

21.9).

+...

+ x ( x + 4 , Y -4)

(n-l,l),(n-3,3),(n-5,5),...

(see T h e o r e m

and

by i n d u c t i o n ,

(n),(n-2,2),(n-4,4),... and

(y)

thus

= x ( x , Y ) + x ( x + 2 , Y -2)

last

p = 2, X

+ X (x,ly) = x ( x + Y ) + x ( x + Y -I,I)

character.

x(X, Iy)

+..+Ix,y3

+ [x,lYl

Rule.

character,

x(X+l, Iy-I)

+ [x+y-l,l]

= [x+l,l y-I]

the L i t t l e w o o d - R i c h a r d s o n

the same

and

= ix+y]

for o t h e r

2-blocks. chapter

g~'

is e a s y

types

of

94 partition, 23.9

for example:

LEMMA

If ~ = (x,y),

g~' = y! g.c.d. Proof:

then

{x: .....(x-1)~l:,

(x-2):2! ..... (x-v):v:}

Let t I and t 2 be p'-tableaux.

Xij = {klk belongs

Let

to the ith column of t I and to the jth column of t2}

Xll u XI2 ~

Xll u X21 X21 u X22

The polytabloids

XI2 u X22

etl and at2 in S~

if an only if no two numbers

have

the tabloid

from any one of the sets

{t 3} in common

Xll u XI2,

X21 u X22 , Xll u X21 , XI2 u X22 are in the same row of {t3}. Any row of {t3} must contain a number from X12 and a number from X21 or no numbers

from XI2 u X21.

Therefore,

< etl,et2

• = 0 unless

IX12 I =

Ix211. Suppose now that IX121 = IX211 . The tabloid {t 3} is common to etl and et2 if and only if each of the first y rows of {t3} is occupied by just one number X21 contains y~

from X21 u X22 and each row containing

a number

from XI2.

Thus,

IXl21: (x - Ix121) ~ common tabloids. Assume that the tabloid representative

{t 3} has been chosen bilizer

of t I.

interchanging numbers

fixed.

t 3 for the common

Let o be the permutation each number

in XI2 with

a number

in X21,

leaving

Then t o = t2~ 2 for some ~2 in the column Therefore,

< et I, at2 • = ±y:

t I ~i O ~ i

IXI21~

sta-

t3

the other

stabilizer

= t2 ' and

of

(sgn 5 )

But {t3} = {tl}~ 1

(x - IXl21) ~

By definition, gU' is the greatest common divisor since 0 s IX12 I ~ y, the Lemma ks proved.

(i)

tabloid

in the row stabilizerof

(sgn ~2 ) depends only on t I and t 2 and not on {t 3} = {t2}~ 2, and hence

23. iO

from

such that t 3 = tl~ 1 for some ~i in the column

t 2, and sgn o = (-i) IX~2 I.

and,

a number

etl and at2 have

of such integers,

EXAMPLES If ~ = (5,2),

then g~' = 2~ g.c.d.(5~,4:l~,312~)

K(hook lengths in [~3) = 23,32,5. Therefore, S (5'2) only if the grom%d field has characteristic 3 or 5.

= 2s.3.

is reducible

But if and

95

(ii) has

Similarly,

characteristic (iii)

field

have

characteristic

is s e l f - d u a l ,

p = 2.

and E x a m p l e

isomorphic

factors

can o c c u r

in e i t h e r

last E x a m p l e 23.1

Then

21.8(ii)

to S (5'2)

of c h a r a c t e r i s t i c

order,

if the

ground

field

the

23.8.

first that

Since

S

Now

example (5,1 z)

S (5'12)

a n d so S (5'12)

let the

ground

proves

S (5'2)

has

compositlon

is s e l f - d u a l ,

is d e c o m p o s a b l e

these over

a

2.

pro%~s t h a t

the h y p o t h e s e s

13.18,

23.11

DEFINITION

The

p-power

lacing

each

hij

in the h o o k

cannot

be o m i t t e d

in

or 23.4.

integer

EXAMPLE

by

shows

and S (7) .

13.17,

23.12

if and o n l y

2 or 5.

factors

The

is r e d u c i b l e

If p ~ 7, S (5'12)

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

field

S (5'3)

If ~ =

9ia~ram

(8,5,2) ,then

[~3 P for ~ is o b t a i n e d

graph

by rep-

for ~ by 9 p ( h i j ) .

the h o o k

graph

is

1 0 9 7 6 5 3 2 1 65321 2 1

0 2 0 1 0 1 0 0 and

[~j 3 =

1 0 1 00 O0 l O O l O O 1 0

and

[~J 2 =

1 0 0 1 0 i0

We n o w 2-part

classify

the

irreducible

Specht

modules

corresponding

to

partition.

23.13

THEO~M

Suppose

x ~ y).

Then

and only

if some

Proof:

S ~ defined

The hook

column lengths

~ =

(x,y)is

over of

the

[~]P

hij

p-regular

field

contains

for

[~3

(i.e.

of p e l e m e n t s ~

are

given

1 ~ j < y

for

hlj

= x - j + 1

for y < j ~ x

h2j

= y - j + 1

for 1 ~ j -< y.

is a j w i t h and

9p(hlj)

let ~p(h2j)

~ Vp(h2j),

= r.

Then

consider

j + pr

= ~p(h2i)

< r

for

if

numbers,

the

largest

< Y + 1 and r

~p(hli)

assume

by

= x - j + 2

If t h e r e

is r e d u c i b l e

two d i f f e r e n t

hlj

property

if p = 2, we

j + 1 -< i < j + p

j with

this

96

But

{hlilJ

Up(hlj) Up(y

~ i < j + pr}

a r = Up(h2j).

- j+l).

Writing

if and o n l y

if Up(b)

23.14

column

Some

if t h e r e

g~'

is a set of pr c o n s e c u t i v e Since

Up(hlj)

b = x - j+2 > Up(X

of

~ Up(h2j),

and n o t i n g

- y+l),

this

Ix,y3 p c o n t a i n s

integers,

we h a v e

t h a t Up(b)

b with

Now,

H(hook

in ix,y3)

= y:

g . c . d . { x : , (x - i) :i~,..., (x - y) :y:} b y L e m m a

23.5

proves

that

S ~ is r e d u c i b l e

x+l x - y+l Since

l.c.m.

=

~ b ~ x+l

numbers

is an i n t e g e r lengths

> Up(b

(y: (x+l) ~)/(x

if and o n l y

and Up(b)

of

On

and

- y+l)

if and o ~ l y

rent

numbers.

Up(b)

s Up(X

- y+l) .

and

23.9,

so T h e o r e m

if p d i v i d e s

#

23.15,

Ix,y] P c o n t a i n s

the o t h e r

- i)

• Up(X

23.15 S (x'y) is r e d u c i b l e if and o n l y if there is an i n t e g e r x - y+l ~ b s x+l a n d Up {~ ~ .x+l. y + l ~ b ;} > 0 .

column

- x+y

x (xXy) (x_l) .... , _ }

{(~),

= D~ b J

_

23.14

>

• ,x+l,

(x+l) (bXl)

Comparing

- j+2)

proves

two d i f f e r e n t

x - y+2

so

Up(X

hand,

Then,

we

see

that

two d i f f e r e n t

suppose

t h a t no

for e v e r y

b with

s(X'Y) is r e d u c i b l e

b with

if some

numbers. column

x - y+2

of

Ix,y] p c o n t a i n s

diffe-

~ b s x+l,

- y+l).

Let

r x - y+l

r+l

= arP

s

+ ar+iP

+

... + asP

(O ~ a i < p, Then

x - y+l
U p ( X - y+l). Thus l ~ _ r + l (ar+ 1 + j~ + ... + aspS. Therefore ... + c r p r

= c o + clP +

+ ar+iP

r+l +

our

supposi-

s • . . + as p (O ~ c i < p)

a n d if

x - y + 1 ~ b ~ x+l, b = bqpq

+ b q + i P q+l

then +

... + b r p r

+ ar+ipr+ 1 +

...

+ asps

(O ~ b i < p, bq Therefore, x+l - b

=

co

+

clP

+

.

+ .

Cq. _ i p q -.i

+.

dqp q

+

+

drpr

(O ~ d i < p)

~ 0).

g7

where

d

qpq

By L e m m a

+

-.. + drP

r

cqpq

=

+...+

CrP

r

bqpq

-

-

... - b r P

r

22.2,

Up[,x+l, b ;

= {~p(b) (bq

=

+

= ~p

+ ~p(X+l

b r + dq

...+

fCqp q +

-

...

+

d r

-

-

Cq

... -

Cr)/(p

i)

-

+ brprj

r - q, by L e m m a Up(X

+

... + c r p r I

%bqp q +

=

- Op (x+l) }/(p - i)

- b)

y+l)

-

22.3

u

(since

b

Up{~

_

~ O)

q

(b).

P Therefore, and S (x'y) 23.16

for x - y+l

is i r r e d u c i b l e , a s

EXAMPLE

if a n d o n l y

S (2p-I'P)

module

believed,

D (3p-l)

f r o m the H o o k

p-modular R.W. 23.17

Formula

has

CONJECTURE

2310).

apparently

d i m S (2p-I'P) - the m i s t a k e which

over

the

This

is i n t e r e s t i n g

field

factors,

H 1 mod would

states

representation

put

x+l, ( b ;} ~ 0

+i

on the e v i d e n c e

two c o m p o s i t i o n

of B r a u e r

irreducible Carter

and o n l y

has

Since

to a c o n j e c t u r e

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

(cf. E x a m p l e

always

by

required.

S (2P-I'p)

if p ~ 2

an earlier a u t h o r that

~ b ~ x+l,

p~

have

that

of p e l e m e n t s

of the

one b e i n g

for p o d d provided

Up(JGj/dim

D of a g r o u p

because case

p = 2,

the

trivial

- this

follows

counterexamples D)

a O for e a c h

G.

forward

No c o l u m n

if p is p - r e g u l a r

of

[~3 p c o n t a i n s

two

a n d S ~ is i r r e d u c i b l e

different

over

the

numbers

field

if

of p

elements. It is t r i v i a l

that

if ~ is p - s i n g u l a r . is n e c e s s a r y proved

the

if and o n l y dered tion

field

neither

that

~ nor

a column

containing

[llj

proved

Specht

in the

23.13).

(This We

module

over

the

given

irreducible,

numbers

condition and has

p = 2. out

is the o n l y

conjecture

two d i f f e r e n t

that

to be

it t u r n s

S ~ is i r r e d u c i b l e

~' is 2 - r e g u l a r .

has

case w h e r e

of 2 e l e m e n t s ,

if x = 1 or 2

in T h e o r e m U such

author

for a p - r e g u l a r

it is s u f f i c i e n t

Over

[~]P has

The

that the

that

2 part (2,2) field

S (x'x)

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

partition

not

is the u n i q u e of 2 e l e m e n t s

consipartibut

24

ON THE DECOMPOSITION There

MATRICES OF

~n

is no known way of determining

the composition

factors of the

general Specht module when the ground field F has characteristic p.

Thus we cannot decide

~n'

which

records

sentation

cases.

THEOREM (i)

the m u l t i p l i c i t y

D ~ (i p-regular)

some special 24.1

the entries

in the decomposition

of each p m o d u l a r

as a composition

irreducible

factor of S ~, except

The theorems we expound give only partial

(Peel [18])

a prime

matrix of reprein

results.

Suppose p is odd.

If p T nt all the hook representations

of

~n

remain irreducible

modulo Pt and no two are isomorphic, (ii)

If p I n, part of the d e c o m p o s i t i o n (n)

1

(n-l, i) (n-l, 12 )

1

•

1 1

I

1

i_ss

~

1

(in) Proof:

~n

q/

1

©

(2,1 n-2 )

matrix of

1

The result is true for n = O, so we may assume that it is true

for n - i.

Note that

x(x'lY)

+

%-1

= x(X-l'lY)

+ X (x'ly-l)

if

x > l, y > O, x+y = n.

Case

(i)

p does not divide n.

In view of T h e o r e m 23.7, we need prove only that no two hook representations

are isomorphic.

non-isomorphic Case

But this follows

restrictions

(ii) p divides

to

at once,

since they have

~n-l"

n.

Suppose x > i, y > O.

Then by restricting

to

%-1'

(x'lY) has at X two, by T h e o r e m

most two modular constituents, and therefore precisely + 23.7. Let ~x be the modular constituent of X (x'ly) satisfying + ~X ~

~n-i

(and let ~

=

x(x-1, ly)

= O and ~i = O).

no other equalities to

~n-l" The following

except

and ~x be that satisfying

~x ~ ~ n - i

= x(x'lY-1)

We must show that for every x, #x-i = #x ;

can hold because

relation between

there are different

characters

(n), in p a r t i c u l a r on all p - r e g u l a r

restrictions

holds on all classes

classes:

99

X (n) - X (n-l'l) (This comes ordinary

from Theorem

character

~n+ ~x-l_

were

this r e l a t i o n , characters

21.7 or d i r e c t

orthogonality

In terms of m o d u l a r

If some

not equal

from T h e o r e m

+ ) (~n-2 + #n-2

by p - r e g u l a r

are zero.

(5)

We w r i t e

(4,1)

1

(3,2)

1

(3,2)

(3,12 )

24.1.

Taking

is

1

1

corresponding

X (5) - X (22'I)

to

(5),

+ X (2'I~)

are i r r e d u c i b l e

Thus, X (22'I) has p r e c i s e l y (22'I) , it follo w s t h a t

(4,1)

= ~(5)

X (2'13)

= 9 (22'1)

X(I s)

-

X

21.7, = 0

we

from

find that

on 3 - r e g u l a r

classes.

by T h e o r e m

24.1.

S i n c e one of these m u s t be

+ X

deduced

(4 i) ' = O

W h e n p = 3, the d e c o m p o s i t i o n

g i v e n in the A p p e n d i x .

(3,12 ) come

+ 9(22,1)

is s i m i l a r l y

(3,2)

and

and i n e q u i v a l e n t ,

two factors.

X(2z,I)

The r e s t of the m a t r i x

EXAMPLE

~5

1

B u t X (5) and X (2'13)

24.3

m a t r i x of

1 1

[9] = [2] and r = 3 in T h e o r e m

and

in character

1

(15 )

The rows

entries

(22,1)

(2,13 )

Theorem

Omitted

c h a r a c t e r of D I.

(3,12 )

Proof:

the e n t r y

of D l as a c o m p o s i t i o n

1

(4,1)

(22,1)

p.

matrices

Thus

X u for the p - m o d u l a r

W h e n p = 3, the d e c o m p o s i t i o n (5)

irreducible

partitions.

is the m u l t i p l i c i t y

of S ~ and ~l for the p - m o d u l a r EXAMPLE

just once in

independent.

f a c t o r of S M o v e r a f i e l d of c h a r a c t e r i s t i c

24.2

appear

= O.

l a b e l the rows of o u r d e c o m p o s i t i o n

and the c o l u m n s

matrices

"''-+ ~ i

the fact t h at the m o d u l a r

are l i n e a r l y

in the Mth row a n d ith c o l u m n

decgmposition

21. 4, by u s i n g the

we h ave

to #~,_ t h e n ~x-i w o u l d

contradicting

F r o m now on, w e s h a l l

- ... ± X (ln) = O.

relations).

characters;

(~n-i + ~ - I ) +

of a g r o u p

by p a r t i t i o n s ,

+ X (n-2'12)

from the e q u a t i o n :

on 3 - r e g u l a r m a t r i x of

classes. ~6

is that

100

(4,2)

Proof:

First

note

that

(22,12

X

)

are

and X

irreducible

by E x a m p l e

23.6(i). By T h e o r e m

24.1,

part

(6) (6)

1

(5,1)

1

of the m a t r i x

is

(4,12 )

(5,1)

1

(4,12 )

1

1

(3,13 )

1

1 1

(2,1 ~ )

1

(i 6 )

Applying turn w e

1 Theorem

21.7,

with

X (5'I) X(4 3-regular

'

classes.

us to d e d u c e

- X (3'2'1)

- X (32)

12)

= [3]

+ X (3'13)

[2,1]

and

[13 ] in

the

+ X (23)

= O

(16)

+X

together

+ X (4'12) remaining

= O

+ X (2'I~)

equations,

- X (5'1) that

- X (23)

X(3,2,1)

-

These

X (6) enable

[93

get, X (6) + X (32)

on

r = 3 and

with

- X (3'13) two

=O

- X (2'I~)

columns

above

- X (16)

should

= O

be

l a b e l l e d (3,2,1) and (32), r e s p e c t i v e l y , and the e q u a t i o n s let us w r i t e X (32), X (3'2'I) a n d X (23) in t e r m s of ~(6) , # (5,1) , ..., in the w a y shown

in the

Note

that

the N a k a y a m a where

complete Examples

ching

Theorem

24.3

in T h e o r e m

matrices

theory, fails

(cf. E x a m p l e

have

been

We

of

~6

but

this

computed

that

that

of

the

using

method

factors rapidly

using

(except

it is q u i c k e r ~5

traditional

and v e r y

without

to i n d u c t i o n

agree

from

to d e t e r m i n e 23.16),

in the A p p e n d i x .

resorting

24.1).

matrix

and b l o c k

for p = 2

and

matrix

and w i t h o u t

the d e c o m p o s i t i o n

decomposition even

24.2

Conjecture,

it is i m p l i c i t

deduce

decomposition

to

the B r a n -

of

finding

of S (2P-I'P) , leads

to f u r t h e r

ambiguities. It seems

to us t h a t

the d e c o m p o s i t i o n the o r d e r

of the

cities

of the

useful

Theorems

(SI,M ~)

factors

composition we k n o w

and C o r o l l a r y

It is u n f o r t u n a t e notation tion,

which

that

has

b u t we e m b a r k

We r e t u r n

if a m e t h o d

matrices

of

%'

of e a c h

Specht

factors.

these

describing two

the

to the n o t a t i o n

task

as w e l l

line

giving

look

the

of s e c t i o n

finding

a basis

rather

concerning

as the m u l t i p l i -

of attack,

simplicity

of e m p l o y i n g

for

information

S ~ as a k e r n e l

results

to be u s e d o b s c u r e s upon

this

13.13,

devised

include

module,

For

are T h e o r e m

17.18,

is e v e n t u a l l y it w i l l

the m o s t

of H o m F ~ n

intersection.

ugly,

and

of t h e i r

that

the

applica-

them.

13, w h e r e

M ~ is d e s c r i b e d

as the

101

space spanned by X-tableaux of type u. The remarks following 17.8 and 17.10 show that the homomorphism Jliv acts on Mu by sending a tableau T to the sum of all the tableaux ob'tained by changing all but v (i+l)'s to i's. e.g.

$l,l

:11122 233

The first result present the special relevant ideas.

*

+

11111 2 33

11112 133

+

11121 133

we prove could be subsumed in Theorem 24.6, case to help the reader become familiar with

24.4 THEOREM Over a field characteristic of prime module isomorphic to the trivial Gn-module S (n) if Ir!Ji 3 -1 mod p% where z i 2ApL”i+lL .4

and only

if

but we the

for

all

Proof: By Theorem 13.13 (or trivially) there is, to within a scalar multiple, a unique element OT in HomP G (S(n),Mu). T is the semistandard (n)-tableau of type u, and OT s?nds {t) to the sum of the (n)tableaux of type u . e.g. if u = (3,2), then wo, =11122+11212+11221+12112+12121+ 12211+21112+21121+21211+22111. Now, the crucial (u1,u2,...,uiDl,ui

step + ui+l

is

that

when Tl is

- v,v,~~+~,...)

an (n)-tableau

there

of type

are

Pi + lJi+l - v ui+1 - v J to T in which

tableaux row equivalent to i's to give Tl

all

but

v (i+l)'s

can be changed

e.g. 1 1 1 1 1 comes from (z) tableaux above, by changing all the 2's to l's, and each of 1 1 1 1 2, 11121,11211,12111, 2 1 1 1 1 comes from (:) tableaux by changing all except one 2 to 1. ui+i-' Therefore, {t)0, belongs to (7 ker Jli I v if and only if each of v=o (uiu~+~i+lj

'

(uiu:+:itll‘

')

'**.'

(uiy)

is divisible by p. This is equivalent to ui Z -1 mod psi where by Corollary 22.5. Thus, Corollary17.18 shows that =i = ap(ui+l), belongs to Su if and only if this congruence holds for all i 2 1. WOT 24.5

EXAMPLES (i)

S(a82'2'1)

contains

a trivial

submodule

if

and only

102

if the

ground

(ii)

field

S (5'2)

(iii)

F has

does

not

characteristic contain

S (P-I'p-I'''''p-I'r)

a n d r < p.

Write

n = x(p

3.

a trivial

contains

- l)+r.

submodule

a trivial

Then

if c h a r

submodule

( ( x + l ) r , x p-l-r)

F = 2.

if c h a r

is the p a r t i t i o n

~'

c o n j u g a t e to ~ = ((p - l)X,r). S i n c e H o m v ~ (s(n),s ~) ~ O, In ~, - --n S ~ 0 S ( ) is i s o m o r p h i c to the dual of S ÷t f o l l o w s t h a t Hom F ~

(S~',S (In))

~ O.

partition

of n s u c h

u n i q u e top c o m p o s i t i o n S (Is) ~ D(3, 2) (iv) that

Consulting

S (4'2)

not have

has

the

F = 2 and

morphisms of t h e m

sends

a tedious

large

the

r e m 24.6 and

I =

proves (cf.

should

below

to p r o v e

(16,2),

(13,5)

that

S (IO'5'3)

Carter's When

that

a semistandard

the

13.13

24.15 13 we

of the

to c o n s i d e r

rows

a tableau

and

cases

the

reverse

of

homo-

17.18. for

fairly

on s m a l l

technique

partiof T h e o -

char

F = 3

Conjecture,

this

characteristic

where

3

Even

below,

for e x a m p l e .

that

there

of type where

decreasing

semistandard.

of S ~ .

over

all

The

usually

H o m F ~ n ( S I , S ~)

~

n

field

;

a

out

F of

in the w a y that

2 elem-

factors

part

shall

of the n e x t

of

we d e f i n e

it is o f t e n

are n o n - i n c r e a s i n g we

is

in

a modification

composition

columns;

second

of

sometimes

choice

It turns

the

the

the

the n u m b e r s down

are

Unfortunately,

matrix

so,

is m u c h

~.

we then

H o m F ~ n ( S I , S ~) is n o n - z e r o

of S (4'2)

) = O.

17.18, since

decomposition

to c l a s s i f y

tableaux

strictly

the

fields

factor

enough

h-tableau

practice

the N a k a y a m a

and C o r o l l a r y

a composition

saw

even

(IO'5'3')) = O w h e n

over

where

combination

of C o r o l l a r y

using

S l is p - r e g u l a r ,

D (6) is a f a c t o r

is g o o d

Using

case

linear

impossible,

a little

can use

or n o t

semistandard

some

see does

23.17).

but HOmF ~n(S(6),S(4'2)

In s e c t i o n

useful

(10,8).

to d e t e r m i n e

see T h e o r e m

after

~, w e

the

we

S (4'2)

24.4.

uninteresting list

no difficulty Hom F~n(Sl,S

case w h e r e D l is

24.5(iv)

of the m e t h o d S~;

or

Theorem

in the

sufficient

ents,

that

I and whether

intersection

altogether

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

completeclassi fication

Example

have

Conjecture

applying

0 implies

not

by T h e o r e m

test w h e t h e r

the k e r n e l but

For example,

reader

interested

not

M ~ and t h e n

task,

rather

for we m a y

where

in the A p p e n d i x ,

to d e t e r m i n e

in the

24.2,

for p = 2, b u t

for any g i v e n

17.18

(except

{t}< t i n t o

partitions.

tions,

factor,

I is 2 - s i n g u l a r ) ,

This

is

composition that

Example

matrices factor

to see

f r o m M l into

Compare

composition

and C o r o l l a r y

H o m F ~ n ( S I , S ~) is zero char

of S~').

decomposition

bottom

It is i n t e r e s t i n g 13.13

factor

a trivial

a trivial

Theorem

and

By c o n s t r u c t i o n , S ~ is p - r e g u l a r , so U' is the , , t h a t D ~ ~ S (In) ( R e m e m b e r t h a t D~ is the

n unique

F = p,

call

most

along such

Theorem

103

probably

classifies

homomorphism of m o r e

t h a n one s e m i s t a n d a r d

horribly 24.6

all cases w h e r e

in H o m F ~ n ( S I , S ~ ) .

there

is a r e v e r s e

homomorphism,

semistandard

the s i t u a t i o n

becomes

complicated~

THEOREM

Assume

t h a t c h a r F = p.

that

Suppose

I and ~ are

(proper)

t h a t T is a r e v e r s e

partitions

(i)

If for all i > 2 and j >- i, N i _ l , j

aij = ~p~(Nij), th e n (ii)

8T b e l o n g s

of n and

semistandard

of type ~lt a n d let Nij be the n u m b e r of i's in the

l-tableaux

jth row of T.

- -i m o d p a ij w h e r e

to H 0 m F ~ n ( M A , S ~ )

and Ker

8 T c S l~.

If for all i -> 2 and j ~ i, Ni_l, j -- -i rood p bij w h e r e

--ib'3 = m i n { £ p ( N i j )' ~'p(ImZ= ( l J +m-I --sZ--j -Nms) )}' then e l e m e n t of HornF ~ n ( s I , S Proof:

combinations

linear

When considering

~) .

S i n c e T is r e v e r s e

and the R e m a r k

L e t t be the

Ker

13.14.

8 T ~_ S l by L e m m a

Therefore,

Ker

13.11

8T c__ S I±

Theorem. l - t a b l e a u u s e d to d e f i n e

{t}8 T is, by d e f i n i t i o n , row e q u i v a l e n t

semistandard,

following Corollary

by the S u b m o d u l e

is a n o n - z e r o

the

~n

a c t i o n on M ~.

the s u m of the l - t a b l e a u x

Then

of type ~ w h i c h

are

to T.

L e t i -> 2, O < v -< ~i - i. S i n c e Z N.. = ~i' w e may c h o o s e j=l iJ Choose a V l , V 2 , . . . s u c h t h a t O < vj• < N i3. for e a c h j and Z v.3 = v. t a b l e a u T 1 row e q u i v a l e n t ~o T, and for e a c h j c h a n g e all e x c e p t vj i's in the jth row of T 1 i n t o By d e f i n i t i o n , in this way,

each

(i-l) 's.

and T 2 a p p e a r s

Since

tableaux

Nik is d i v i s i b l e

- vk

Corollary

the h y p o t h e s i s

then

- Vk!

by p, by C o r o l l a r y

of the T h e o r e m h o l d s ,

is an i n t e g e r k w i t h

.

Ni_l, j ~ -i m o d p aij Ni_l, k + N i k

Under

to T.

~ N . = ~i > v = ~ v t h e re j=l 13 j=l J'

If for all j

from

3

row e q u i v a l e n t

O ~ v k < Nik

tableau.

in { t } S T ~ i _ l , v is c o n s t r u c t e d

in {t}0 T ~ i - l , v

J-3

different

L e t T 2 be the r e s u l t i n g

tableau T 2 involved

of p a r t

22.5.

Thus

17.18 proves (ii),

if the h y p o t h e s i s t h a t M I S T ! S~

it a g a i n

follows

that

of p a r t

(i)

as r e q u i r e d .

104

{t}Kt ~i-l,v

does not

i n v o l v e T2, e x c e p t i f i-i - vk > Z (Ik+m_ 1 ~ Nms ) m=l s=k

Nik But

f o r m < i - i, T 2 h a s

since T 2 has at l e a s t since

come

~ N numbers equal s= k ms from a tabIeau row equivalent

s=kZ N i - l ' s

Nik

+ Nik

- v k i's h a v e b e e n

therefore,

T 2 has

less

that

this

ains

two

shows

than

excedes

in p a r t

maximal

24.7

COROLLARY

composition second

(ii)

24.8 Then

shall

the

D (32)

zj

use

assume

is a f a c t o r

of S p;

Theorem

24.6

also

of SP;

factor

factor

are

Let

S 1 n S I± is t h e

of S ~.

24.6,

Under

the

24.7.

to f i n d

We give all

to 2 - p a r t

(3,2,1) of S~;

the

just

compo-

partitions.

and char

F = 3.

take T = 3 2 2 1 1 . 1

take T = 3 2 2 1 1 1 take T = 3 2 1 1 2 1

gives

If for all

- P i + l ) , t h e n S~

i >_ 2, lji_ 1 -__lji - -i m o d is i r r e d u c i b l e

over

a field

p zi

where

of c h a r a c t e r i s t i c

p.

Proof: Nij 24.6

The

= ~i+j-i

unique - ~i+j

show that

every

of S ~ if I is p - r e g u l a r .

later

~ =

i ~ 2 to p r o v e .

we have

of C o r o l l a r y

factors

when

(i) of T h e o r e m

corresponding

24.3).

of S (5'I)

of S ~, a n d

of p a r t

again

k or j < k - r + 2). 3 i, (r) (i) + - d Pi = ~i di+l i

and

be

of n w i t h

•

×

X

X

1 ,S ~)

~ 0 for

I ~ ~ and

I,~

any p a i r

from

(7,3,1),

106

(52,1), (5,32 ) and

(5,3,2,1).

of the d e c o m p o s i t i o n

matrix

D(7,3,1) S (7'3'I)

the

Compare

of

~ii

the

D(52,1)

D(5,32)

1

1

S (5'32)

1

1

1

S (5'3'2'I)

1

1

1

Note

that

same

for e a c h

the n u ~ b e r

the C o r o l l a r y

we

are

il +

~(2)

2

24.12

next

zero

Theorem

we

Given

a = ao+

ro'~ above

ir-i (r) + ~

nodes

need

24.10;

with

not

be

in p a r t i -

i I > i2>... >ir- 1

at any stage.

since when

char

The

hypothesis

F = 2,

X X X X

and H O m F ~ 4 ( S ( 3 ' I )

a contains

is zero

+

,S(22))

are n o n - z e r o

(by T h e o r e m

24.4).

require

two n o n - n e g a t i v e

alp +

b = bo+ blP that

+...

omitted,

2

to the

in C o r o l l a r y

Horn F ~ 4 ( S (4) ,S (22))

DEFINITION

We say

raise

case

(3)

~

H O m F ~ 4 ( S ( 4 ) ,S(3'I))

our

1

X

(Dy the C o r o l l a r y ) , For

be

we

the

i2 +

X X X

X X and w h i l e

D(5,3,2,1)

~(k-j+2)

to raise

~_icannot

X X

~

includes

allowed

i I > i 2 >...>

of n o d e s

~(k-j+l)

Z(1) since

4 by 4 s u b m a t r i x

3.

1

S (52'I)

cular,

following

for the p r i m e

arP r

...+

...+ b s P

b to b a s e

integers

a and b,

(0 _< a i < p,

s

a r ~ O)

(0 -< b i < p, b s

p if s < r and

let

~ O).

for e a c h

i b i = 0 or

b. = a. l 1 24.13

E~LE

0,2,9

= 1.32

24.14

65 = 2 + 0.3 and

DEFINITION

n + 1 contains Since D (n-j'j)

evaluating 24.15 factor

The

p,

THEOREM

composition

(James

of s(n-m'm)

is

[6] f

precisely

3.

fp(n,m)

is d e f i n e d

by

fp(n,m)

= 1

if

and = O, o t h e r w i s e . factors

j _< m, by C o r o l l a r y

the d e c o m p o s i t i o n

+ 2.33 , so 65 c o n t a i n s

to b a s e

function

m to b a s e

the o n l y

with

+ 1.32

ii = 2 + 1.32

12.2,

matrix and

[8]) .

for

of S (n-m'm) a sensible ~n

have

first

the step

form towards

is to p r o v e

The multiplicity

of D (n-j'j)

as a

(n-2j,m-j) . P

Proof

Since

the r e s u l t

is t r u e w h e n

n = 0 or

i, w e m a y

assume

it for

107

n' < n.

L e t t be the

M (n-m'm) the

(n-j,j)-tableau

L e t T be the

(n-j,j)-tableau

(l,l) th, ( l , 2 ) t h , . . . , ( l , m ) t h

24.6

, the ~ maps

defined

{t}0 T ~

u s e d to d e f i n e of type

places.

(n-m,m)

~ n a c t i o n on h a v i n g 2's in

As in the p r o o f of T h e o r e m

on M (n-re'm) h a ve

m-i I'~ k e r ~l,i i=r

the

the p r o p e r t y

if n-m-j

that

- -i rood p ~

(m-r)

Also k e r 0 T c_ S (n-j'j)± m-i Therefore,

all the c o m p o s i t i o n

But,

factors

by the s e c o n d i s o m o r p h i s m

m-i m-i {~ k e r ~l,i / ~ ker s i=r i=o ~i ,i

of S (n-j'j)

occur

in

ker i=r

~l,i

t h e orem, m-i ( ~ k e r 91,i i=r

+

r-i {] k e r ~i, r ) / i=o

r-I {~ k e r ~l i i=o M(n-m'm)/

Thus,

every

S (n-m'm)

composition m-i = ~ ker i=o ~l,i

r-i ~ ker i=o ~l,i

m-1 ~ ker ~l,i is e i t h e r a f a c t o r of i=r r-i M(n-m'm)/ ~ ker By T h e o r e m ~l,i i=o

f a c t o r of or of

17.13 we have: 24.16

If n-m-j

H --I m o d p ~ ( m - r )

f a c t o r of S (n-m'm) NOW s u p p o s e contains

m-j

or of one of that

to b a s e p.

, then e v e r y

f a c t o r of S (n-j,j)

is a

{S (n-i'i) IO < i < r-l}

fp(n-2j,m-j)

= i.

T h e n m -> j a O

If m > j, then there

is a u n i q u e

and n-2j integer

+ 1 Jl

such that n-2j+ and

1-

(m-j)

O ~ jl- j < m-j

B u t then n-2j

+ 1 contains

+

(jl-j)

rood p£p(m-j)

. jl- j to base p.

H e n c e we may

find i n t e g e r s

s u c h that m = Jo > Jl >''" and Then, or one of

n-

Js > Js+l = j

Jk - Jk+l - -i rood p ~ ( J k - J )

by 24.16

every

f a c t o r of S (n-j'j)

{S (n-i'i) IO < i -< j-l}.

S (n-i'i) for 0 1 and integer

case

To prove

24.18,

The

such

contains

+

let

~

r-i

O)

... p, so

> f

(n,m),

trivial

= i.

prove

D (n-jpj)

the

a

fp(n,m)

n and m, we

that

p,

to b a s e

and

+ fp(n-l,m-l)

~n-I

fp(n-l,m)

m-i

= O

of g e n e r a l


fp(n,m)

of p,

easily

i m p l i e s t h a t pr d i v i d e s n + i, and the a r g u m e n t above p r o v e s t h a t r+l p does n o t d i v i d e n-m+l. T h e r e f o r e , ~ p ( n - m + l ) = r. H e n c e S (n-re'm) is i r r e d u c i b l e Since by

in this

case,

S (n-re'm) % ~ n - i

the B r a n c h i n g

plicity

Theorem,

fp(n-l,m)

that we m a y

has

take

a power

D(n-m'm)+

that

shows

that

j -> i, s i n c e

fp(n-l,m)

is

that n + 1 contains

n contains

m-I

induction,

factor

to b a s e (n-l,j)

there

of S (n-j'j)

a power

rood p ~

that

only

The

rood p ~ ( m ) above

the

fp(n-l,m)

+

= %

> %

= fp(n,j).

(n-l,m)

1 _< i -< j < m such has

D (n-l)

shows

j to b a s e

j-I to b a s e

j to b a s e + %

implies

congruence

if n+l c o n t a i n s

if n c o n t a i n s

if n c o n t a i n s

~n-i

shows

t h a t m is n o t

n -= m - 1

(n-l,j-l)

is an i w i t h

Since fact

Now

+ %

and D ( n - i ' i ) %

This

(m)

p if and o n l y

(n,m)

multi-

of p.

o f p.

p.

with

j with

p if and o n l y

p if a n d

D (n-l)

= S (n-m'm) .

• S (n-re'm-l)

hypothesis.

a power

to b a s e

a unique

shown

contains

m is

= fp(n,la) .

m to b a s e

m to b a s e

%

By

we h a v e

and D (n-m'm')

as S (n-m-l'm)

induction

n ~ m+j-i

+ fp(n-l,m-1)

and n contains

when

m is n o t

O -< j < ra Further,

~n-I

m or m - i

there

23.1~3,

factors

by the

j = m in 2 4 . 1 8

-> i, n c o n t a i n s

of p n o w

same

+ fp(n-l,m-l),

Suppose, t h e r e f o r e , fp(n-l,m-l)

by T h e o r e m

the

p,

p,

and

p. T h e r e f o r e ,

(n-l,m-l)

that

D (n-i'i)

as a f a c t o r w i t h

is a

109

multiplicity fp(n-l,m) + fP (n-l,m-l). (n-j,j)But, since n -= re+j-1 m o d plp(m! In 24.16 shows that e v e r y f a c t o r of S is a f a c t o r of S (n-m'm) particular,

D (n-i'i)

is a f a c t o r of S (n-re'm)

The m u l t i p l i c i t y fp(n-l,m) with

of D (n) as a f a c t o r of S (n-m'm)

+ fp(n-l,m-l),

since s(n-m'm)+

this m u l t i p l i c i t y ,

shows

%-1

by our induction

24.19

This proves

has

when

o u r n e x t m a i n result,

The m u l t i p l i c i t y

is p r o v e d .

is at m o s t

D (n-l)

hypothesis.

that D (n) is n o t a f a c t o r of S (n-m'm)

> fp(n,m).

and so 24.18

as a f a c t o r

Further,

fp(n-l,m)

24.18

+ fp(n-l,m-l)

namely

of D (n) as a f a c t o r of S (n-m'm)

is at m o s t

f (n,m) . P F i n a l l y we p r o v e 24.20 most

If j > i, D (n-j'j)

is a f a c t o r of S (n-m'm)

at

fp (n-2j,m-j) . The w a y we s h o w this is to c o n s i d e r

a modular factor, m-j).

representation

but s(n-m'm)% 24.20

-1

H

then f o l l o w s

as our s u b g r o u p

1

a subgroup

}I of

~n'

and find

D of H s u c h t h at D (n-j'j)+ II has D. as a 3 3 has D as a f a c t o r w i t h m u l t i p l i c i t y f (n-2j, at once.

We s h o u l d

H, so that we

p r i m e 2 is e x c e p t i o n a l e we c o n s i d e r Case

with multiplicity

like to c h o o s e

can a p p l y

induction.

n-2 or

Since

the

first

p is odd.

The o r d i n a r y i r r e d u c i b l e r e p r e s e n t a t i o n s of ~(n-2,2) are given ~(2) i e~ = ~(12) by S ~ 8 ~ and(2~ ~ ~ ~]2 as ~ v a r i e s o v e r p a r t i t i o n s of n-2. Since

p is odd,

the p - m o d u l a r

D' " and

irreducible

D ~ 8 D (2) , D ~ 8 D (12)

D '~ ) are i n e q u i v a l e n t representations

as ~ v a r i e s

and the m u l t i p l i c i t y

of D (n-j-l'j-l) is

8 D (I~)

fp(n-2j,m-j)

when

Now, by the L i t t l e w o o d - R i c h a r d s o n same c o m p o s i t i o n modules

of the

D (n-j-l'j-l)

factors

as

f o r m S ~ 8 S (2) .

8 D (12)

of n-2,

s(n-m'm)%

~(n-2,2)

has

~ S (12) , t o g e t h e r w i t h

In p a r t i c u l a r ,

s(n-j'J)+

partitions

as a f a c t o r of

as a f a c t o r of s ( n - m ' m ) +

On the o t h e r hand,

~(n-2,2)

IIence

are given by

j > i, by i n d u c t i o n .

Rule,

S (n-m-l'm-l)

representations.

~(n-2,2)

over p-regular

S (n-m-l'm-l)

® S (12)

of

the m u l t i p l i c i t y ~(n-2,2)

the

some of

is f p ( n - 2 j , m - j ) ~

has D (n-j-l'j-l)

8 D (I~)

as a f a c t o r w i t h m u l t i p l i c i t y one (since f~(n-2j,O) - i), and for i < j s ( n - i ' i ) + ~ ( n 2,2) does n o t h a v e D (n-j-l'j-~l) 8 D (12) as a f a c t o r (since f p ( n - 2 j , i - j ) D (n-j'j),

has

D(n-j'J)+

~(n-2,2)

l i c i t y one.

= O).

Now,

the f o r m D (n-i'i)

every with

has D (n-j-l'3-1)

f a c t o r of S (n-j'j) , b e s i d e s i < ~, so it follows

8 D (I')

as a f a c t o r

that with multip-

110

The Case

results

2a

of the

two p a r a g r a p h s

prove

24.20

in this

case.

p = 2 and n is even.

s(n-m'm)+ By i n d u c t i o n ,

%-1

equals

has

this

f2(n-l-2j,m-j)

case

Case

2b

the

since

factors

factor

< n,

one,

D (n-j-l'j)

• S (n-re'm-l)

with

It is s i m p l e

multiplicity

to v e r i f y

that

this

with

s(n-j'J)+

~n

" has

D (n-j-l'j)

as a

and

for i < 3, ' s ~ n - i ' i ) + ~ n-i does not As b e f o r e , D ( n - j ' J ) + % - 1 t h e r e f o r e has

as a factor.

as a f a c t o r

.

as S (n-m-l'm)

n is even.

for 2j

multiplicity

D (n-j-l'j)

this

same

contains

In p a r t i c u l a r ,

D (n-j-l'j)

the

+ f2(n-l-2j,m-j-l)

f2(n-2j,m-j),

factor with nave

last

multiplicity

one,

and

24.20

is p r o v e d

in

too. p = 2 and n is odd.

s(n-m'm)+ S (n-m'm-2) f2(n-2j'm-j+l)

~n-2

has

the same

This

contains

w h e n m-j is even, Thus, s ( n - 3 ' 3 ) + ~

city

2,

a factor. D (n-i'i)

for i < j-2, But

with

D(n-j'J)+

every

~n-2

has

The

results

Now

24.17,

D (n-3-I'3-I)

.

by

~n

Remark

last

equals

as a f a c t o r w i t h

~ does

of S (n-3'3~,

the

D (n-j-l'j-l) of the

multiplicity 2f 2(n-2j, multipli-

.

s(n-l'l)+

factor

i < j-2,

with

@ 2 S (n-m-l'm-l)

+ f2 (n-2j,m-j-1) , w h i c h

~ has n - z

and

as S (n-m-2'm)

D (n-j-l'j-l)

+ 2f 2 (n-2j,m-j)

m-j)

factors

n o t have

besides

following

D (n-j'j), has

Theorem

as a f a c t o r w i t h two

paragraphs

D (n-j-l'j-l)

23.7,

the

24.20

form

so

multiplicity

prove

as

2.

in this

final

case.

24.21

24.19

COROLLARY

By

the way,

24.22

respectively, n+l

to b a s e

same

we

theorem

EXAMPLE

24.20

give

as the m u l t i p l i c i t y

conjecture involving

Suppose

p = 3.

n, n+l w r i t t e n 3,

together

If j a l, the m u l t i p l i c i t y

of S (n-m'm) is the of S (n-m-l'm-l) .

of a g e n e r a l

and

that the

The

to b a s e

rows

24.15.

of D (n-j'j)

as a f a c t o r

of D (n-j-l'j-l)

Corollary

removal

Theorem

24.21

of the of the

is

first

a special

case

column.

following

3, and the n u m b e r s

as a f a c t o r

table

record,

contained

in

for O ~ n ~ 13.

0

1

2

3

4

5

6

7

1

2

lO

ll

12

20

21

22

0

0

0

0

0

0

0

0

1

2

i

2

8

9

i0

ll

12

13

100

iO1

102

llO

iii

ll2

0

0

0

0

0

0

1

2

lO

i

2

i0 ii

i0 12

111

Under to b a s e

3.

places

There

(counting

following base

n = 13,

3.

pair

are

l's

from

are

we have

in the

l's

which

(2+l)th,

in the

Another

in the

0,2,10,12

(O+l)th,

the d i a g o n a l )

of m a t r i c e s .

There

labelled

for e x a m p l e ,

column

example:

(O+l) th and

(3+l)th

(5+l)th

13 in the

contains places

integers

and

labelled

i0+i

(2+l)th

are

0 and

of the

2 to

column

i0.

1

1

1

1

1

1

1

1

The 2-part

part

6

of the

partitions

at once. the

8

Simply

rows

and

4

n = 9

2

truncate by

(9)

1

(8,1)

1

1 1

0

decomposition

13 matrix

the m a t r i x

2-part

(9)

e.g.

1

1 ii of

~n

for p = 3 and n ~ 13 can be

columns

1

1

1

lO

1 1

1

1 12

1

1

1

at the

partitions

(8,1)

(7,2)

9

7

corresponding

read off

column

these

labelled

in d i c t i o n a r y (6,3)

5

3 to

matrices n,

and

label

order.

(5,4)

1

(7,2)

1

(6,3)

1

(5,4) F o r p an o d d p r i m e of

%

is g i v e n

24.33

EXAMPLE

the

column

are g i v e n above,

(9)

(9)

1

(8,1)

1

labels

(8,1) 1

1

(5,4) U,l 2 )

(6,1 ~ ) ~,i ~ )

Suppose can be in

24.1

and

most

Applying

as in E x a m p l e

[9] p a g e

52.

decomposition

matrix

24.15.

p = 3 and n = 9. found

of the

24.2

Combined

with

Peel's

Theorem

Alternatively, the

24.1, they

information

gives

U,2) (6,3)

by T h e o r e m s

explicitly

this

and n small,

1

(7,2)

(6,3)

(5,4)

(7,12 )

(6,2,1) (5,22 )

(4,3,2) (42,1)

112

(4,1 s )

!

(3,16 )

1 1

(2,17 )

1

(19 )

1

1 1

A p p l y i n g T h e o r e m 8.15 to the first five rows,

another part of the

d e c o m p o s i t i o n m a t r i x is (5,4)

(19 )

1

(2,17 )

1

(42,1)

1

(22,15 )

1

(23,13 )

1

1

(2~,i)

1 (The rows c o r r e s p o n d i n g to

(19 ) and

1 (2,1 ~) already occur above).

Using T h e o r e m 21.7 we find that the last three columns should be l a b e l l e d (4,3,12),(32,2,1)

and

(9).

Incidenta~y, we do not know how to sort out

e f f i c i e n t l y the column labels once we have taken conjugate partitions as above

(although T h e o r e m A in [9] gives some partial answers).

We have now a c c o u n t e d for 12 of the 16 labelling columns. 23.6(i),

S (5'3'I)

so we have two

those c o r r e s p o n d i n g to

and S (3'22'I)

more

3-regular p a r t i t i o n s

are irreducible,

3-modular irreducibles

(4,22,1)

and

(5,2,12).

by Example

to find, namely

But

X(7, 2) _ X(4,22, 1) + X (4,2,13) on 3-regular classes

(using T h e o r e m 21.7

with

Iv] = [4,2]).

Appealing

to the theory of blocks of defect 1 (or to the N a k a y a m a Conjecture) part of our d e c o m p o s i t i o n m a t r i x is (7,2) (7,2)

1

(4,22,1)

1

(4,2,13 )

(4,22,1) 1 1

By taking conjugate partitions, we get (5,2,12 ) (5,2,12 )

1

(4,3,12 )

1

(2~,i s )

(4,3,12 ) 1 1

Now T h e o r e m 21.7 enables us to complete the d e c o m p o s i t i o n matrix, since we can write every o r d i n a r y c h a r a c t e r w h i c h corresponds to a 3s i n g u l a r p a r t i t i o n in terms of ordinary characters c o r r e s p o n d i n g to 3regular partitions,

on 3-regular classes.

113

When rows are

of known

(p = 2 , n

p = 2

sources

[13] =

24.1

ii),

3,

8 -< n

James

[123) . M a c

James

the

and

[21]

most

difficult

Stockhofe

used

a computer

employing

Theorem

11.6.

= 2,

(p = 2 , n

=

gives

to

matrices

[6](p

Aog~in

cases

applied.

partitions

of

However,

all

the

(n-m-l,m,l)

form

the

[6]).

James

Mac

Aog~in[15]

be

for

decomposition

_< 9),

Stockhofe

< iO)

cannot

matrix

(see

for

(p = 2 , n

(p =

The

Theorem

decomposition

for

Our Kerber

p = 2,

the

12,13),

in

the

iO),

are

Kerber

decomposition

p = dim

2,n

=

12

D (5'4'2'I)

Appendix

Mac

[15] (p = 3 , 1 1 < n < 13 the

find

n =

Aog~in and

and

Peel

,completed

matrices and

are [153

13, dim

and

[14] by

for

p=

for

these

D (7'4'2) ,

5,n_

forms

are

for u , v

inner

t with

n removed,

products,

p-tableau

the

last

since

< fz@i,fr0i

is,

there

n in the

e t for e~

standard

U i is

an a b s o l u t e l y be

a multiple

is a real

constant

c is p o s i t i v e . rith

and ft

for

~-tableaux w h i c h

row, f~

have

let

•

Suppose

n in the

r.thl

then

fr = u + ap Pe + a p + l e p + 1 +

Since

• •.

u,v in U i.

t having

and write the

Since

f o r m on U i m u s t

That

> for all

u in Vi_ 1 and a r > O. fr0i

Vi_ 1 = U 1 @

@l is an ~ ~ n _ l - i s o -

in U i,

on SI~ .

Lemma.

= c< u,v

t p , t p + I,.. .,tq are

for some

row etc.)

on U i by

f o r m is that

by S c h u r ' s

If p -< r < q

that

our n e w b i l i n e a r

standard

row.

e know

r2th

an ~ n _ l - h O r a o m o r p h i s I L ~

Since

is Vi_ I.

therefc

>* = < u0 i, v@ i >

For each denote that

form

~n_l-mOdule,

of the

n in the

SI~

a bilinear < u,v

with

constructed

kernel

Ui'll we

1~torpnism f r o m U i o n t o

those

+ a p + l e-p + 1 +

= apep

tabloid

here

> = c


9.4,

.. . + a r e r

{tr } w i t h

fz,fr

by

a positive

coefficient,

for p -< z < r, we

deduce

and

that

fr 0i = /~ fr We

are

assuming

I~ ~ n _ l - m O d u l e

S Iz,

so

that

Young's

Orthogonal

Form

is c o r r e c t

for the

for x < n,

f r ( x - l , x ) @ i = /~ fr(x-l,x) = /c Here,

t s = tr(X-l,1) , and

statement are the have

of Y o u n g ' s

same

proved

as t h e i r

Step

2

The proof We k n o w

that

the

real

Orthogonal positions

the d e s i r e d fr(x-l,x)

(Plfr + P2fs ) =

in tr).

result = Plfr

of T h e o r e m there

are

numbers

Form

of S t e p

(Plfr + P 2 f s ) @ i

Pl and

P2 are t h o s e

(the p o s i t i o n s Since

of x-i

i, n a m e l y

that

25.3. numbers

in the

and x in t r

0 i is an i s o m o r p h i s m ,

+ P2fs , for x < n.

real

.

al,a2,...,a r with

we

118

fr Theorem {tj}

and p r o v e Case

1

alel + a2e2 + "'" + a r e r

combination

of s t a n d a r d

the c o r r e s p o n d i n g

result

polytabloids

than x-1 in t

= t k.

Then

{t k}

fk = Clel + "'" + Ckek 25.1,

and a p p l y i n g

of p o l y t a b l o i d s

1 shows

fr = a m u l t i p l e in this fr 2

Since

and not in the same

r

4 {tr}.

Therefore,

where

ci = O unless

25.5,

fk(X-l,x)

{t i} ~ {tk}-

is a l i n e a r

combi-

that

of fk + a m u l t i p l e

of fk(X-l,x) .

case, alel + "'" + a r e r

where

a 3 = O unless ,

{t i } ~ {t r }

•

than x-1 in t r or is in the same

as x-l.

t r is s t a n d a r d ,

2 implies

down successive We m a y

{t i} 9 {tj},

{t i} 9 {tr}.

F o r e v e r y x < n, x is h i g h e r

row or c o l u m n

Case

=

Lemma

e i for w h i c h

S i n c e x < n, S t e p

Case

e i with

as x-l.

L e t tr(x-l,x)

Therefore

O.

for fr"

For some x < n, x is l o w e r

Using

>

f o l l o w if we can show that a. = O u n l e s s 3 By i n d u c t i o n , w e m a y a s s u m e that w h e n {tj} ~ {t r} , fj

row or c o l u m n

nation

and a r

25.3 w i l l

~ {tr}.

is a l i n e a r

=

t h a t ~r

it is e a s y

to see that the h y p o t h e s i s

(= tr' w i t h n removed)

has

1,2,...,n-i

of

in o r d e r

columns.

certainly write fr = b l f l + "'" + b r - l f r - i

+ brer

where

br ~ O.

L e t x be the s m a l l e s t i n t e g e r such t h a t b. ~ 0 for some j and 3 m x u ( t r) < m x u ( t j) for some u, if such an i n t e g e r x exists. We aim to produce a contradiction. First,

1 < x < n, s i n c e

tj b e i n g

standard),and

tableaux

t r and tj

mnu(tr)

By the m i n i m a l i t y L e t x be in the

for all u, mlu(tr) = mnu(tj)

of x, m x _ l , u ( t r ) (y,z) p l a c e

of t r.

= mlu(tj)

= ~i + "'" + ~u

z mx_l,u(tj)

= 1 (t r and for all ~-

for all u.

T h e n y > 1 (otherwise,

for all

u, mxu(tr) = mx_l,u,(t r) + 1 _> m x _ l , u ( t j ) + 1 -> m x u ( t j) , c o n t r a d i c t i n g the d e f i n i t i o n of x). S i n c e t has 1 , 2 , . . . , n - i in o r d e r d o w n s u c c e s s i v e r c o l u m ns, x-i is in the (y-l,z) p l a c e of t r. T h e r e f o r e , u s i n g Step i, er(x-l,x)

= -e r

F o r u a y, mxu(tr)

and

fr(x-l,x)

= mx_l,u(tr)

= ~fr

+ 1 a mx_l,u(tj)

+ 1 a mxu(tj).

119

The d e f i n i t i o n

of x t h e r e f o r e

mxu(tr)

shows

that

< m x u ( t j) for some u < y.

B u t m x _ l , u ( t r) = uz for u < y (since tr successive

has

1,2,...,n-i

in o r d e r down

columns),

and the f i r s t row of t. c o n t a i n s at m o s t z n u m b e r s 3 less than or e q u a l to x - i (since m x _ l , l ( t j) ~ m x _ l , l ( t r) = z). Because t 5 is s t a n d a r d , this m e a n s t h a t x m u s t be in the (l,z+l) p l a c e of tj, and x-i is in a c o l u m n of t. no l a t e r 3

than the

zth column. z

z

z+l

IxC t

r

t.

=

=

I

3 x-1

Y

If t k = t j ( x - l , x ) , fj(x-l,x)

then Step

1 gives

= o l f j + o2f k

where

O < o I < i.

Therefore, b l f I + ... + bjf.3 + =

f

=

r

-f

r

- ... - b j ( O l f j + ~2fk ) - ... + bre r

S i n c e b 3, ~ 0 and ~i ~ -i, This m e a n s

mx_l,l(tr),

bj = 0 u n l e s s

{tj}

2 shows

{t i} ~ {tr}.

Step

3

9 {tr}.

This

Calculation

{tl~}

permutation

in the last

of x.

3.16.)

= 1

has norm i).

{t 2} @ {t3} , e 2 is not involved

(4 5) with respect

(i 2) ~-~

that < fi,fi

involved

(1 2~, (2 3),(3

fl,f2,...,f5,

1/2

/3/2

/3/2

-1/2

1/2 /3/2

(2 3)

4)

we have:

/-3/2 -1/2 1

"-i

1/2

/3/2 1/2

(3 4)

(4 5)

1 1/3 2/2/3

It is interesting basis 23.3

is always

2/2/3

/3/2 -1/2

/3/2

-1/2

-1/3

to see that the last element

a multiple

(cf. Example

/-3/2

23.6(iii)

fixed by the Young subgroup

of the vector and f5 above). ~p

{t}KtP t used in definition This

and to within

~ x e s a unique element of S ~ , by Theorem dim H o m ~ n ( M ~ S ~ ) = i).

of the orthonormal

4.13

is because

both are

a scalar multiple (Theorem

4.13 shows

~p that

125

26

REPRESENTATIONS The

general

representation

permutation

character another, Remember

table of

G24

G L d(F)

results

~n

of the theory,

of n o n - s i n g u l a r has

image

o v e r F.

G Ld(F) o v e r F.

Let

in G Ld(F),

Hence

inside

i, 2 , . . . , ~

d over

Although

for W (I) .

the same vector

If g =

the

linear

~e p l a n

of any group,

over

on a d - d i m e n s i o n a l

be a b a s i s

of d i m e n s i o n

G Ld(F).

thereof.

from any r e p r e s e n t a t i o n

is

over a field F.

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

many new representations

acts n a t u r a l l y

There

from a s t u d y

in terms of the g e n e r a l

to any s u b g r o u p

for ea c h n and e a c h p a r t i t i o n

infinitely

g r o u p M24

d x d matrices

of this s e c t i o n w i l l be s t a t e d

G Ld(F)

u s e d p a r t of the

following

a representation

a homomorphic

they a p p l y e q u a l l y w e l l

produce

in the study of m o r e

Frobenius

to find t h a t of the M a t h i e u

that any g r o u p w h i c h

construct,

GROUP

is u s e f u l

For example,

application

(by d e f i n i t i o n )

group,

t h e o r y of

groups.

less o b v i o u s

of the g r o u p

F has

O F THE G E N E R A L L I N E A R

to of

we can

field.

space, (gij)

W (I) say,

is a m a t r i x

then !g

The g e n e r a l

= Z gij ~ • 3 e l e m e n t of W (I) ® W ( 1 ) m a y

E a.. [ i,j__l~o(l,~) I ,

of S~

p#,~,

'Pc then choose

Ac,R c leading For each p r o p e r

as a f a c t o r of

of S O'v X w i t h

at least

a

17.14).

Therefore, the n u m b e r

of M - t a b l o i d s

>- Z a

o

of type

~ = dim S O'M

dim S °

o= ao

I G'n

A t l e a s t one of the i n e q u a l i t i e s our kernel Recall

is too big,

and the s e c o n d

that a O is the m u l t i p l i c i t y

is s t r i c t is s t r i c t

(the first if

is s t r i c t

if

dim S l > I ~ o ( l , ~ ) I ) •

~ = S O,M of S C as a f a c t o r of M c ¢

129

Therefore, a

dim HornC ~n(S~,Mc)

u

= the n u m b e r This

contradiction

26.4

X runs

Let W ~

over

Let the ~ maps p is a p r o p e r

of ~ - t a b l o i d s

completes

DEFINITIONS

w here

= dim Hornc ~ n ( M C ' M C ) of type

~ be the v e c t o r

all n o n - d e c r e a s i n g

space

sequences

act on W p~'~ by acting on each partition

~, by T h e o r e m

13.19.

the proof.

of n,

direct

whose

sum of S p~'~

terms

component

are 1,2,...,d.

separately.

When

let W p = W p'p

We now have 26.5

THEOREM

Let

I be a proper

(i) d i m W 1 equals entries

This

direct

follows

identify

the action

of G Ld(F)

~-maps

with

the

that W 1 is a G Ld(F) From Theorem THEOREM

module,

W ~ ' p has

no " i n d u c i n g tric group

up"

case).

of G Ld(F),

w h i c h we

a series,

power,

on W O'I

since W 1 is the

call

We have

and hence shows

G Ld(F)

that the

and then T h e o r e m

26.5

shows

a Weyl module.

in a

series

Specht

factors

equals

series

takes place

of times %jl occurs

here,

justifies

indeed,

are W e y l modules.

the n u m b e r

of times

for S p~'p in a Weyl

...® W (pn) is given by Young's

This

defined

... 8 W (pn) .

all of w h o s e

in this

the n u m b e r

W (~2)@

ing of the section;

Rule.

series

(Notice

as it did in the c o r r e s p o n d i n g

all the e x a m p l e s

we have p r o v e d

their

analogues. isomorphic holds

with

we have

S 1 occurs

In particular, = W(~I)8

W(P2)8

on a s y m m e t r i c

action

26.3,

the S p e c h t m o d u l e

26.3,

use of suffix n o t a t i o n

The number of times W 1 occurs

wO'~

l-tableaux

S1 X .

An u n p l e a s a n t

26.6

Then

of ~-maps

from T h e o r e m

W O'p w i t h W ( ~ I ) ~

acts on W O'~. commute

of kernels

immediately

sum of the spaces Next,

defined

of n.

of s e m i s t a n d a r d

frora {1,2,...,d}

(ii) W 1 is an i n t e r s e c t i o n Proof:

partition

the n u m b e r

we gave

for that

symme-

at the b e g i n n -

characteristic-free

For example, W ( 1 ) 8 W ( 1 ) 8 W (I) has a G Ld(F) series w i t h factors to W (3) ,W (2'I),W (2'I) (13) ,W ,in o r d e r from the top,and this for every field F.

We now i n v e s t i g a t e

g =

character

~2

values.

• G L d (F) ed

Let

130

If F is a l g e b r a i c a l l y one of the above

closed,

form,

every elements

of G Ld(F)

and so it is s u f f i c i e n t

is c o n j u g a t e

to s p e c i f y

to

the c h a r a c t e r

of g on a W e y l m o d u l e . 26.7

DEFINITION

symmetric

For an i n t e g e r k,

function {k}

of e l , . . . , ~ d .

=

26.8

{O} = 1

EXAMPLES

the kth h o m o g e n e o u s

T h a t is,

~

ig il~ (By c o n v e n t i o n

let {k} d e n o t e

...~ iks d ~il ~ 1 2 " ' ' ~ i k

and

{k} = O

if k < O)

{i} = ~i + ~2 + "''+ed

{2} = el2 + ~22 + . . . + e2d + ele2 + ~i~3 +''+ e d - l a d

{3}

+ 3d + b2

2 + d-l d

+'"

+

d-l d2

+

~i~2~3

+ ...+ ~ d _ 2 ~ d _ l ~ d 26.9

THEOREM

{k} is the c h a r a c t e r

Proof

[ g = ~i -[ + a c o m b i n a t i o n

if 1 s i I s...~ is

eil...ei k

form

of ~'s

with

i k s d, then the c o e f f i c i e n t

.

S i n c e W (k) has

i l . . . i k , the r e s u l t

26.10 COROLLARY w ( I n ) = wO, 1 Now,

of ~ on W (k) .

a basis

of

j < i.

Therefore,

il...i k

consisting

in

il...i k g

of e l e m e n t s

{_~l}...{In } is the c h a r a c t e r of ~ on W ( I I ) ®

recall

of the

follows.

f r o m 6.1 t h a t m =

are i n d e x e d by p r o p e r p a r t i t i o n s ,

(ml~)

is the m a t r i x w h o s e

... @

entries

g i v e n by

[ l l ] [ 1 2 ] . . . [ l n] = Z ml~[ ~] From Theorem 26.11

26.6, we h a v e

{ l l } { 1 2 } . . . { l n} = E m l ~ { ~ } . Since

the D e t e r m i n a n t a l

F o r m gives

the i n v e r s e

of the m a t r i x m,

we h a ve 26.12

THEOREM

If I is a p r o p e r p a r t i t i o n

on the W e y 1 m o d u l e W l is We w r i t e

{I} =

of n, then the c h a r a c t e r

of

l{li-i+~} I .

l{li-i+j} I = the c h a r a c t e r

of g on W I.

Then

immediately 26.13

THEOREM

{l}{~}

is the c h a r a c t e r

The L i t t l e w o o d - R i c h a r d s o n as a l i n e a r

combination

of

{~}'s

Rule

of ~ on W 1 ® W Z.

t e l ls us how to e v a l u a t e

(where i is a p a r t i t i o n

{I}{~}

of r, Z is a

131

is a partition of n-r and 9 is a partition the L i t t l e w o o d - R i c h a r d s o n

Rule follows

of n), since we know that

from Young's

Rule.

It is worth noting that were we to define {k} = where

{~i,~2,...}

Z ~il ~i2... isi I s...si k is countable

~ik

set of indeterminatess

then

{ll}{12}...{l n} = Zl ml~{~} and

{l} = l{li-i+j} I

are equivalent results work identities

definitions

for

of {l},

el,...,ed

for i a partition

in an infinite

in the indeterminates

~l,...,~d

{l} is called a Schur function, is thus isomorphic over partitions to multiply Schur 26.14

Note;

n.

(since our

).

and the algebra of Schur functions [l]'s, where I varies

The L i t t l e w o o d - R i c h a r d s o n

Rule enables

functions.

functions

can be evaluated explicitly by

THEOREM

If ~ is a proper p a r t i t i o n Vl ~2 ~n {~} = Z m Z' ~ ~il~i2"''~i n

In all that follows,

of n indices il,i2,...,i n from {1,2,...}

of n

the above must be

to the algebra generated by the

of various

Schur

field,

depending

Z' denotes

of n, then

the sun over all unordered

(no two equal)

chosen

from {1,2 .... ,d}

on w h e t h e r we wish to define

sets or

{p}in terms of

{~l,e2 ..... ~d } or of {~i,~2 .... }. Proof of T h e o r e m 26.14

(m m')iv =

= (~ mlo X ~, ~ m y X T)

~ mlam , this being an inner product of characters

= (x[ll][12]'''[In],x[Vl][~2]'''[Vn])

of

%"

, by the definition of m.

= dim Hom C ~n (MI,M ~ ) = the number of l-tabloids = the coefficient

considering how this c o e f f i c i e n t Therefore,

of type 9, by T h e o r e m

~ ..~n of ~ii~22. is evaluated,

{ll}...{l n} = Z (m m')

Z'

~i ~2 ~n ~i I ~i 2 "''~i n"

l~ But {~}= Z l (m -I ~l {ll}'''{in} =

Z

l,~,o

in {ll}...{In},

by 26.11, Z , ~.9 1

( m -1)

~I ml~ mvo

iI

~ ~.2. . .

12

~.9 n in

13.19. by

us

132

m

=

Z,

91 ~2 . Vn ~il ei2" " ~in

~U 26.15

k Let s k = i~ Gi

DEFINITION

We can now p r o v e 26.16

THEOREM

Let

Proof

denote

on p.

the

of

s

(ii)

{~} = 7. p IC~)i

~

n

how this

with

cycle

of p in

corresponding

s

Pl

P2

... s

lengths

~n"

XP(OlSoI Sp2

of type

Let XU(p)

to the p a r t i t i o n

...Spn

of t a b l o i d s

(in) w h e r e

row of the tabloid. ~i ~2 ~n of el ~2 ... a n

coefficient

each

in ~ f i x e d cycle

in spl Sp2...

Spn, by c o n s i d e r i n g

is e v a l u a t e d .

spl

,

Sp2

• . .

Spn

=

~ X[~I][~2]'''[~n](P)

= E x [ V l ] [ 9 2 ]'''[~n ] (p) (mu~)-I{u}, v,U = Z XU(P) P This

proves

part

{P},

of

~n'

ZP

Ic- 71

1

and this

×

1

¢p)

Spl

is the s e c o n d

from the d e f i n i t i o n

relations

Sp2

G, then

for all n >- O and all p r o p e r

= 7. P ~

The c e n t r a l i s e r

If G is any group,

1

26.14

of m.

1

X

~

of the c h a r a c t e r

¢P)

Xp

¢P){U}

and 0 is an o r d i n a r y partitions

p of n, e U

X p (p) @(gP 1 )@(gP2).. .0(gPu)

order

IC(p) I and the c h a r a c t e r

~.n and the sum i s

= {l},

character

of

is a

over

all p r o p e r

p-tableaux

with

entries

from

(g~ G)

X U refer

partitions

(pl,__pp2,...,pu) , w h e r e Pl >- P2 a'''.~.Pu > O. If e has d e g r e e d, then @P has d e g r e e equal

standard

~n

of G, w h e r e

@~(g)

as

v2

~ z.2. . . . el n

p a r t of the Theorem.

COROLLARY

group

~i ~il

by T h e o r e m

for the columns

Z ~ "Spn = U,P

26.17

character

7~

(i) of the Theorem.

By the o r t h o g o n a l i t y table

by p.

of p is cont-

,

Therefore

QI'

= ~ X ~ (p) {~}

Pn

x [ 9 l ] [ ~ 2 ] ' ' ' [ ~ n ] (P) = the n u m b e r

= the c o e f f i c i e n t

~n

centraliser

of

(i)

in a single

and sO = i.

Then

= the nunfoer of ~ - t a b l o i d s ained

k -> 1

p be a p e r m u t a t i o n

of the c h a r a c t e r

, evaluated

if

the u s e f u l

p2,...,p n and let C(p) be the value

, as required.

{1,2,...,d}

to the s y m m e t r i c

p of n;

p is w r i t t e n

to the n u m b e r .

of semi-

133

Proof:

There is a homomorphism # from G into G Ld(C). If g e G, let k k k #(g) have eigenvalues ~i' e2'''''~d " Then ~i' e2''''' ed are the k eigenvalues of g , and so @(gk) elk + ... + ~ . The result now follows

from Theorem

26.18

EXA~ZPLES

26.16(ii)

Referring

6 3 , the last of which

and Theorem

to the character

tables

of

50' ~i'

~2

is (13 )

Centralis er order: X(3) I X (2'I) X (13) we have,

26.5(i).

for any ordinary

(2,1)

(3)

6

2

3

i

01 -i

-I1 1

character

0 of any group G, and any g in G,

0 (0) = the trivial character of G 0 (i) = 0 1 )2 + ~@(g2) * 8 (2) g) = Y(O(g) 8(12 (g) = ~(0 i (g) )2 _ x0 (g2)

• )3 + ~0(g2)@(g) = ~(O(g)

8 (3) (g)

+~@(g3)

I 3 + O.@(g2)@(g) O (2'I) (g) = q(O(g))

0(13 ) (g) Note that 0(1)8

= ~(@(g)) 3 _ ~0(g2)0(g) @(i)

= 0(2)

(cf. Young's

Rule)

= (2 d) + d = d(d+l) 2 = (d) = d(d-l) 2 = (d)

deg 0 (12) deg 0 (13) deg e (2'I) deg 0(3)

= (d+l) d(d-l) 3 .d+2. =

Similar

+ 0 (3) , etc.

d, then

des 0 (2)

(The last two degrees Theorem. )

+ ~@(g3).

+ 0(12 )

0(2)8 0( 1 ) = 8(2,1) If @ has degree

- ~ O ( g 3)

( 3

)

are most easily

to the Hook Formula

calculated

for dim S 1

by using the next

we have

26.19

THEOREM

dim W 1 =

~ (d+j-l) (i,~)c[13 K(hook lengths in [I]) .

Proof:

We prove

first that dim W (k) = L,k+d-l. d-i ) if k is a non-negative

integer. The natural

basis

of W (k) consists

of

(k)-tabloids

with entries

and

134

from

{1,2,... ,d}.

sequences

of

There

"bars"

is a I-i

(i)

eg

*l

~-~

and

"stars"

are

Since

{I} =

(*)

l**q*l

1

~ith

between

d-i

bars

this

basis

and

and k stars

I i**'I*

33

4

.k+d-l. ( d-1 ) s u c h

There

correspondence

777

sequences,

8{I i + j - i}l,

8

so this

is the d i m e n s i o n

of W (k)

we ]lave

+ d - l + j - i d

=

-

1

3_

Id(d+l)... (d + I i - 1 + j - i) I = f(d) , say. (I i + j - i)'

Let

I have

h non-zero

an h × h m a t r i x ) . Ii + 12 +

"''+lh I

and

leading

the

result

"how

will

ensure

the n u m e r a t o r

that

1

far r i g h t

taking

1 lengths

~(hook follow

k -> - h + l r and i* is the

(k m e a s u r e s

are

the p o l y n o m i a l

the

determinant

f(d)

has

of

degree

coefficient

I. > k+i t t h e n (d+k) i* d i v i d e s -l f(d) for k < O.

Case

(so w e

that

1 I (I i + j - i)' =

Therefore, When

parts

It is c l e a r

if w e

largest

f(d)

statement

19.5

and

20.i.

[13)

can prove:

integer

i such

for k >- 0 and

of the d i a g o n a l in the

, by in

we

that

(d+k) i*+k

are",

a n d the

of the T h e o r e m

divides

above

will

is correct.)

k -> O. For

i -< i*,

expression

for

the e n t r i e s

d -< d+k

f(d)

_< d+ li- i.

above,

in the ith

we

see

Examining

that,

the

third

determinantal

for i < i*,

(d+k)

divides

row of o u r m a t r i x .

Therefore,

all

(d+k) i* d i v i d e s

f(d). Case

2

k

< O.

Here we (i,j)th

claim

entry

that

for all i,

f(d) and

= det(Mk(d)) for all

where

j _> -k,

~(d)

is

a matrix

whose

is

( I i + d + kj - i + k ) This f(d)),

is

certainly

so assume,

subtract

the

inductively,

jth c o l u m n

the n e w m a t r i x ,

true

for

for k = -i that

of Mk(d)

j _> -k+l,

the

(by o u r

it is

from

the

(i,j)th

true

first for k.

(j+l)th entry

is

expression For

column

all

for j -> -k,

of Mk(d).

In

135

(li+d

Thus,

+ j-i + k) d + k our

_

new m a t r i x

(li

may

+ d + j-l-i d + k

be

taken

+ k)

as M k _ l ( d ) ,

=

li+d

and

the

+ j-i + k - l ) d + k-i result

claimed

is c o r r e c t . + j-i li 0

-Since and

i i + j-i

=

0 1

if if

I i

> 0 for i -< i*

and

li I. + j-i l

j >_ -k,

>_ 0

~(-k)

has

the

form

I I

i

O's -k-i Therefore, whence

the

the n u l l i t y

d e t ( M k(d)) 26.20

= f(d),

rank

i*

l's

and

l's

l h-i*

h+k+l

of ~ ( - k )

of Mk(-k)

is at m o s t

is at l e a s t

(-k-l)

i* + k.

+

Thus

(h-i*

+ i),

(d+k) i*+k

divides

as r e q u i r e d .

EXA~?LES (i)

If I = d i m W (2)" " = d (d+l) 2.' (ii)

If

(k)

then

dim W l =

[I] = X X X , then

d (d+l) ... (d+k-l) k:

the h o o k

graph

is

X X Replacing

the

(i,j)

In p a r t i c u l a r ,

4 3 1 21

node

in

[I] by

j-i,

we h a v e

O

1 2

-i 0 Then

the T h e o r e m

As w i t h S I, the

W 1 .

the H o o k

formula

of s e m i s t a n d a r d

gives

d i m W ~ = d(d+l) (d+2) (d-l)d 4.3.2.1.1. Formula

of T h e o r e m tableaux

for the

26.19

when

dimension

is m u c h

calculating

more

of the

practical

dimensions

Specht than

of W e y l

module

the

count

modules

APPENDIX

THE D E C O M P O S I T I O N M A T R I C E S OF THE S Y M M E T R I C GROUPS ~ n FOR THE PRIMES 2 AND 3 WITH n < 13

We have d e l i b e r a t e l y p r e s e n t e d these d e c o m p o s i t i o n m a t r i c e s w i t h o u t sorting the c h a r a c t e r s

into blocks.

p a t t e r n s w h i c h m i g h t hold in general;

This makes it easier to spot for example,

compare the part of

the d e c o m p o s i t i o n m a t r i x of 013 c o r r e s p o n d i n g to p a r t i t i o n s h a v i n g 3 parts w i t h the d e c o m p o s i t i o n m a t r i x of 510 , and see the remark following C o r o l l a r y 24.21.

137 The

decomp, o s i t i o n

matrices

o f ~ n f,or t h e

,-4

n = 0

*I

(2

(

*I

~

(3) I)

n = 4

I

I 3 "2 3 I

i

)

(i)

H

m ~

n = 3 1 '~2

n = I

I

H

2

,-4

.~

(0)

prime

I

(4) (3 i) ( ~2 ) (212 ) (i ~ )

,-"

n = 2

i

i

(2)

i

(! ~ )

~

~m i I i i I I i

~O zt

= 5

n i 4 5 -6 5

(5) (4,1) ( 3 2) ( 3 ~2 ) (221) (213 ) (i s )

4 i

1 I I i i I i 2 i 2 i I I I

1 1 i 1 1 I i i i

I (7) 6 ( 6,1) 14 ( 5,2) 14 (4,3) 35 ( 421) 15 (512 ) 21 ( 321) 21 (322 ) ~20 (413 ) 35 ( 3 212 ) 14 ( 231) 15 (31 ~ ) 14 (2213 ) 6 (21 s ) ! (I T )

I i -2 i

i I i i i i

i i i

2

I i

i i

I i

I

i i i

i i i

I

i i I i

i

1

i

09

I

n = 6 i (6) 5 (5,1) 9 (4,2) ~'~16 ( 3 2 1 ) I0 ( 412 ) 5 (32 ) i0 (31 ~ ) 5 (2 3 ) 9 (2212 ) 5 (21 ~) i (16 )

v

138

The decomposition,,matrix, o f ~ , , f o F the prime, 2

H

(~) -.1"

oo ['--- ~o

1 7 20 28 64 70 14 21 56 42 35 go

(8) (i e ) (7,1) (21 ~ ) (6,2) (221 ~) (5,3) (2312 ) (521) (3213 ) (431) (3221) (42 ) (2 ~ ) (612 ) (31 s) (422 ) (3212 ) *(322) (513 ) (41 ~) *(4212 ) Block number:

The d e c o m p o s i t i o n

matrix

oo-1-0

u9

L..O .-~

I 1 1 1 1 1 1 1 i 2 1 i i i 2 2 1 2 2 2

1 1 i i i

1 i i

1 1 2 1

i

I ! i 1 2 1

of ~9 for the prime

,.-t

co

,,..o

co

co

eo

C

0 r--I

I 8 27 48 42 105 162 168 28 84 120 42 56 189 216 70

(9) (19 ) (8,1) (217 ) (7,2) (221 s ) (6,3) (2313 ) (5,4) (2~i) (621) (321 ~) (531) (32212 ) (432) (3221) (712 ) (316 ) (421) (323 ) (522 ) (3213 ) *(33 ) (613 ) (41 s ) (5212 ) (4213 ) (4312 ) (4221) *(51 ~ )

1

Block number:

1 2 1 2 1 1 1 2

1 1

1 1

1

1 1

2

I

i 1 1

1 1

1

2

1

! 1

2

1

2 2

1 Z

3

1

2 1

2

1 11

I i I 1

2

1 1

2

139

The decomposit, ion ,matrix of

,--I

i 9 35 75 9O 160 315 288 450 768 42 36 225 252 210 84 35O 567 300 525 126 448

(i0) (9,1)

(8,2)

(I I° ) (21 ° )

(2216)

(7,3) (231 ~) (6,4) (2~I 2) (721) (321 s ) (631) (32213 ) (541) (3231) (532) (32212) *(4321) (52 ) (2 s )

(812)

(31 ~)

(622 ) (321 ~ ) (422) (3222 ) (432 ) (331) (713 ) (416 ) (6212 ) (421 ~) (5312 ) (42212 ) (4212 ) (423 ) (5221) (4313 ) (61 ~) (515 ) *(5213 ) Block

number:

eo

~i0

(o

for the 2 r i m e

oo ~:) O e o _,1- H ~ : D O b rqr--I

co Cq ~

O C; ~

oo (O ¢"-

i i i

i i i i

I i I i I

!

2 1 !

i i i 2 1 !

I

I

i

I !

I I

2 1 1 i 2 ? 2 2 3 2 3 2

1 I 1 1 1 1 1 1

I 1 1 3 3 1 2 2

I I 1 1 1 1 1 1

I i

1 2 I. 1 1

i I

i i i

I 2

i

i I i i 1 2 1 2 1 3

2

140

The d e c o m p o s i t i o n

matrix

H

of ~ll__~for the prime

C) r-'t

_-I-'~

G C'

..1~ C~,I (4D O0 ~D e o O o m

~ - O0 ~D O0 --~...~- ,--I(.0 r.--i

~

i i0 44 ii0 165 132 231 550 693 990 99O 2310 45 33O 385 660 462 120 594 1232 1155 ii00 1320 1188 825 210 924 1540 252

(ii) (I0,i) (9,2) (813) (7,4) (6,5) (821) (731) (641) (632) (542) (5321) (912 ) (521) (722 ) (532 ) (423)

(111 ) (21 e ) (221 ~ ) (231 s ) (2~i 3 ) (2Sl) (3216 ) (3221 ~) (32312 ) (32213 ) (32221) (43212 ) (318 ) (32 ~ ) (321 s) (3312 ) (332)

(813 )

(417 )

(721 z) (421 s) (6312 ) (42213 ) (5412 ) (4231) (6221) (431 ~) (4221) (4322 ) ~(4321) (523 ) (4213 ) (71" ) (516 ) (6213 ) (521 ") (5313 ) (52212 ) *(61 s ) Block number:

~

f-q

co

..~

up

r-t

r-t

~-t

~.~

t--,!c,4

i i i i

i

i

I i

I 2 I

i

I

i I i i i i

i i i

2

i

i

i 3

i

I i ! i

2

2

i I

i

I 2 2

! i i I i

i i

2 2

i i

2 3

i 3

i 2 2 2

3 2 2 4

i i 2

i 2 I

I i i i 2

I i

I i i i i I

2

i I i i

i i

I

i

2 1 1 2 2 1

I

i

1 2 1 2 1 2 1 1 1 2 1 2

2

141 T_he d e c o m p o s i t i o n

,--I 0 ,--I

,--I ,--I

1 ii

(12) (Ii,I)

(3. 12 ) (21 *o )

54

(10,2) (9,3)

(2218 ) (2316 )

(8,4) (7,5) (921) (831) (743.) (651) (732)

(2~I ~) (2Sl 2) (321 ~)

154 275 297 320 891 1408 1156

(3221 s) (32~I)

(642) (543) (632!)

( 2 s ] 2) (329212) (3321) (4323_3 )

5775

(5421)

(43221)

132

(62 )

(26 )

55

(10] 2 )

(31 s )

616 1320 1650

(822 ) (522) (632 )

(3216 ) (3~23 ) (3313 )

462 1SS 945 2376 3080 1485 2079 4158 2970 1925 4455 2640

(43 ) (913 ) (8212 ) (7312 ) (6432 ) (52] 2 ) (7221) (5321) (4231) (693 )

3564

7700 462 2100 1728

(3 ~ ) (41 s ) (4216 ) (4221 ~) (42312 ) (42 ~ ) (431 s ) (43212 ) (4322) (421 ~ ) (5322 ) (42212 ) 2(4222) (81 ~ ) (517 ) (6313 ) (52213 )

(5413 )

Block

number:

for the p r i m e

_n~- ¢-4 ~o eo ~

C> 0 o.~ t - eOuO

co 0 ~

c o (.o oo ..~('40

,,.o ¢~ ,--I o-~ ~t['-

o.~ eo ~O

0 r--I

oo

09 o~oo

CO L'~ £0 r'--

~ _~~D uO

CO c.O uo

o'~

t'--

co ~o CO

i I I I i I i i i i i I 1 1 i 1 I 31111

I

1 Iii 31 Iii 311111

i i i

I i ! iii I

I

i 552312 I I iii 2 i 2 I 2111 i 2 I 1211 32211 421322 421322 3 i i 321211 2512 i 2411 321211 351312 4 2 i 22111 622322

(5231)

(62212 ) (531 ~) ~(53212 ) (71 s ) (61 G ) ~(621 ~ ) (7213 ) (5215 )

of ~ 1 2

_~- 0 _n~ 0 rq

(32213 )

3.925 2673 211.2 5632

330 3696 3520

matrix

i

112

i i

I

i ii I

I I I i i

I Iii i i I ii I ii Ii

2 i

i i i I

2 2

622221 862423 221211 4 2 2 2 2 1

i i i 122

2 i

i i

I i I i i 1 2 1 2 1 1 1 2 3 1

2

142

The

decomposition

matrix

of~13

for

{'4 t',l

e-d

1 (13) (1'3) 12 (12 ,I) (21' I ) 65 (11,2) (2219 ) 208 (10,3) (2317 ) k29 (9,4) (2~i s ) 672 (8,5) (261 ~ ) ~29 (7,8) (261) ~29 (1021) (321 s ) 1385 (931) (32216 ) 257~ (841) (32~I ~) 2860 (751) (32"12) 3k32 (832) (3221 s ) 6006 (742) (322213 ) 51~8 (652) (322~i) 8~35 (643) (33212 ) 12012 (7321) (4321~) 17160 (6421) (432212 ) 15015 (5431) (43221) 66 (1112) (31'°) 1287 (621) (325) 938 (922) (3217) 3575 (73z) (3~i~) 3~32 (523) (3622) 257~ (5~") (3"1) 220 (i01~) (419) 1~30 (9212) (~217) ~212 (8312) (42ZLS) 686~ (7412) (~2s13) 5720 ¢651") (42~i) 38~0 (8221) (4316) 8580 (5221) (4323) ii~0 (6321) (4321~) 3q32 (~31) (~3~) 4004 (72~) (4215) 12012 (6322) (42213) 12870 (5~22) (42221) 11583 (5322) (~231z) 8580 *(~,= 32) ~95 (91~) (51a) 3003 (8213) (52!s) 7800 (7313) (52~i~) 10296 (641~) (52~1z) 5005 (5~i~) (52") 7371 (72212) (5316) 20592 (6321~) (5321~) 21~50 (5~21~) (53221) 16016 ~(53212 ) 9009 (62~i) (5~i~) 729 (81~) (617) ~290 (721~) (621~) 9360 (631~) (62z1~) 92~ ~(71~) Block numbe.~:

the

prime

,"-te'~

eO~

1 1 1

1

1

1

1 1

1

1

1 1 3 4 2

1 1 1

1

1

1 1 1

1 1 1 1 1 1 1 1 1

2

1

4

i

1

i

i

i i

I I i

3 I

3

i

1

I i 1

1

I

1

1

I 1 1 I I I i I

i

I

3

2

1

I i i

2

3 3 2 2 2 2 2

I 1 3 3

2

3 2 2

2

2 i

I

I i

I i 1 i

i i i

i 1 1 1 1

I

I i

1 2 1 1

1 i

i i i

i 1

1 2

3 1 3 1 2 2 2 i I 1 2 1 1

1 2 2 i

1 1 i

i I

i

i 2

i 1 2 1 ! 3

I 2

I

3

i

~-

i

I I 2 1 3 2 1 3 1 1 1 1 3

2 1

~ ~ 2

i I 1 I

i 2 I

2

2 1 1

2 1 i

I

6

~

i

i 2

1 i 1 I

3

6

1

i

2 3

2 ? 3 2 2

12 8 7

i 1 1

I

i I

i 2

6 5 u, 3 5 8 8 3 7

I

1 2

~

i i I

2 2 1

I

i

2 2

1

7 2 1 2

I

i

3 4

8

i

I

3

4

2 1 1

2

2

i

i

1 2 1 2 1 2 1 1 1 I I 2 1 2 1 2 2 1

2

143

The

decomposition

matrices

,--{

~.

of

for the

3

nrime

,-J

,-4,--{

-2 n

el

:

0

n

(~)

1

:

n = 2

1

(I)

*i

I

1

r-4

t-4

t-JeO

,-~ O O

t'-{ ..'~ r H

~D

cO

¢w

.:t C O

Cw

UD

COC'~

n = 3

C'~

n = 4

1 (3) *2 ( 2 , 1 ) i (13 )

1

i i

1 1 1 1

1 3 *2

n -- 5

(4) (3,1) (2 ~ ) (212 ) (! ~ )

3 i

I 4

1 1 I

5 "6

i i i

4 1

r-~ = r

O%, ~

[O

(5)

i

(4,1)

I 1 1

(3,2) (312 ) (2~ i)

1 1

i 1

~'-UD

(6) (5,1)

9

(4,2)

5

(3 z )

I0 "16

(41 = ) (321) (2212 ) (2 3 ) (313 ) (21 ~ ) (16 )

5 1

u~, = ~

~O

=t CO

O9('O

n = 7

1 5

5 i0

1

(213 ) (15 )

=t OD

n = 6

9

:teO

.n~

1 1 1 1 1 1 1 1

1 1 1 1 1 1

1

1 1 1 1 1 1

1 8 14 14 15 35 21 ?I 35 "90 14 15 14 6 1

(7) (6,1) (5,2) (4,3) (512 ) (421)

1 ! 1

1 1 1

1

1 1

1

(413 ) (231) (31 ~ ) (2213 ) (91 s ) (i ~ )

1

1

(322 )

(321 ~ )

1 1

(321) 1

1 1

1 1

1

1

1 1 1

1 1

1

144 The

decomnosition

i 7 ?n 28 1~ 21 64 7n 58 ~42 *90 5E ?n 35 14 35 64 98 21 2~ 7 1 Block

(8) (7~I) (6,2) (5,3) (42 ) (612 ) (591) (431) (4? 2 ) (322) (4212 ) (32] 2 ) (3221) (513 ) (2 ~ ) (41 ~) (3213 ) (2312 ) (31 s ) (221 ~ ) (2] 6 ) (18 )

number:

matrix

~8

o{

fgr

the

prime

I 1 i 1 ] l

1 l

I

l 1 I I

I I ! 1

],

I

I ]

i

i i !

i 1

i

! I 1

1 I I ! ? o ! ? 3 1 1 2 3 4 1 2

3

145 The d e c o m p o s i t i o n

matrix

r-~ t-- t-('w

~4

1

(9)

8

(8,1)

27 48 42

(7,9) (6,3) (5,4)

28 le5 162

8U 120 168

169

(32212 )

*42

(3 3 )

56 84

(613 ) (323 )

"70 ]89

(51 ~ ) (4913 )

]20

(3213 )

49

(9~I) (4! s )

(321 ~ )

t--- Lo cO

r-~ o % C-40~

~- o~ r-4 C4 (~4 CO _~- ~D

,-4

r-4

r-i~

¢~,r-4

,-4~

¢w~

1 1 1 ! 1

I ! i

(4921) (3291)

~ (.0

1 I

(592 ) (432)

916 168

3

! I .]

ii Ii Iiii

(4312 )

for the p r i m e

,-4 ub~ ¢~ co

-I-~

1

(5912 )

Block

CO

I I I

916

1

r-4 ,-4 .zt

(712 ) (691) (531) (421)

189

56 ln5 48 98 27 8

C'4

of ~ 9

].

! 1 ]

1 1 iii ! I I

I

1

i !

1 1

1 1 1

1 i I

I

] I 1

1 1

1

1 !

1

! 1

! ]

1 1

I I I

i I ! I i i

] i

I

(23! ~ )

(316 )

! i

(2213 )

(217 ) (19 ) number:

1 1 2 1 ! 1 ! 3 1 1 ! 4 ~ 2 1 5

146 The d e c o m p o s i t i o n

matrix

o f (~10 f o #

the

prime, 3

o~

I 9 35 75 90 42 36 160 315 288 225 450 252 210 350 567 300 525 *768 252 567 450

(I0) (9,1) (8,2) (7,3) (6,~) (52 ) (812 ) (721) (631) (541) (622 ) (532) (422) (432 ) (8212 ) (5312 ) (4212 ) (5221) (4321) (3222 ) (42212 ) (32212 )

84

(713 )

i i 1

1 i

i I

I i i i i

i i i

1

(I l° )

Block

1 1 i i i i

I

I ! i i 1

1

1 1

1

1

1

i 1

i

1 1 1

i 1

i

1 1

i

1 1 1 1

i

1

i

1 1 i

1 i

1 1

1

1 1

1 1 1 1

i

(31 ~ )

numbers:

1

i I

I

(61 ~ ) (5213 ) (4313 )

(2216 ) (218 )

1

i

126 *448 525

36 35 9

i i 1

I

I

160 75

i

1 I

i

9~ 84

I i I

i

i i

(423 )

(2 s ) (SI s ) (421 ~ ) (321 ~ ) (32213 ) (2~i 2 ) (416 ) (321 s ) (231 ~ )

i i

!

(331)

(3231)

i i

I

300

42 126 350 225 315

i i

i

210

288

1 1 1

1 1

1 1

1

I i 1 ! 2 1 1 2 1 3 1 3 3 2 3 2 1 1 4 1 1 1 3 5 2

147

The

,-~C

1 i0 qW ii0 165 132 q5 231 550 693 330 385 990 990 660 462 594 1232 1155 ii00 2310 1320 "1188 1320 1540 2310 990 120 825 ~62 210 92~ 1540 825 660 1155 330 *252 924 ii00 1232 990 693 132 210 594 385 550 165

120 231 ii0 45 4~ I0 1

m a,,t r i x

decomposition

.

. ~

tn

o~ii

¢-J co

,for t h e

m~ tn

,,~

3

prime

:t c~

m

,,, o9

¢o

~'~

co

,,~

(ii) 1 (i0,i) 1 (9,2) 1 1 (8,3) 1 1 (7,4) 1 1 (6,5) 1 1 (912 ) 1 (821) 2 1 1 (731) 1 1 1 1 (641) 1 (521) 1 1 (722 ) 1 1 1 1 (632) 1 1 1 (542) 1 1 1 (532 ) 1 1 1 1 (423) 1 1 1 1 (7212 ) 1 (6312 ) 1 1 1 1 (5412 ) 1 1 1 1 (6221) 1 1 1 1 1 1 1 1 ! 1 1 1 1 (5321) 2 1 (4221) 1 1 2 1 1 1 1 (4321) 1 1 (4322 ) 2 1 1 1 1 1 1 (52212 ) 1 1 1 1 2 1 1 1 1 1 1 (43212 ) 1 1 (3z221) 1 1 (813 ) (52 ~) 2 1 (3s2) 1 1 I (71 ~ ) 1 (621 s ) 1 1 I I 1 (5313 ) 1 (4213 ) 2 1 1 1 (3312 ) 1 1 1 1 1 (~231) ! 1 1 (32 ~ ) 1 (61 s ) 1 1 (521 ~ ) 1 1 1 1 1 (431 ~) 1 (42213 ) 1 1 1 1 1 1 1 (32213 ) 1 (32312 ) (2Sl) 1 (5i t ) i i (421 s ) (321 s ) 1 1 1 1 ( 3 2 2 1 W) 1 1 1 1 1 (2"13 ) (417 ) I (3216 ) 2 1 1 (231 s ) 1 1 (31') (2217 ) 1 (219 ) 1 (l'l) 1

Block numbers:

1 2 2 1 2 2 3 1 1 3 1 2 3 3 1 2 4 1 1 2 1 2 4 1 2 2 3

148

£L9g

EIOl lEhl

0')

T68 9~61 h9££

~ ~

L6g h9SE Sh6

ii-i

~

8~LI

,r,,l ,-I 8~LI x

9£61

~

~

~

~

~ ~

4-

Ogl

~

~

~

~

~,~

,-4

,~

,-I

,-I e,l

e,J

~o

e,,

2.

r-~N

-,4

8

,¢I

I68 0~I "[

L6g

r'l

,-4

'-~ ,-4

,-4

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References

i.

R.W. CARTER and G. LUSZTIG,

On the modular representations

the general linear and symmetric groups, Math Z. 136 2.

C.W. CURTIS and I. REINER r

groups and associative algebras,"

"Representation Interscience

(1974),

of

193-242.

theor V of finite

Publishers, New York,

1962. 3.

G.H. HARDY and E.M. WRIGHT,

"An introduction to the theory of

numbers," Oxford Univ. Press, Oxford, 4.

1960.

J.S. FRAME, G. de B. ROBINSON and R.M. THRALL,

of the symmetric group, Canad. J. Math. 5.

H. GARNIR,

sym~triques,

6 (1954),

Th6orie de la representation

The hook graphs

316-324.

lineaire des groupes

M~moires de l a Soc. Royale des Sc. de Linger

(4), I0

(1950). 6.

G.D. JAMES, Representations

of the symmetric groups over the

field of order 2, J. Algebra 38 (1976), 280-308. 7.

G.D. JAMES, The irreducible representations

of the symmetric

groups, Bull. London Math. Soc. 8 (1976), 2~9-~32. 8.

G.D. ~TkMES, On the decomposition matrices of the symmetric groups

I, J. Algebra 43 9.

G.D. JAMES, On the decomposition matrices of the symmetric groups

II, J. Algebra iO.

(1976) , 42-44.

43 (1976), 45-54.

G.D..TAMES,

A characteristic-free

theory of ~n' T. Algebra 46 (1977) ii.

G.D. TAMES, On a conjecture of Carter concerning irreducible

Specht modules, 12.

Math. Proc. Camb. Phil. Soc. 83 (1978),

G.D. JAMES, A n o t e

A. KERBER,

14.

to appear.

"Representations

Notes in Mathematics,

11-17.

on the decomposition matrices of 512 and ~ 3

for the prime 3, J. Al~ebra, 13.

approach to the representation

430-450.

of permutation groups I," Lecture

no. 240, Springer-Verlag.

A. KERBER and M.H. PEEL, On the decomposition numbers of symmetric

and alternating groups, Mitt. Math. Sem. Univ. Giessen 91 (1971), 45-81. 15.

E. MAC AOGAIN, Decomposition matrices of symmetric and alternating

groups, Trinity College Dublin Research Notes, TCD 1976-10. 16.

J. McCONNELL,

Note on multiplication

theorems

for Schur functions

"Combinatoire et reDresentation du groupe sym4trigue,

Strasbourg 1976,"

154 Proceedings 579, 17.

1976, Ed. by D. Foata,

Springer-Verlag,

N. MEIER and J. TAPPE, Ein neuer Beweis der Nakayama-Vermutung

8 (1976),

Symmetrischer

Gruppen,

Bull. London Math.

Soc.

34-37.

M.H. PEEL, Hook representations

Math. J. 12 (1971), 19.

no.

252-257.

~ber die Blockstruktur

18.

Lecture Notes in Mathematics,

of s~nnmetric groups,

Glasgow

136-149.

M.H. PEEL, Specht modules

and the symmetric

groups,

J. Algebra

36 (1975), 88-97. 20.

M.H. PEEL, Modular

renresentations

of the symmetric groups,

Univ. of Calqar Z Researcb Paper no. 292, 21.

D. STOCKHOFE,

Die Zerlegunqsmatrizen

S12 und S13 zur primzahl 22.

W. SPECHT,

Gruppe, 23. group,

2, Communica%ions

Die irreduziblen

Math Z. 39 (1935),

R.M. THRALL,

1975.

Young's

Duke J. Math.

der Symmetrischen in Al~ebra,

Darstellunqen

Gruppen

to appear.

der Symmetrischen

696-711. seml-normal

8 (1941),

611-624.

representation

of the symmetric

Index

Basic

combinatorial

basis,

orthonormal

lemma

, standard

-

bilinear

form,

-

-

9

29, invariant

r non-singular

binomial

coefficients

block Branching

Theorem

Hook diagram

73,

77,

89

80,

92,

98

115

-

69

-

formula

1

-

graph

73

2

-

, skew-

73

77,

135

87 84,

85,

93

Involve

13

34,

62,

79

irreducible

representation

16

39 S 40 e 71 Carter

Conjecture

97,

102,

character column

23,

stabilizer

composition

factor

16,

-

-

cycle

diagram

9 9,

type

Littlewood-Richardson

decomposition

matrix

Maschke's

42,

43

8

- , conjugate

9

t hook

- , r-power dictionary

80 r 92,

98

95,

97

order

9

order

Murnaghan-Nakayama

vector

80p

Conjecture

Order, -

-

space

power

9 8 i0

irreducible representation

Orthogonal

16

Form

orthonormal

3

~-maps p-power

27

group

125

graph

18

Gram matrix algebra

102

on t a b l o i d s

2

126

relations linear

85,

dictionary

ordinary

-

pair

group

85

, dominance

114

basis

115

67 diagram

p-regular

general

79

Rule

8

dual module

Garnir

1

Theorem

25

I13 t 136

diagram

Exterior

52

74

98 t iii,

-

Rule

62 r 130

Nakayama Form

dominance

104

6

Determinantal

-

89,

51

i01

partition

Specht module

i04 r ii0

, trivial

conjugate -

42,

-

79 I0

60, -

105

3 16,

41

-

95,

partition

class

s, p a i r

"

w proper

permutation

54 5

, 2-part

-

36 36

of partitions

~artition -

97

94 t 95 r 97, of

106 54 54 5

156

permutation

module

polytabloid

standard

29

13

Submodule

Theorem

, standard

~9

symmetric

group

exterior

126

, symmetric

126

-

power, -

13

-

15 5

power

126

Tableau Row

stabilizer

i0

-

9

, standard

29

tabloid Schur

function

semistandard -

-

131

homomorphism

tableau t reverse

signed

46

transposition type

column

sum

13

skew-hook

73 13

-

-

29 5

of

tableau

44

of

sequence

54

5

Specht -

lO

, standard

45 102

signature

-

module dimension

30,

76

Weyl -

module -

129

dimension

Young's

natural

179,

representation 114

52 w 77 -

-

Specht

, irreducible series

stabilizer

89, 65,

104 69 i0

133

-

Orthogonal

-

Rule

Youn~

subgroup

Form

114 51,

69 13