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unpublished.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,

<

linear

, >

<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

way

=



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,


with

the

tableau

t is g i v e n

by

e t = {t}K t The spanned

Specht

module

A polytabloid, just

the

tabloid

it m u s t

{t}.

± 1

(If v e M ~, t h e n

the

tabloid

4.4

S ~ for the p a r t i t i o n

~ is the

submodule

of M ~

by p o l y t a b l o i d s .

{t}

EXAMPLE

All

be noted, the

v is a l i n e a r

is i n v o l v e d If t = 2 5 1

depends

tabloids

combination

in v if its then

Kt =

on the

involved

tableau

in e t have

of t a b l o i d s ;

coefficient (1-(23))

t, n o t coefficient

we

say

that

is n o n - z e r o . )

(1-(45)).

34 (We a l w a y s

denote

et = 2 5 1 34 The

practical

the -

identity 351 24

way

-

permutation 241 35

of w r i t i n g

down

+

et,

by

i~.

Also

341 25 given

t, is to p e r m u t e

the

14

numbers

in the

nature and

of the

then

draw

Since 4.5

columns

lines

~t ~ = ~
It we w i s h

domain,

.

When

M U is

Specht are

immediate.

U =

(in),

S U is the We 4.6

if

When

alternating use

LEMMA

Let

a n d t* is

I = ~ then

Proof:

Let

select

signed

column

stabilizer

then

of t* b e l o n g Also,

, then

by c o n s t r u c t i o n .

4.7

and

Thus,

multiple Now

let

< { t l } , { t 2}

cases

FGn-module. of

When

~n'

and

3.7

to p r o v e

Suppose

{t*}< t ~ O.

that Then

t is a ~ ~ £ ~,

and

row of t*.

Then

- { t * ( a b)} = O. of t, o t h e r w i s e

~i,...,~< on

1 and

for the

(a,b)

and

we

could

subgroup

of the

obtain

+ "''+~<)"

{t*}
O,

contradicting

for e v e r y

columns

of t,

is one

in this

and L e m m a

of the

case,

our

i, the n u m b e r s

tabloids

{t*}

= {t}~

hypothesis. in the

3.7 g i v e s

ith

row

I ~ U.

involved

in

for some

permutation

{t}

t is

a ~ - t a b l e a u r then

of e t.

u is a l i n e a r of e t

of the

~ + sgn~).

of n.

colunm

It u is an e l e m e n t

u__~
{t*}

{t*}< t = {t}z

COROLLARY

same

(l-(a b ) ) ( ~ i

to d i f f e r e n t

of

for the

special

representation

in the same

= {t*}

t consisting

that

trivial

most

(= ± et) ~

in the

that

that

results Two

have

is a r b i t r a r y .

independent

many

Lemma

that

that we must

surprising

are

shall.write

an i n t e g r a l

field

field.

(i.e.

representatives

follow

the

regular

two n u m b e r s

now proved

if I = U

z in Ct,

be

of


We h a v e

of

F, we

over

11.3

ground

that

combinatorial

(l-(a b))

coset

is

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

a and b be

a and b c a n n o t

It w o u l d

to the

{t*}< t = ±{t}< t

field

work

its d i m e n s i o n )

a ~ - t a b l e a u F and

{t*}

the

representation

I and

ground

(n) , S U = M U = the

the b a s i c

one p o l y t a b l o i d .

it is h a r d l y

independent

U =

any

modules

remarkable

sig-

t h a t way,

tableau.

4.8 a n d L e m m a

module,

M U is i s o m o r p h i c

now

tableau

also

by

then

the

obtained

so

to the

in T h e o r e m

is m o r e

are

= eta,

generated

(for i n s t a n c e ,

What

module

et~

attaching

tableau

of e a c h

for S p e c h t

a permutation

field.

rows

attention

results

ways,

to e a c h

F is u n s p e c i f i e d ,

its p r o p e r t i e s base

modulef

to d r a w

Many

the

we have

and it is o n l y

a field. Since

permutation

between

S U is a c ~ c l i c

and S~

of t in all p o s s i b l e

relevant

, by

of u - t a b l o i d s

{t*}

and

{t*}K t is

the Lemma.

< , > be > = 1 if

combination

the u n i q u e {t I} = {t2},

bilinear O

if

form on M ~ for w h i c h {t I}

~ {t 2}

a

15

Clearly,

this

is

f o r m on ~,ip, w h a t e v e r inner

product We

F o r u,

shall v E M ~,

a symmetric, the

field.

(cf. E x a m p l e often

use =

following

~ <(sgn ~£C t

=

[

field

is ~,

then

the

bilinear

form

is an

trick:

~)u~,

v>



~eC t tile f o r m is

(since

If the

non-singular

2.5).

the



~n-invariant,

~

-invariant.) n

The 4.8

crucial

Proof:

e t ~ U.

That

SP~

that

using

our bilinear (James

u e U a n d t is o f e t.

t so t h a t

and

n

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

is,

0 =
T~EOREM

u a n d t, {t}>

S P / ( S p n S p±)

is

zero

S p n S pi

itself,

o r is c o n t a i n e d

in S p n S p±.

follow

except

at o n c e , remains

a basis

see

later

d i m ( S P / S p n S p±) But

Theorem,

the

-+ i.

Therefore,

how

is the

coefficients the

Using have

Further

submodule

when

we

o f S p is e i t h e r

1.5,

all p a r t s

still

extend

of the G r a m m a t r i x has

entries

of tabloids

involved

so S p n S ~i d o e s n o t ) is a l w a y s

of

to p r o v e

of

the

S~

the

that

field.

e i is a p o l y t a b l o i d . By T h e o r e m respect

f r o m the p r i m e

of the Grara m a t r i x

S P / ( S p n S pi

turn

submodule

with

rank

show

irreducible.

rank

F.

shall

then

t

maximal

way.)

we extend

We

or absolutely

to do tllis in a s p e c i a l

and

Remark

4.7,

et>

for S p wilere e a c h

subfield, Since

is n o n - z e r o ,

for all u a n d

=
tha~t w e

the p r i m e

it is a b s o l u t e l y

any

irreducible

el,...,e k

the G r a m m a t r i x

since

multiple

by C o r o l l a r y

is s e l f - d u a l .

Theorem

Choose

Then

is the u n i q u e

By t h e S u b m o d u l e

that

of I,iPf

±lave U _~ S p.

then

{t}

Proof:

S ~ / ( S p n S p±)

this

b y et, w e

u< t = O,

=
is n o n - z e r o t t h e n

ant/ S P / ( S p n S pi)

basis.

If U is a sub[lodule

a p-tableau.

can c h o o s e

Since

(We s h a l l

f o r m is

[7]).

u< t = a m u l t i p l e

for e v e r y

4.9



THEOREm4

If,

if t h i s

=

U m S ~ o r U c S p±.

Suppose

If w e

[ ~eC t

result

Tile S U B M O D U L E

then either

=

is the

same

increase

to this

subfield

in a p o l y t a b l o i d

1.6,

are

over

of F, all

F as o v e r

in d i m e n s i o n

irreducible,

if

it f o l l o w s

irreducible. that

all

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

up as S P / ( S p n S p±) ; the T h e o r e m

means

representations

that we

can work

of

~n

over ~ or

16

the

field

where

of

char

from

the

the

more

more

4.10 Ker

p

elements.

F = O,

subtle

general LE~

by

Remark

I ~

Ker

Proof:

~.

in go

of

section

If

I = Ft

immediately

the

on

completing

this

section

Ii.

The

to

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

0 c S i± b y

later

be

Suppose

t is

~ et 0 = =

Lemma

the

the

iO

0 to

follows

impatient

M 1 into

of

case

also

reader

sections

fro~

restriction

COROLLARY

Hom F~

(SltM~) i

in

I ~ If

p,

I ~

if

and

ii.

M ~ and

S l is

for

Sl

multipli-

I = ~,

If

@ $ S I.

and

e t % Ker

~-tabloids)K

then

G is

I = ~I

Ker

11.3

The

13.17).

0,

~t of

F = O t and

~.

(cf.

Since

{t}Q

combination

and

since

ways

a i-tableau.

char

then

Theorem,

several

{t}< t 0 =

(a l i n e a r

4.6,

4.11

Submodule

improved

that

O

By

can

0 is

concentrate

remainder

a constant.

will

Lemma

now

the

approach

result If

0 t then

cation

We

although

then

e t @ is

t.

a multiple

a non-zero

of

element

e t.

of

0 is m u l t i p l i c a t i o n

by

a

n constant. Proof:

When

matrix

with

field

of

F = Q,

respect

Any

4.12

all

Proof: so

the If

I = ~.

and

a basis

defined

~

ordinary ~

S

inner

of

on

it be

are

product.

S l therefore

The

rank

equals

of

dim

the

= O

S l can

zero

on

irreducible

and

S l for

be

Now

extended

apply

Lemma.

OF

~n ) .

REPRESENTATIONS

and

absolutely

representations

to

the

irreducible of

M ~ is

completely

be

The

I and

~n"

, then I ~ ~ by Corollary 4.11. Similarly, n S ~l± _ - O, t h e T h e o r e m follows from Theorem

S Ql

any

M 1 = S l @ S l± .

therefore S l±.

IRREDUCIBLE self-dual

Since

Gram

Thus

ORDINARY

over

SQ

O.

letting

(THE

an

F = O r S l n S II

char

M l by

modules

~ive

to

homomorphism on

THEOREM

Specht

, > is

characteristic when

defined

<

p %

l

4.9

2.4. Since

also 4.13

THEOREM

(once)

and

Some so we

reducible

when

char

F = O,

Corollary

4.11

gives If

some

of

authors

explain

how

char

F = O t the

{S 1

II ~ ~}

prefer to

to w o r k

find

composition

(possibly inside

a right

ideal

the of

factors

with

group the

of

M p are

S~

repeats). algebra

group

of

algebra

~n'

and

of n

corresponding Given

to

the

Specht

a ~-tableau @:

Pt

~ +

t,

module. let

{t}~

Pt

= [ ~ oER t

, so

(n ~ ~ n ) .

that

pt e F ~ n ,

and

let

17 This is clearly

a well-defined

F ~n isomorphism

ideal Pt F ~n onto M ~ (It is well-defined, <=> {t}n = {t}.) isomorphism

Restricting

from Pt Kt F ~ n

result can be interpreted the Specht module S p depends depends

since Pt ~ = Pt <=> ~

onto S ~.

Using this isomorphism,

for two reasons.

only on the partition

~, whereas

~-tableau

t.

First,

{t}; this greatly simplifies

examples,

the right ideal Pt Kt F ~ n

manipulations

some examples

every

~Je prefer

Perhaps more important

as will be seen in the next section,

develop~l~nt to work through points.

an

the Specht module

in place of Pt' which is a long sum of group elements, object

Rt

@ to the right ideal Pt
in terms of the group algebra.

approach

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

from the right

we have

is that a single

with particular

where we pause

in the

i l l u s t r a t i n g many salient

18

5.

EXAMPLES

5.1 EXAMJ?LE r o w of t h e

Reverting tabloids S (n-l'l)=

(5 - i ) F ~

S (n-l'l)±= Clearly, the

to the n o t a t i o n

in M (n-l'l)

Sp([

S (n-l'l)z

Submodule

S (12)

5.2

EXAg~?LE

row.

To

(without by

i to p o i n t

a I + ...+

a n = O}

that when

on

S (n-l'l)± M (3'2) pair

j, w e h a v e

-- S (n)

not

A T i

allow

By i d e n t i f y i n g

constructed

351

241

24

an e d g e

i~ w i t h

an i s o m o r p h i c

+

~___~s

"quadrilateral

for t h e

with

34

25

alternate

=

1 3 5 24

~_i1

i

make

S (n-l'l)

the

is

set of

the e d g e copy

= n-l.

u p its

to b e

second graphs

"weighted" joining

point

o f M (3'2).

corresponds

edges weighted

Then e t l , . . . , e t 5

1 2 5 34

For

to

t

5-

-Z

± l" is a g e n e r -

1 g

5"e

12 35

4

12 45

t -

!

41.3

3

1 3 4 25

3

correspond to

4

4

n.

Specht module S (3'2)

tl,t2,t3,t4,t5

respectively.

~.

n > 2

n,

(3,2)-tabloid which

o f ~(3,2) , c o n s i d e r we

divide

a n d d i m S (n-l'l)

in d e t a i l .

where

F divides

~.

34

Let

By

@ S (n-l'l)±

of numbers

picture

5 points,

coefficient.

251

Any

n.

if char F divides

char F does

= S (n-l'l)

cases

if c h a r

example,

ator

first

if c h a r F = 2 a n d n = 2

c M (n-l'l)

shows

a geometric

= M (12)

for M (n-l'l)

We examine

loops)

a field

the

if c h a r F d i v i d e s

series

by the u n o r d e r e d

get

and only

S (n-l'l)

a n d M (n-l'l)

t h a t i n all

determined

n = {~a i ~ [ a i ¢ F,

c s(n-l'l)if

composition

same Theorem

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

where

+ ~ + ...+ n).

0 c s(n-l'l)'c the u n i q u e The

2.5,

we have

Theorem

0 c S(12)±=

are

of Example

is o m i t t e d ,

s"

t 2..

4" I

'

,4,"

< 45

.

-~3

respectively. The

iO e d g e s

The

last edges

12

are

< 13

ordered

<23


involved

by

3.10:

4 < 24 in

< 34

< 15

< 25

< 35

e t l , . . . , e t5 are 2 4,3 4,2 5,3 5,4 5

19

(which

correspond

to

{tl},...,{t5}.)

different,

etl,...,et5

from

clear

that

this

later.

respect

they

checks

they

Specht

do

give

dimension

Let

us

find

module,

a basis,

last

edges

are

Note

that

it is

b u t we the

shall

far

prove

Gram matrix

with

is 2

2

1 -i

4

1

2

2

1

4

2

1

1

2

2

4

2

-i

1

1

2

4

1

if c h a r

F = 0 or c h a r

if c h a r

F = 3, rank

if char

F = 2, rank A = 4.

dim(S(3'2)/S(3'2)n

the

these

independent.

the

2

that

Therefore,

span

4

=

Since

linearly

that

basis

A

when

also

Assuming

to this

One

are

is

4 or i,

S (3'2)±

F > 5, rank A = 5

A = 1

S (3'2)±)

= 5 unless

char

F = 2 or

3,

respectively.

Certainly,

I

F =

~

and

4" are o r t h o g o n a l

like

F(-I)

=

-3 to " q u a d r i l a t e r a l s

(An u n l a b e l l e d to S (3'2)"

5 graphs

with

edge

is a s s u m e d

to h a v e

(F(-i)

is d e f i n e d

by

alternate weight

F(-i)

edges

i).

= F(-I)(i

weighted

That i)

is,

for

they

F(-I)+

F (-i),...,F (-5) span

a space

F(-2) are

S (3'2)I

When n S (3'2)±

has

dimension

We

the 5.3

5

char

by F,

It is e a s y if c h a r

F = 3.

I

to v e r i f y

F ~ 3, and

that

that

they

Hence

F (-I) ,F (-2) ,. .. , F(-5)

(by 1.3).

c h a r F = 2, et2 + et3 + et4 + et5 = F. T h e r e f o r e , F E S (3'2) in this case, and by d i m e n s i o n s it spans S (3'2) n S (3'2)±.

do n o t y e t h a v e

belongs

two

= 3F.

independent

4 when

is s p a n n e d

W h e n c h a r F = 3, etl S (3'2) n S (3'2)± .

graph

F(-5)

linearly

of d i m e n s i o n

S (3'2)I since

+...+

belong

1 < i < 5.) m

NOW,

+ i".

+ et2

= F(-5),

a convenient

to S (3'2).

However,

way

every

and now

F (-I) ,.. . ,F (-5)

of c h e c k i n g such

graph

whether certainly

span

or not

a

satisfies

conditions:

(i)

The

s u m of the

(ii)

The

valency

coefficients

coefficients

of e a c h p o i n t

of the e d g e s

at e a c h

of the e d g e s is

point

zero. is

is

(Formally:

zero.)

zero. the

s u m of the

20

These

conditions

the c o n d i t i o n s . us r a p i d l y of edges, when

hold because

In fact,

to c h e c k

t h a t F ~ S (3'2) w h e n

(F(-5)

has

6 edges

So far, w e h a v e h i g h l i g h t e d

above.

Find a basis

(N.B.

char F = 2

satisfies

S (3'2)

(F has

and e n a b l e

an e v e n n u m b e r

and that F(-5)~

S (3'2)

and e a c h p o i n t has v a l e n c y

two p r o b l e m s

for the g e n e r a l

It is n o t o b v i o u s

for S (3'2)

characterize

and e a c h p o i n t has e v e n v a l e n c y ) ,

char F = 3

(a)

a generator

the p r o p e r t i e s

even

to be d i s c u s s e d

Specht module

O or 3). later:

like t h a t g i v e n

t h a t d i m S U is i n d e p e n d e n t

of the

field.) (b) module Example

Find conditions

as a s u b m o d u l e

similar

of M~(cf.

to 5.3 c h a r a c t e r i z i n g

the s e c o n d

expression

the S p e c h t

for S (n-l'l)

in

5.1).

We h a v e p r o v e d in the g e n e r a l This example

that etl,...,et5

are l i n e a r l y

it is a lot h a r d e r

is c o n c l u d e d

form a basis Define

case,

of S (3'2)

by a s i m u l t a n e o u s

and t h a t

~o ~ H o m F ~ 5

to p r o v e proof

conditions

(M(3'2)'M(5))

independent;

here,

as

that they span S (3'2). that e t ,...,e~

5.3 c h a r a c t e r i z e

and ~i e H O m F ~

5

S (3'2/5.

(M(3'2),M(4'I))

by ~o:

91

Now,

:

abc de

~

abcde

a b c d e

+

a b c e d

conditions

5.3(i)

or

(ii)

+

a b c d e

hold

(i.e. d-~ ÷ d + e)

for an e l e m e n t

v of M ( 3 ' 2 ) i f

and o nly if v ~ K e r @o or v E K e r @i' r e p e c t i v e l y . Therefore S(3,2) Ker ~o n K e r @i (cf. L e m m a 4.10), and we w a n t to p r o v e Write

S (3'I)' (3'2)

for the s p a c e

s p a n n e d by g r a p h s

of the

equality. form

i

+~~ j k NOW,

S (3'I)'(3'2)

=

~ K e r ~o and

(since ~i : i-~ - ~---K+ i + j - i - k f o l l o w i n g s e r i e s for M(3'2) :

i

j

-

91 sends

= ~ - k).

i---~

S (3'1)'(302)

Therefore,

o n t o S (4'I)

we have

the

21 Dimensions

M(3,2)

1 Ker ~o S(3,1)

I , (3,2)

-> 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 <~, = [{(sgn ~)~I ~ E Ct,}, as usual. fln~ for S ~ ", so that u~ = (sgn z)u w h e n 6 @n" It is routine to verify that there is a w e l l - d e f i n e d ~ ~n-epiA' A (~n) m o r p h i s m @ from M~ onto S~ ~ S sending {t'} to ({t} @ u) Pt,; Let u be a generator

0 is given bM 6.8

@: {t'w} + 0 sends Now,

({t} ® u)Pt,~ =

{t'}Kt,

<{t}
to

({t}K t ® u)~ =

({t'} 8 u) Pt,
(sgn z ) { t ~ } K t ~ 8 u.

= {t}<tp t 8 u.

{t}> = <{t}< t ,{t}Pt> = <{t}K t ,IRtl{t}> = [Rtl.

Since IRtl is a n o n - z e r o e l e m e n t of ~, {t'}Kt, @ ~ O. l' I'± Ker @ ~ S ~ , and, by the S u b m o d u l e Theorem, Ker 8 ~ S ~ A t

,.At

.-All,

dim S~A = dim Im 8 = d i m ( M ~ / K e r 8) >- 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'

<

and [5]).let GX, Y =j=l [ (sgn ~j)~j. In all a p p l i c a t i o n s ,

GX, Y is c a l l e d

X w i l l be t a k e n

of t and Y w i l l be at the b e g i n n i n g tations poses

Ol,...,Ok

note

are,

X u Y, and w h o s e

7.1

agree with

entries

(i+l) th column.

n o t unique,

The p e r m u -

b u t for p r a c t i c a l

so that t o l , t O 2 , . . . , t o k

t except

increase

. (Garnir

at the end of the ith c o l u m n

of the

t h a t we m a y take ~ l , . . . , O k

the t a b l e a u x w h i c h occupied

of course,

a Garnir element

in the p o s i t i o n s

vertically

downwards

purare all

occupied

by

in the p o s i t i o n s

by X u Y.

EXAMPLE

if t =

1

2

4

3

, X = {4,5}

and Y = {2,3}

then t ~ l , . . . , t o k

5 m a y be t a k e n

as

t = t I = 12

t 2 = 12

t 3 = 12

t 4 = 13

43

34

35

24

25

25

5

5

4

5

4

3

when 1 -

sgn ~i = 1 for i = 1 , 3 , 4 , 6 , (34)

+

(3 5 4) +

(2 3 4) -

t5 = 1 3

sgn a i = -i for i -- 2,5 and (2 3 5 4) +

t 6 = 14

GX, Y

(2 4)(3 5).

!

7.2

THEOREM

Proof:

If

(See P e e l

IX u YI [19])

> ~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

<...<

b u t we

If t has

then

{ti}

independent.

then

last

of M ~ and t h a t

If the t a b l o i d s

Therefore,

involved

If t' = t~ w i t h

involved

a m = O, k < j.

(3.11)

Proof:

of t,

< {t2}

is all we n e e d

{t'}

are e l e m e n t s

linearly

independent,

tabloids

izer

{t I}

in no v k w i t h

clear

are

Vl,V2,...,v m involved in v i.

V l , V 2 , . t . tv m are

and

and

that

tabloid

columns~

element

there

of the

are n u m b e r s

~ {t' (wx)}.

that

then

all

the

9 {t}.

{t' (w ~ }

Since ~ {t}.

column w

stabil-

< x with

{t' (w ~}

w

is

Therefore,

{t'}

{t}. 8.4

THEOREM

Proof:

(See P e e l

polytabloids tions nation the

!etlt

are

to p r o v e

a standard

[19])

linearly

that

of s t a n d a r d

reader now

is

We h a v e

already

independent,

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

to do this.

p-tableau}

is a b a s i s

proved

that

and we n o w

can be w r i t t e n

- a glance

use

for S ~.

the s t a n d a r d the

Garnir

as a l i n e a r

at E x a m p l e

7.3

rela-

combi-

should

show

30

First we write It]

={tllt

totally lence

for the

colur]n e q u i v a l e n c e

I = t~ f o r sor~e z E Ct}.

ordered

in a w a y

similar

The

column

class

o f t;

equivalence

to the o r d e r

that

classes

is are

3 . 1 0 on the rov~ e q u i v a -

classes. Suppose

et,

[t]

that

t is n o t

c a n be w r i t t e n

when

It']

when

~ ~ C t, w e m a y

down

coluF~s.

the

jth

bs

with

< [t]

and

and prove

the

suppose

Unless (j+l)th

aq >

standard.

as a l i n e a r

By i n d u c t i o n ,

co~]ination salne r e s u l t

that

columns,

for e t.

the e n t r i e s

t is s t a n d a r d , have

we may

of s t a n d a r d

some

Since

in t are

adjacent

entries

assume

that

polytabloids etz

=

(sgnz) e t

in i n c r e a s i n g

pair

order

of columns,

say

a I < a 2 <... < a r, b I < b 2 <... <

b q for s o m e q al

~I A

a ^q

>

b

A

a

Let

s

r

X = {aq,...,a

Garnir

q

r}

and

Y = {bl,...,b

eler~ent GX, Y = ~ ( s g n 0 = e t [(sgn

= -[

8.5

b 1 <...< (sgn ~)et~ , the

COROLLARY

~round

ofJthe Remark 8.6

= [(sgn

<.. . < a r ,

result

proof

comes

[ta]

from

the

corresponding

7.2

. < It]

from

our

for ~ ~ 1 •

induction

Since

hypothesis.

of the S p e c h t

module

the

standard

~-tableaux.

8.4 is g i v e n

in s e c t i o n

number

In S ~ any p o l y t a b l o i d

result

~)et~

of T h e o r e m

of s t a n d a r d

consider

By T h e o r e m

follows

dimension

and equals

combination This

say.

and

of

S ~ is i n d e p e n d e n t

can be written

17.

as an i n t e g r a l

polytabloids. the p r o o f

of T h e o r e m

8.4;

alternatively,

8.9 b e l o w .

8.7 the

fieldf

COROLLARY

Proof: see

The

An independent

linear

~)a

bq
Because

et

~)~,

q}

COROLLARY standard

Proof: 8.8

et~

= etz

COROLLARY

tabloid

The matrices

basis

of S ~ .

Now

Since

result

follows

apply

order

v is a l i n e a r from Lemma

gn

integer

Corollary

If v is a n o n - z e r o

(in the p a r t i a l

Proof:

representin~

all h a v e

element

combination

respect

to

8.6.

~ on t a b l o i d s )

8.3.

over Q with

coefficients.

of SU~

then

involved

of standard

every

last

in v is s t a n d a r d . polytabloids,

the

31

8.9

COROLLARY

If v e S ~

and the c o e f f i c i e n t s of the tabloids

i n v o l v e d in v are all inte~ers~

then v is an i n t e g r a l linear combina-

tion of s t a n d a r d p o l y t a b l o i d s . Proof:

We may assume that v is non-zero.

the < order)

Let {t} be the last

(in

tabloid i n v o l v e d in v, with c o e f f i c i e n t aE Z, say.

By

the last corollary,

{t} is standard.

last tabloid in v - a e t is before

Now L e m m a 8.3 shows that the

{t}, so by i n d u c t i o n v - a e t is

an i n t e g r a l linear c o m b i n a t i o n of s t a n d a r d p o l y t a b l o i d s .

Therefore,

the same is true of v. 8.10

COROLLARY

If v E S~ and the c o e f f i c i e n t s of the tabloids invol-

ved in v are all integers t then we may reduce all these integers m o d u l o P and obtain an e l e m e n t S~, w h e r e F is the field of p elements. Proof:

By the last Corollary,

standard polytabloids,

v = [a i ei, say

the tabloid c o e f f i c i e n t s p.

(a i ~ Z).

in v, we obtain

The e q u a t i o n ~ = [~i ei

Remark

v is an i n t e g r a l linear c o m b i n a t i o n of R e d u c i n g m o d u l o p all

~, say.

shows that ~

S~

Let ~i b~ a i m o d u l o

.

If we knew only that the s t a n d a r d p o l y t a b l o i d s

proof of C o r o l l a r y

span S~, the

8.10 shows that any p o l y t a b l o i d can be w r i t t e n as

a linear c o m b i n a t i o n of s t a n d a r d p o l y t a b l o i d s over any field.

There-

fore, we can deduce that the s t a n d a r d p o l y t a b l o i d s span S ~ over any field, k n o w i n g only the same i n f o r m a t i o n over Q. 8.11

COROLLARY

If F is the field of p elements t then S~ is the

p - m o d u l a r r e p r e s e n t a £ i o n of Proof: 8.12

~n

o b t a i n e d from S ~

O

A p p l y the last Corollary. COROLLARY

There is a basis of S ~ f all of w h o s e elements

inv-

olve a unique s t a n d a r d tabloid. Proof:

Let {t I} < {t2} < ... be the s t a n d a r d ~-tabloids.

the only s t a n d a r d tabloid i n v o l v e d in etl by L e m m a 8.3. involve

{tl} , w i t h c o e f f i c i e n t a, say.

{t I} is

et2

may

Replace et2 by ft2 = et2 - a etl.

Then {t2} is the only s t a n d a r d tabloid i n v o l v e d in ft2.

Continuing

in this fashion, we c o n s t r u c t the d e s i r e d basis. C o r o l l a r y 8.12 is useful in n u m e r i c a l 8.13

EXAMPLE

Taking etl,...,et5

just one s t a n d a r d tabloid, e x c e p t et5

calculations.

as in Example

5.2, each involves

w h i c h involves 2 4

as w e l l as

5 . Replace et5 by ft5 = etl + et5. Then e t l , e t 2 , e t 3 , e t 4 , f t 5 involve respectively 2 4,3 4,2 5,3 5,4 5 w i t h c o e f f i c i e n t i, and no other s t a n d a r d tabloids.

32

Consider

the

following

vector I

3

4 v v belongs zero, -2

to S (3'2) , s i n c e

and e a c h

point

4, -3---~, -2 5,

-3

all

in

these

= 2etl

- et2

- et3

- et4

+ 3et5.

the

Suppose {t}0 h a v e

a basis

l-tabloids.

We m a y

pond all

obtain

of S QI±

fl,...,fm Define

that

N has

and~that

((t} E M~).

all

the

Then t reducing

0 of H o m F G ( ~ l± then n 0 = S~

If k e r

the e n t r i e s

and extend I of M~. Let

the m a t r i x

N =

by

the

standard

basis

{tl}, .... ,{t m} be (nis)

the

by

> 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 <... < r m be the i n t e g e r s

can be r e m o v e d

@i:

where

(ii)

[~] = [4,22,1], obtained

general:

to a S p e c h t m o d u l e

above S lj in the s e r i e s

[19])

by the F r o b e n i u s

f r o m the m o r e

o v e r an a r b i t r a r y

isomorphic

[~]. Define

+ [4,23 ] + [4,22,12]

THEOREM

The two p a r t s

Reciprocity

irre-

the

for i n d u c i n g to % + 1 "

+ [4,3,2,1]

(ii) S ~ +---~n-i ~ @ { S ~ away f r o m [p]}.

node

and ~ G n + l

an o r d i n a r y introduced

[ 4 , 2 ~ , i ] ~ G 8 = [3,22,1]

THE B R A N C H I N G ~

Gn_l

We have

[4,22,11% ~ i O = [5,22,1]

9.2

(i)

to

~n-l"

like t h a t in 6.4, we h a v e

are s p e c i a l

to [~]

us how to r e s t r i c t

~ n to

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

These

node

tells

from

the s t a n d a r d

e t where

t is

a n d n is in the r l t h , r 2 t h , . . . , or rith row of t. and

V i 8 i = S II,

li-polytabloids

span S li.

In the s e r i e s 0 ~ V 1 n K e r 01 c V 1 ~ V 2 n K er 02 ~ V 2 ~

...

35

...c Vm_ 1 ~ V m n Ker 0 m = V m = S ~ we h a v e d i m ( V i / ( V i n K e r 0i)) But

since

= d i m V i @i = d i m S xl.

m [ d i m S ~I = d i m S ~, i=l the d i m e n s i o n

leaux.

Therefore,

above,

and Vi/Vi_ 1

of a S p e c h t m o d u l e there

is e q u a l i t y

is F ~ n - i

is the n u m b e r of s t a n d a r d

in all p o s s i b l e

- isomorphic

to S II.

places This

tab-

in the s e r ~ s

is our d e s i r e d

result. 9.5 EXAMPLE As an F ~ 8 - m o d u l e , S (4'22'I) has a s e r i e s w i t h factors, r e a d i n g f r o m the top, i s o m o r p h i c to S ( 4 ' 2 2 ) , S ( 4 ' 2 " 1 2 ) , S ( 3 ' 2 2 ' 1 ) (cf. E x a m p l e

17.16.)

36

i0.

p-REGULAR We h a v e

it can be

PARTITIONS

seen

In o r d e r

~i+l Otherwise,

A partition

= ~i+2

For example,

LEMMA

(62,5~,i)

The

of p - r e g u l a r

Writing

see

w has

divisible

sible

those the

zero has

or i r r e d u c i b l e , a n d prime

partitions

that

characteristic for w h i c h

p.

S ~ is or

following

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

= ~i+p

if for some

i

> O.

is p - r e g u l a r

of a g r o u p

in t h a t

number

partitions

Proof:

equals

= "'"

class

of an e l e m e n t

that

between

is

field

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

A conjugacy

10.2

ground

in S ~±, we m a k e

DEFINITION

order

if the

to d i s t i n g u i s h

is not c o n t a i n e d i0. i

t h a t S ~ / ( S ~ n S ~±)

zero o n l y

class

by p.

coprime

Therefore,

the n u m b e r

is c o p r i m e

of p - r e g u l a r

if p a 5.

a p-regular

class

if the

to p.

classes

of

~n

equals

the n u m b e r

of n.

a permutation

order

if and o n l y

is c a l l e d

~ as a p r o d u c t

of d i s j o i n t

cycles,

to p if and o n l y

if no c y c l e

has

the n u m b e r

of p - r e g u l a r

of p a r t i t i o n s

~ of n w h e r e

the

ratio

classes

we

length

of

n ~i of ~ is d i v i -

no p a r t

by p. Now

simplify

(i) ator.

following

(i

-

xP) (i - x2P)...

(i

-

x) (i

Cancel

-

equal

in two ways:

x2)...

factors

leaves (i - xi) -I = ~ p~i p~i

(i - xmP) in the n u m e r a t o r

and d e n o m i n -

This

and the c o e f f i c i e n t summand

(xi) 2 +

of x n is the n u m b e r

is d i v i s i b l e

to t a k i n g

(i + x i +

by p.

x a f r o m the

first

of p a r t i t i o n s

(The p a r t i t i o n bracket

(xi) 3 + ...) of n w h e r e

( . . . 3 c , 2 b , i a)

(x2) b f r o m

no

corresponds

the s e c o n d

bracket,

and so on.) (ii)

For each m divide

in the n u m e r a t o r ,

(i - x m)

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

into

(i - x mp)

to give

(i + x m +

(xm) 2 + ...

+

(xm)p-l).

m=l Here part

the

coefficient

of the p a r t i t i o n Comparing

reader

who

of x n is the n u m b e r occurs

coefficients

is w o r r i e d

section

19.3

of H a r d y

Remark

Like

most

about

p or m o r e

of p a r t i t i o n s

and W r i g h t

combinatorial

no

times.

of x n, w e o b t a i n problems

of n w h e r e

the d e s i r e d

of c o n v e r g e n c e

equality

is r e f e r r e d

(The

to

[3]). results

involving

p-regularity,

Lemma

37 10.2 does not require p to be prime,

and it is only w h e n we come to

r e p r e s e n t a t i o n theory that we m u s t not allow p to be composite. We next w a n t to i n v e s t i g a t e 10.3

~

the integer g~ d e f i n e d by

= g . c . d . { < e t , e t , > l e t and et, are p o l y t a b l o i d s

in S~}.

The i m p o r t a n c e of this n u m b e r is that it is the g r e a t e s t common d i v i s o r of the entries in the G r a m m a t r i x w i t h respect to the s t a n d a r d basis of the Specht module.

(Corollary 8.6 shows that any p o l y t a b l o i d

can be w r i t t e n as an integral

linear c o m b i n a t i o n of s t a n d a r d p o l y t a b -

loids). 10.4

LEMMA

(James [7]) Suppose that the p a r t i t i o n ~ has z. parts

equal to j. Then j~l= zj: Remarks

and ~

divides

3~l(Zj:)J~= _

Since O~ = i, there is no p r o b l e m about taking i n f i n i t e

products.

Some of the integers i n v o l v e d in the d e f i n i t i o n of g~ may

be zero or negative, g.c.d.

divides ~

but we adopt the c o n v e n t i o n that,

for example,

{-3,0,6} = 3.

Proof:

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

r e l a t i o n ~ on the set of ~-tabloids by

{t I} ~ {t 2} if and only if for all i and j, i and j b e l o n g to the same row of {t 2} w h e n i and j b e l o n g to the same row of {tl}. Informally, s h u f f l i n g rows. Now,

this is saying that we can go from ~t I} to {t 2} by The e q u i v a l e n c e

classes have size

~ z I j=l 3

if {t I} is i n v o l v e d in e t and {t I} ~ {t2} , then the defini-

tion of a p o l y t a b l o i d shows that {t 2} is i n v o l v e d in et, and w h e t h e r the coefficients

(which are ±i)

depends only on {t I} and {t2}. m u l t i p l e of j~izj: Example 5.4). Next,

are the same or have opposite signs Therefore,

tabloids in common,

let t be any ~-tableau,

the order of the numbers if t =

1 2 3 4

any two p o l y t a b l o i d s have a

and j~izJ:

divides g~

(cf.

and obtain t* from t by r e v e r s i n g

in each row of t. then t* =

567

For example,

4 3 2 1 765

8 9 iO

iO 9 8

ii

ii

Let ~ be an e l e m e n t of the column stabilizer of t h a v i n g the property that for every i, the numbers i and iz b e l o n g to rows of t w h i c h have the same length.

(In the example,

w can be any element of the

group __~{5,8} x ~{6,9} x ~{7,10}). Then {t~} is i n v o l v e d in e t and et, w i t h the same c o e f f i c i e n t in each. It is easy to see that all tabloids

common to e t and et, have this form.

(In the example,

every

38

tabloid common be

involved tabloid

in the

et.

>

10.5

has

first

= j~l

in et, has

this

(zj ~)j' The

10.6

COROLLARY

of the n u m b e r s

this

Proof: say.

the

prime

if and o n l y

order and

and

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

happens

8 in the

row of a c o m m o n

COROLLARY

Proof:

5 or

1 in the

first

lemma

if and only

Corollary

4.7 shows

Going

and

back

so on.)

g~ if and o n l y if p d i v i d e s

if p d i v i d e s

is c o p r i m e

Looking

at et, to et,,

Therefore,

no 2 must < et,

is p r o v e d .

p divides

in e a c h

row.

row.

tabloid,

If t* is o b t a i n e d

multiple

first

if ~ is p - s i n g u l a r .

z ~ for some 3

j, and

g~.

the

~-tableau

row of t r then

to p if and o n l y t h a t et,< t is

t by r e v e r s i n g

et,~ t is a m u l t i p l e

the of et,

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

a multiple

of et,

et.< t = h e t

Now r h = h < et,

{t}

> = < 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'}
to

{t}<tp t @ u, and K e r ~ = S I'±. Thus, if I' is p - r e g u l a r , the s u b m o d of S 1 g e n e r a t e d by {t}KtP t is i s o m o r p h i c to D I'. In terms of the

ule

group

algebra

PtKtPt give

all

(p p r i m e

F ~n'

this

(choosing one the

irreducible

or = ~).

means

that

t for e a c h

the

right

partition

representations

of

ideals

generated

whose

conjugate

~n

over F when

by

is p - r e g u l a r ) char

F - p

42

12

COMPOSITION We next

FACTORS

examine

M ~ and S ~ in general zero,

what

can be said about

terms.

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

The p r o b l e m of finding of prime

the complete

groups

incorrect

give

First, 12.1

X > ~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
if s o m e

column

of T contains

two identical

numbers. ^

If we d e f i n e

@T b y

^

@T = the then

13.5

suggests

To eliminate

such

restriction

that

sometimes

trivial

of ~ T to S ~T is

elements

zero,

, since

e t ~ T = { t } @ T < t.

of H o m F ~ n ( S I , M U ) ,

we make

the

following ~.6

DEFINITION

decreasing umns

of T.

13.7

along Let

EXAMPLE

of the t w o

A tableau the

rows

~o(l,~) If I =

tableaux

T is s e m i s t a n d a r d

of T and strictly be the

(4,1)

1 1 2 2 3

if the n u m b e r s

increasing

set of semistandard

and

~ =

and

(2,2,1),

then

down

tableaux ~'o(l,U)

are n o n the

col-

in

consists

1 1 2 3 . 2 ^

We

a i m to p r o v e

that

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

eT with

T in

~(i,~).

~o(I,~)

46

usually called

give

a basis

semistandard

the S p e c h t

for H o m F ~ n ( S I , M P ) . homomorphisms,

module,

the d i f f i c u l t

These

and, part

~somorph~ms

span

Hom F G

linearly

independent

uses

a partlal

order

~(l,p)

(cf.

13.8 by

[T] of t a b l e a u x DEFINITION

interchanging

x and w < x. 13.9

Let

IT I] ~

~

When

the p a r t i a l

(SI'M~)"

[T 2] if

w and x, w h e r e

Then

EXAMPLE

indicates

in

I =

(3,2)

order

on

3.11

standard whether

proof

that

column

and

and ~ =

order

(2,2,1),

1

be of

they

are

equivalence

3.15) :

to a l a t e r

column

will basis

the s e m i -

IT 2] can be o b t a i n e d

a partial

the

0

The

on the

w belongs

generates

the

is to d e c i d e

standard

classes

homomorphisms

as w i t h

4

from

column

[T I]

of T 1 than

.

the

following

equivalence

classes:

tree

1

! 1 l/ /

The

crucial,

13.10

but

\

trivial,

property

It T is s e m i s t a n d a r d ,

[T']

~ [T]

13.11

unless

LEM~MA

T'

of this

and T'

partial

order

is row e q u i v a l e n t

is:

to T,

then

= T.

{~T[T ~ ~O(l~p)}

is a l i n e a r l y

independent

subset

of

H o m .F ~ n (sl rMZ) " Proof:

(cf.

Lemmas

homomorphisms equal Then

zero, from

with choose

the

8.2 T in

8.3).

~Yo(l,~)

T 1 such

definition {t}

and

that

If Za T @T and n o t aTl

of 8T and

Za T QT = aT 1 T1

all

~ O, b u t

is a l i n e a r the

field

aT = 0

combination

if

IT1]

4 IT].

13.10, + a linear

combination

T 2 satisfying

of

coefficients

of t a b l e a u x

[T I] @ IT2].

47

Since classes,

the

and

coluntn s t a b i l i z e r T 1 K t ~ O, {t}
Therefore,

Ea T ~ T

We now have has

LEMMA

to b e

where

careful

column

equivalence

that

7 a T @ T = {t} 7a T e T < t element

about

~ 0

o f H o m F ~ n ( S I , M ~) , as r e q u i r e d .

the

case where

our ground

field

2:

Suppose

and write

shows

is a n o n - z e r o

characteristic

13.12

this

of t preserves

{t}KtO

t is the g i v e n

t h a t ® is a n o n - z e r o

element

= ~c T T

(c T E F, T £ ~ ( l , ~ ) )

l-tableau.

Unless

of HOmF~n(SlrM~)

char F = 2 and

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

then (i) C T .

= 0 for e v e r y

tableau

T* h a v i n ~

a repeated

entry

in s o m e

column. and

(ii)_CTl

~ 0 for some

Proof:

Part

(i)T* =

(j)T*.

Since

(i) S u p p o s e We wish

that

TI~

in the s a m e

that CT,

column

of

t,

and

= O.

Kt(i,j ) =-Kt,

T*(i,j)

If c h a r sing

tableau

i # j are

to p r o v e

7 c T T(i,j) Because

semistandard

= T*,

it

F = 2 and

the o r d e r

of

= {t}K t 8(i,j) follows

I is

that CT,

2-regular,

the n u m b e r s

in e a c h

= --E C T T = 0 when

let ~ be row of

c h a r F ~ 2.

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

t.

By C o r o l l a r y

rever10.6

,

{t}K t ~ K t = {t}K t • T h e r e fore Z CTT= By entry

13.5,

appears Part

tableau

{t}
which

in 7~ c T T ~
(ii)

annihilates

and

no

{t}K t ® =

has

a coltura c o n t a i n i n g

so C T ,

If z is in the

Kt = Z C T TZ K t . a repeated

= O.

column

stabilizer

o f t,

then

1 -(sgn

~)~

{t}< t . T h e r e f o r e 7 c T T = Z CT(sgn

~)T~

,

CTl

T 1 a n d T 2 are

so

Since

= + CT2

8 ~ O, w e m a y

c T = 0 if

IT I]

Lemma

that we may

show

the c o l u m n s We

~ IT].

when choose

a tableau

The previous assume

that

column

T 1 such

equivalent.

t h a t C rl ~ O, b u t

paragraph

and part

(i) of the

the n u m b e r s

strictly

increase

down

of T 1 .

shall

be home

if we

can derive

a contradiction

from assuming

48

that

for

TI,

b I < b 2 < ...< b s

a

q

> b

some

j, a I < a 2 <...<

for some

q

are

a r are

the e n t r i e s

the e n t r i e s

in

in the

jth c o l u m n

(j+l) th c o l u m n

of

of T 1 and

q.

aI

b1 A A

a

•

q

b

q

A

s

^ a r

Let let

xij

Z(sgn

be

the e n t r y

~)~

be

in the

aGarnir

{ X l , j + I, . . . ,Xq,j+l}.

For every

(q,j+l)th, T 1 Z(sgn zero, with

tableau

agreeing

there

with

have

= {t}< t ~ ( s g n

must

tableau

t, and

{Xqj,...,Xrj}

a)~@

and

= O.

T, T Z (sgn ~)~

is a l i n e a r the

places.

be a t a b l e a u

+- CTl,

have

Unless

IT I]

char

All

F = 2 and

{~TIT ~ ~;'o(l,~) } is a b a s i s

the

with

tableaux

involved

in

Z c T T Z (sgn ~)~

is

above.

and

this

of

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

c T ~ O such

described ~ IT],

combination

(l,j+l)th,

and s i n c e

T ~ T1

the p l a c e s

< aq <.. . < a r , we m u s t i n i t i a l c h o i c e of T I. THEOREM

of the

T on all e x c e p t

coefficient

T 1 on all e x c e p t

13.13

~)~

(q,j)th,...,(r,j)th

~)~

place

for the sets

Then

cT T ~(sgn

tableaux

(i,j)th

element

that T a g r e e s

Since

b I <...< bq

contradicts

our

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

for H O m F ~

(SI,M~). n

Proof:

Suppose

@ is a n o n - z e r o

element

of H o m F ~ n ( S i , M ~) .

By L e m m a

13.12, {t}K t @ = Zc T T, w h e r e We m a y

assume

13.10,

{t}
ETI3 @

IT2].

standard follows 13.14

that

c T = O if

- CT] e T l )

homomorphisms,

~ - CTI~TI

and so the

from Len'~la 13. ii. COROLLARY Unless

d i m H o m F ~ n ( S l , M ~) of type

~ .

Remark

If ~ is o b t a i n e d

char

equals

from

~ O for some

T E ~o(l,~)

is a l i n e a r

;~y i n d u c t i o n ,

CTl

and

[TI3

T 1 E ~o(l,~).

~ IT].

Then,

by

combination

of t a b l e a u x

T 2 with

is

combination

of s e m i -

same

a linear is true

F = 2 and the n u m b e r

of e. The

Theorem

now

I is 2 - s i n g u l a r , of s e m i s t a n d a r d

~ by r e o r d e r i n g

the p a r t s

l-tableaux

(e.g.

~ =

49 (4,5,O,1)

and 9 =

(5,4,1)),

then visibly

dim HOmF ~n(SI,MP) Equivalently, 13.6.

we m a y c h o o s e

Therefore,

shape

the

an u n u s u a l

different

we

112

13.15

list b e l o w

orderings

tableaux

of the o r d e r we c h o o s e

the e l e m e n t s

in

in d e f i n i t i o n

of a g i v e n

on the entries.

~ o ( ( 4 , 1 ) , (2,2,1))

for

of {1,2,3}:

2

11

23

3

2

3211

3221

2

1

1132

1122

2

3

COROLLARY

o r d e r of i n t e g e r s

n u m b e r of s e m i s t a n d a r d

and size is i n d e p e n d e n t

For example,

= d i m H o m F ~ n ( S I , M ~)

Unless

when

1 < 2 < 3

when

3 < 2 < 1

when

1 < 3 < 2

c h a r F = 2 and I is 2 - s i n g u l a r r e v e r y e l e m e n t

of H o m F ~ n ( S ~ r M p) can be e x t e n d e d

to an e l e m e n t

of H o m F ~ n ( M ~ M p ) .

^

Proof:

0 T can be e x t e n d e d

Of c o u r s e ,

Corollary

of no d i r e c t p r o o f That Theorem

13.15

is t r i v i a l

for the g e n e r a l

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

13.15

can be false if char F = 2

by the easy:

i

EXAMPLE

if char F = O, b u t we k n o w

case.

13.13 and C o r o l l a r y

and I is 2 - s i n g u l a r 13.16

to ®T"

If c h a r F = 2, ~ + T

+

12

defines

an e l e m e n t of

( S ( 1 2 ) , M (2)) which H ° m F ~ 2 ( M ( I 2),M(2 ) H°mF ~ 2 )"

c a n n o t be ~ x t e n d e d

13.17 implies

COROLLARY

c h a r F = 2 and I is 2 - s i n ~ u l a r ~

Proof:

There

and n o n e 13.14

Unless

to an e l e m e n t of

I ~ p

H o m F ~ n ( S l t M ~) = O t and H o m F ~ n ( S l r M l) ~ F. is j u s t one s e m i s t a n d a r d

at all u n l e s s

gives

I ~ p .

of type ~ if I = p, 3.7).

Corollary

the result.

Corollary I is p - r e g u l a r

1 3 . 1 7 has

already been proved

(Lemma 11.3),

for the case w h e r e

u n d e r the h y p o t h e s i s

and we n o w p r o v i d e

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

that

proof

c h a r F ~ 2.

Let 8 ~ H o m F ~ n ( S I , M P ) , a p-tableau.

h-tableau

(cf. the p r o o f of L e m m a

and s u p p o s e

If ~ ~ u , or if i = p and

that t is a h - t a b l e a u {tl} is not i n v o l v e d

and t I is in et,

50

then

some

column

pair of numbers

of t.

a,b b e l o n g

to t h e

same

row of t I and the

same

Therefore

< e t @ , { t I} > = -< e t ( a , b ) 0 , { t I} > = -< e t 0 , { t l } ( a , b ) = -< e t 0 , { t I} Since

c h a r F ~ 2,

I # Z , and that et0 If I = ~

and z belongs

< et0,{t}z et0

> = < et@

= < et@i{t}

13.18

< e t 0 , { t I}

involves

> e t.

COROLLARY

This

tabloids

column

> = sgn

Thus

Unless

> .

> = O.

only

to t h e

~-l,{t}

>

proves

involved

stabilizer

0 = 0 if I = ~.

o f t, t h e n

~ < et@,{t}

0 is m u l t i p l i c a t i o n

char F = 2 and

that

in e t w h e n

> by

and this

shows

that

a constant.

I is 2 - s i n ~ u l a r l

S 1 is i n d e -

composable. Proof: one

If S 1 w e r e

component,

and produce

contradicting Remark:

group, 13.19

we

are

shall

each

could

a non-trivial

decomposable

investigate need

THEOREM

alon~

we

take

the p r o j e c t i o n

element

onto

of HomFGn(SX,MI),

the l a s t C o r o l l a r y .

There

When we

decomposable,

the

{0TIT

r o w of T}

Specht

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

- see E x a m p l e

theory

of

23.10(iii).

the g e n e r a l

linear

simple e [Y(lr~)

is

modules

a basis

a n d the n u m b e r s for HornF ~

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

(MXrM ~) . n

Proof:

Our

s e t of h o m o m o r p h i s m s

representative The

linear

independence

Suppose

has been

TI,T2,...,T k from each

that

of the

set

0 is a n e l e m e n t

constructed

by

row equivalence

follows

f r o m the

of HOmF~

taking

class

one

of

definition

~(l,~). of 8 T.

(MX,M ~) • If T a n d T'

are

n row equivalent,

t h e n T'

< {t}0,T'

= Tz

> = < {t}0,T~ =

Hence

and

since

for some

<

{t}@,T

z in Rt,

a n d so

> = < {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

<

that ~i+l

- ~i"

15.7

If ~i # = ~i

every

1 in a s e q u e n c e Thus we

15.8

~ in w h i c h

0 for the p a r t i t i o n

of type

is at m o s t

~

of p a r t i t i o n s

give

since

is good. the

s(0,(3,2))

columns

last column

l~a = ~i~ for i > i, then s(l~,~) = s ( ~ , ~ ) ,

can absorb

EXAMPLE

and third

and

below

first part

of ~ into

= s((3),(3,2)).

give

The

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

~#. sequences

in the s e c o n d

and the sequences

in the

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

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

~

22111

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

s((3,1) ,(3,2)) 21211

12121

21121

12112

21112

11221

12211

11212 11122

Compare the factors Given

Example

5.2, w h e r e

of d i m e n s i o n s a pair

~,

to a diagram.

lose ~

by v e r t i c a l

row

(cf.

15.9

a series

we

lines.

The picture

obtained

of s u b m o d u l e s

with

is no coincidence' record

them in a p i c t u r e

We shall draw a line b e t w e e n

the p i c t u r e

each

for ~4~ ,~ w i l l

by e n c l o s i n g

ro%~ and enc-

always

all the nodes

be ident-

in the

first

15.7).

EXAMPLE

Referring

xx

This a node

has This

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

similar

i fied w i t h

M (3'2)

1,4 and 5.

xx

nesting

suggests

from ~ to ~ .

not in the

to Example

first

row.

15.8, we have

"

that we

should have

We n e e d only c o n s i d e r

some n o t a t i o n w h i c h

absorbing

a node w h i c h

adds is

56

15.10

DEFINITION

1 such that ~

<

(i) obtained ~Ac,

from ~

and

~c-l~ >

~

~ Z.

Let c be an integer greater

than

~ = ~c-l" ~c-i Z~, then

,Z by changing

~

Ac, ~ is the pair~ of partitions to

~c

~c~ + i.

If

~c-i = ~c

'

then

~ is the pair 0,O.

by changing

~ ~ ,~R c is the pair of partitions Zc

to

The operator of the row above and ~

Dc

and

~c-i

R c merely moves (R stands

to

some nodes

for "raise"

that

~c-i

equals

EXAMPLE

~c-i

obtained

from

Z # ,Z

~c-i + ~c - Zc " lying outside

and A stands

are involved in the definitions

we stipulate 15. ll

~c

If

(ii)

~

Suppose

~# to the end

for "add").

Both

of A c and Rc, and note that

"

R2 Ix x xlx x

=

~xxxxl

X X

t a2

xx,x.

=

~xx~

t' A2

Other examples

are given in 15.13,

Since R c raises some nodes, the first row,

15.12

to a pair of partitions

i is a proper partition.)

EXAMPLE

of the form

To obtain

from O,v to ~

((4,3,1),(4,5,22)),

to

apply

(O,(4,3,1,2,1,2)):

fxxxxl ~A3 xx~1~ R4R3~xxxl x x x

I,i

(when, per-

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

of operations A c , R c leading

A 23 A3 R4 R3 R5 R6 R4 R5

all the nodes in

A c , R c on a pair of partitions

It is also clear that

Given any pair of partitions,

and a sequence 15.13

and we always enclose

any sequence of operations

leads eventually force,

17.15 and 17.16.

IxJ

x xlx x

x

X X

X X

X

X

X

X X

X X

X X

m

,~.

57

xxx

R5 R 6 ÷

X X x

R4 R 5

X

÷

Ix X x l x

X

x

X X

X X

The

critical

15.14

theorem

THEOREM

s(~r~)

for s e q u e n c e s

The

\ s(~ ~ Ac,~) Given

following and

is

gives

a i-i

correspondence

between

s(~ ~ r~Rc).•

a sequence

in the

first

set r c h a n g e

all

the b a d

c's

to

(c-l) 's. Proof: ~c-i

Recall

= ~c-i

"

that

~c-i

c's

are

For

all

of the o p e r a t o r s

a sequence

(c-l) 's, all

= ~c-i

pc~good The bad

o u r definition

Therefore,

c's

and

changed

# Pc-~c to

A

s(~,p)\

and R c r e q u i r e d

c

s(~Ac,~)

contains

good.

bad

(c-l)

s I in

's

c's. to

give

a sequences

s 2.

We c l a i m

that

15.15

j,

s 2 ~ the n u m b e r This

is c e r t a i n l y

15.15

is o b v i o u s l y

(c-l)

which

ality

in 15.15

is

true

15.15 the

c's

true

shows

j replaced term

s 2 has

in s 2 are good.

and o n l y

if

i is g o o d

It is m o r e Given

the

by i)

c's by

Hence,

the

for

to p r o v e

because

the i n e q u of

Therefore,

also.

~ Pc-i

good

(c-l)'s , and

the

i-i

given

map

(c-l) 's by

brackets,

).

that

is g o o d

to s(p ~

the

Then is a

the n u m b e r

s 2.

so s 2 b e l o n g s

all

right-hand

term

case,

s I and

for i # c-i or c, i

and

j = i.

the ith

in this

in b o t h

jth t e r m of

in s I.

true

when

But

case

at l e a s t

replace

before

jth t e r m

is s t r i c t ,

is the same

in Sl,

difficult

any s e q u e n c e

( , and all

in s 2.

j = i + 1 in this

that

the

j = i, so a s s u m e

s I but bad

the ith for

for

(c-l)'s

before

for j = i + i, e x c e p t

in

(with

of g o o d

(c-l)'s

true

is g o o d

(c-2) 's b e f o r e 15.15

the n u m b e r

of g o o d

all

in s 2 if

,~R c)-

and onto.

left-hand

brackets,

For example,

if c = 3

1 2 1 2 3 1 2 3 3 2 2 1 1 3 1 1 2 2 3 goes

to Now,

Therefore, kets

than

1

( 1

( ) 1

in any s e q u e n c e every

belonging

right-hand

right-hand

( ) )

bracket

brackets,

( ( 11

) 11

( ( )

to s(p # ,pR c), is p r e c e d e d

and the

sequence

all

the c's

by m o r e looks

with

r = Pc-i

good. brac-

like #

Po(PI(P2(...(P r

are

left-hand

+ Pc - 2Pc'

58

where

e a c h pj

with

i ~ c-i

is a c l o s e d

to g i v e Let

Let

are

s belong

s* b e t h e (c-l)'s c-i

The Theorem s

will

s(~

We 15.17

nun~er

proves

15.17

The nu~er work

same

c-l,

then

i

map;

(precisely

right-hand

the

brackets,

a c-i

is b l a c k

in s if it

it is w h i t e .

from s by

changing

the

first

~c -

be

of

c appearing

element

in two steps.

in b o t h

s and

o f s(~ ~ ,~)

a black x

shows

that

for the is the

that

15.17

The

(by t h e

number

every

x is a b l a c k the n u m b e r Thus,

x

of

is t r i v i a l l y except

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

"black")

c in s is good.

•

the

This

c-l.

of white

we may

case we have

of w h i t e

definition

since

jth term,

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

x.

c-1.

x,

in s.

First

the nturd3er o f w h i t e

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

(c-l)'s

noting

every

the u n i q u e

15.16

t e r m x in s,

proof

since

to s.

in the c a s e w h e r e

which

that

say

otherwise

follow,

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

s if i t is t r u e black

We

obtained

o f c's b e f o r e

of good back,

terms

for an i n v e r s e

brackets

unpaired

We must prove

let x b e

of good

"extra"

to c's.

of good

number

hope

become

bracket;

ti%e p r o o f

Initially, x = the

some

image.)

a n a s* w i l l

For every

-< the n u m b e r

containing

in s* is good.

mapping

tackle

one

sequence

will

be g o o d ,

Ac,~)

must

to s(~ ~ ,~Rc).

to an e x t r a

Every

is o n l y

~c - ~c#

first

reversed

us an i n v e r s e

black

15.16

that there

the

which

corresponds

~J

clear

reverse

brackets

system,

o r c.

It is n o w namely,

parenthesis

start true

when

already

(c-l) 's in s < the at the e n d o f s a n d

f o r the the

(j-l)th

(j-1)th

term of

t e r m is a

done.

Next we have 15.18 good

Either (c-2)'s

For • ~c

-

c = 2, o r

> 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


Sg

AC'U Oc-l,p~

a ~-tableau,

= E {sgn n ) ~ l ~ a set B of Pc

get

=

'~ ~ c _ l ~ p c

fixes

,PRc

= O.

let

the n u m b e r s f r o m the

first

tR c say,

Uc~

in t o u t s i d e

cth

cth row of t i n t o

of the

a tableau,

S U~

and

numbers

in the

If B c o n s i s t s we

with

S~

t be

of the n u n ~ e r s

-- O.

last Example

LE ~V~A

Proof:

~l,l

row of t, and m o v e

the

numbers

[U ~ ]}.

(c-l)th in the

the

rest

row.

cth

row of t,

then

and , uR c

{ t R c }
any o t h e r

a ~R o - t a b l o i d , been

moved

up

is

inside

a factor

{tl}
et

except

~c~

is the

Since S 0'0 shows

p~ Ac, ~

that

~A =t

has

] .

the

cth

row of t, we

of the n u m b e r s ,

say

L e t y be the n u m b e r

still

x, w h i c h

above

get

has

x in t.

' and so

the

the

cth =

one m o r e

node

= ~c ) ' the p r o o f ~ = 0 c_l,Uc

.

•~

= {t} ~ c _ l , ~ c#
tabloids

row of

'~ ~ c - l , p ~

,U~ c

from

one

= {t}
from

~= et

= 0 if Pc-i

[~

of
sum of all

numbers

Therefore,

numbers

but now

= O.

'~ ~ c _ l , ~ c~

{t} ~ c _ l , ~ c

etR c

set of ~c

{t I} say, lies

Then (l-(x y))

=

{t} i n t o

e~ ~ ,~Rc tR c enclosed we

obtained

used

the

and

by m o v i n g

all

(c-l) th row.

•

in the

cth

to d e d u c e

row (or S ~ A c ' ~ that

{tl}
= O

69 17.13

THEOREM (i) S ~ ' ~

S~'~

(James [i0]) ~ = S~'uRc ~c-l,~ c

(iii) {et~

= s ~ A c ,~

~ ker ~ c _ l ~ c

(ii) S ~ '~/S ~ Ac'~ ~ S ~

and

'~Rc

dim S ~#'~ = Is(~#,~) [; indeed,

'~ i t corresponds

to a sequence

in s ( ~ , ~ )

by 17.6}

is a basis of

S~,P (iv) S ~ ' ~ given by Proof:

has a Specht series.

The factors in this series

Let O,v be a pair of partitions

,~ by a sequence

from which we can reach the pair

of A c and R c operators

(cf.

15.12)

dim S 0'~ = dim M ~ = Is(O,~) I by 17.8. assume

that dim S ~ ' ~

I s ( ~ , ~ ) I and prove

is(~*Ac,~) [ and dim S Now,

We may therefore

that dim S ~ A c ' ~

=

'~Rc = Is(~#,~Rc) I.

i s ( ~ , ~ ) I = dim S ~ ' ~ > 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
H(hook

n~ lengths in Ill)

i hil"

20.2

EXAMPLE

The hook graph

for [4,3,1]

is

6 4 3 1 421 1 Therefore,

dim S (4'3'I)

8' = 6.4.3.4.2

= 70.

The hook formula is an amazing result.

It is hard to prove directly

even that n~ is divisible by the product of the hook show that the quotient

is the number of standard

Proof of T h e o r e m

20.1

non-zero

It is transparent

parts.

dim S

let alone

We show that the result is true when I has 3

a full proof obscures By Corollary

lengths,

l-tableaux.

that the proof works

the simplicity

in general,

but

of the ideas required.

19.5, l

1

1 (hll - 2) '

1 (hll - l) '

1 hll'

1 (h21 - 2)'

1 (h21 - i)'

1 621'

1 (h31 - 2)'

1 (h31 - i).'

1 h31'

1

hll(hll - i)

hll

1

h21(h21 - i)

h21

1

h 3 1 ( h 3 1 - i)

h31

1

1

hll! h21-w h31.

(hll - h21) (hll - h31)(h21 - h31) hll~ h21: h31~

giving the first result.

78

1

1

hll h21 h31

1 hll h2l h3l

H(hook

1 lengths

(hll-

i)(hll-

2)

hll - 1

(h21-

i)(h21-

2)

h21 - 1

(h31 - i ) ( h 3 1 -

2)

h31 - 1

1

1 (hll - 3)'

1 (hll - 2)'

1 (hll - i) :

1 "Ch21 - 3) ~.

1 (h21-

2)'

1 (h21 - i)'

1 (h31 - 3)'

1 (h31 - 2)'

1 (h31 - i)'

~(hook

in [~])

lengths

1 , in [11-1,12-1,13 -i]) by induction

, as required.

79

21

THE

IiURNAGiIAN-NAKAYAI~hk

The M u r n a g h a n - N a k a y a m a of c a l c u l a t i n g In the be

the

same

21.1

a single

statement

RULE

Rule

entry

below,

as that

of the

in the

the

zp

E ~l

remainin~

X

= E {(-i) i Xg(W) ] [I]

As

usual,

a 1-cycle 21.2

is

an e m p t y

sum

and e f f i c i e n t

table

of

way

~n"

of a s k e w - h o o k

is d e f i n e d

to

hook.

RULE

numbers.

the ~ r a n c h i n g

character

leg-length

where

of the (nP)

n-r

beautiful

corresponding

TiiE i 4 U R N A G H A N - N A K A Y A ~ Su__up.pose that

is a v e r y

p is an r - c y c l e

and w is a p e r m u t a t i o n

Then \ [~]

is a s k e w

is i n t e r p r e t e d

r-hook

as zero.

of The

le~

len@th

case where

i}. p is

Theorem.

EXAMPLES (i)

Suppose

we w a n t

to find

the v a l u e

(5,4,4)

of X

on

the

class

(5,4,3,1).

;2 j

x

There

are

two w a y s

Murnaghan-Nakayama

of r e m o v i n g

Rule

gives:

on

(5,4,3,1)

X (5'4'4)

a skew

5-hook

= X (3'3'2) =

-

X(2,12)

-

from

X (5'3)

X(3,1)

[5,4,4]

on

the

(4,3,1)

+ X (22)

applying

and

on

the

(3,1),

rule

again

(22 ) remove

a skew

(ii) 13-cycle,

3-hook

X (5'4'4) since

is

we

from either

zero

cannot

on any

= X [2,12] =

-×

=

--1.

class

renLove h o o k s

on (3,1), or [3,1].

(1)

on

because

on

(7,3,3)

=

X (32)

containing of t h e s e

=

_X(2,

on I)

an 8 , 9 , 1 0 , i i , 1 2

lengths

(3,3) +

cannot

(i)

(iii)

(5,4,4)

we

X (3)

on

(3)

from

or

[5,4,4].

80

The only character ter table of

X

=

2,

(o)

+

X

(O)

table of

is unnecessary in

Rule needs several p r e l i m i n a r y

case where

following

p is an n-cycle,

Rule gives

these pieces of information

See the remarks

GO"

~0 ~

We first prove the special

finally we combine

of the charac-

Rule is that of

~ O is a group of order i, and a computer

the character

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

on

table required in the construction

Our proof of the M u r n a g h a n - N a k a y a m a lemmas.

(o)

Gn using the M u r n a g h a n - N a k a y a m a

Remember that evaluating

=

~I.12

then

for [u][x,lr-x],

and

to prove the Rule in

for an alternative

approach.

A hook diagram is one of the form [x,lY]. 21.3

LEMMA

Unless both

~O hook diagrams.

[~][8] = [a + b,l n-a-b] Proof:

[~] and [~] are hook diaprams ~ [~][8]

If Is] = [a,l n-r-a]

+ [a + b - l,ln~a-b+l]

If one of [el and [8] contains

[a][8] = [u][8]* Suppose,

= [83[u]'.

therefore,

the

This proves

+ some non-hook diagrams.

(2,2)-node,

the so does

the first result.

that [~] = [a,l n-r-a]

and [8] = [b,lr-b].

order to obtain a hook diagram in [a] [83", we have to put the places

shown,

then 2,3,...

contains

and [8] = [brl r-b] then

b

In

l's in

in order down the first column: b

E

J~lx . . . . .

I

X[ *

* ...

*

U The second result 21.4 Suppose

THEOREM

follows.

(A special

Xv(0)

=

Let

Rule). Then

{ (-l) n-x i f [v] = [x,l n-x] O

Proof:

case of the M u r n a g h a n - N a k a y a m a

that p is an n-c~cle F and ~ is a partition of n.

otherwise

Is] and [8] be diagrams

for

G r

and

~n-r

with O < r < n.

Then the character inner product (X[U][8] is zero, since coefficient

[u][8]

x(n)-(n-l,l)+(n-2,12)contains

... ± (in))

two adjacent hook diagrams,

i, or no hooks at all by Lemma 21.3.

each with

81

By the F r o b e n i u s restricts

to be zero on all Y o u n g

0 < r < n; except

in particular,

perhaps,

vector which characters except lar,

Reciprocity Theorem,

has

X (n) is i.

Theorem

21.4

but the above p r o o f 21.5

LEMMA

of n-r.

Since

Suppose

that

of

all o t h e r

table of

table

O n,

is n o n - s ~ g u -

of the p-column.

using

with

~n'

the column

and O o p p o s i t e

it is the p-column,

can also be p r o v e d

(in)

~(r,n-r)

Therefore,

the c h a r a c t e r

Therefore,

is more

form

of the c h a r a c t e r

column v e c t o r must be a m u l t i p l e

entry o p p o s i t e Remark:

P.

p.

X (x'In-x)

to all columns

with

of the

"'" ±

zero on all classes

our n-cycle

(-i) n-x o p p o s i t e

that a s s o c i a t e d

this

subgroups

it has value

that c o n t a i n i n g

is o r t h o g o n a l

X (n)-(n-l'l)+

But

the

as required.

the D e t e r m i n a n t a l

Form,

elegant. I is a p a r t i t i o n

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

Then

(i)

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

of

[I] in [~][xfl r-x]

is zero unless

[13\

[~3

is a u n i o n of skew-hooks. (ii)

The m u l t i p l i c i t y of [I] in [9][xtl r-x] is the b i n o m i a l coeffim-i (c_x) if [ I ] \ [9] is a union of m d i s j o i n t skew hooks h a v i n g (in

cient total)

c columns

(and r nodesl.

Proof:

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

appears

in [9][x,l r-x]

can replace

the nodes

in such a way (i)

if and only in [I] \[9]

Rule

assures

us that

the d i a g r a m

if [9] is a s u b d i a g r a m by x l's,

one

2, one

of

[I]

[I] and we

3,...,

one

(r-x)

that

Any c o l u m n

containing

a 1 has

just one

i, w h i c h

is at the top

of the column. (ii)

For i > i, i+l is in a later row than i; in particularp

two numbers (iii) (iv)

greater

The

than

1 are in the same

first n o n - e m p t y

Any row c o n t a i n i n g

row contains a number

no

row. no n u m b e r

greater

than

greater

than

i.

1 has it at the end

of the row. Suppose Then

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

[i] \ [9] does not contain

of [i] ih [9][x,l r-x]

four nodes

is non-zero.

in the shape

X X X X since n e i t h e r (by

left hand node

can be r e p l a c e d by a n u m b e r

(iv)); nor can they both be r e p l a c e d by

1 (by

(i)).

greater

than 1

Therefore,

[I] \ [~] is a u n i o n of skew hooks. Suppose

that

[I] \ [~] is a union of m d i s j o i n t

skew-hooks,

having

82 c columns.

When we try to replace the nodes

in [I] \ Iv] by numbers, we

notice that certain nodes m u s t be r e p l a c e d by l's and others by some numbers b > i, as in the f o l l o w i n g example 1 1 b 1 1 X 1 1 b b c = ii, m = 4

1 1 X

b X

b b Each column contains at most one 1 (by (i)).

Also, each column

contains at least one i, e x c e p t the last column of the 2nd, mth components forced.

(by (ii),(iii)

There remain

m-i spaces

and

(x-c + m-l)

(iv)).

Therefore,

(c-m+l)

l's are

l's w h i c h can be put in any of the

left at the top of the last columns in the 2nd,

components.

3rd,...,

3rd,...,mth

The p o s i t i o n of each number g r e a t e r than 1 is d e t e r m i n e d

by

(ii) once the l's have been put in. m-i [v][x,l r-x] is therefore ~ - c + m-1 > =

The m u l t i p l i c i t y of [I] ()m-i "c-x" as we claimed.

in

P r o o f of the M u r n a ~ h a n - N a k a y a m a Rule: Let X [v][~]) av~ = (XI + ~(n-r,r) , and v is a p a r t i t i o n of n-r.

w h e r e ~ is a p a r t i t i o n of r

If p is an r-cycle and z is a p e r m u t a t i o n of the r e m a i n i n g n-r numbers,

then

X (~P)

But

au,(x,lr-x)

=

7~

avp x v(#)

=

VZ XV(#)

XU(p)

r x=£1av' (x,lr_x)(-i) r-x,

by 21.4.

= (X l, X [v][x'Ir-x] ) by the F r o b e n i u s R e c i p r o c i t y Theorem =

(m-l)

by L e m m a 21.5.

c-x

The d e f i n i t i o n s of m and c give r • c > m, so m-i = (-l) r - c { ( % 1 ) x=l (c-x) (-i) r-x =

{(-1)r-c

O

-

(mll) + ... _+ (m-l) } m-i

if m = 1

if m ~ i.

83 However, w h e n m = i, Ill \ Iv] is a single skew r-hook of leg length r-c.

Therefore, X (~P) =

Z {(-l)ixg(~) I Ill \ [9] is a skew r-hook of leg length i},

w h i c h is the M u r n a g h a n - N a k a y a m a Rule. 21.6

COROLLARY

Suppose p is a prime.

for Ill is d i v i s i b l e b? Pr then X

If no entry in the hook graph

is zero on all p e r m u t a t i o n s w h o s e

order is d i v i s i b l e by p. Proof:

The h y p o t h e s i s

shows that no skew k p - h o o k can be r e m o v e d from

Ill, so the M u r n a g h a n - N a k a y a m a Rule shows that X tations c o n t a i n i n g a k p - c y c l e Remark that

is zero on all permu-

(k > 0).

The h y p o t h e s i s of C o r o l l a r y 21.6 is e q u i v a l e n t to the s t a t e m e n t

I ~ n I / deg X ~

is coprime to p, by the Hook Formula.

ary t h e r e f o r e illustrates

irreducible c h a r a c t e r of a group G and

IGI / deg X

then X is zero on all p - s i n g u l a r elements of G. m o d u l a r theory,

The Coroll-

the general t h e o r e m that if X is an o r d i n a r y is coprime to p,

(In the language of

X is in a b l o c k of d e f e c t O.)

The M u r n a g h a n - N a k a y a m a Rule can be r e p h r a s e d in a way w h i c h is useful in n u m e r i c a l calculations, 21.7

THEOREM

ter of

e s p e c i a l l y in the m o d u l a r theory for ~ n "

If 9 is a p a r t i t i o n of n-rf

then the ~ e n e r a l i s e d charac-

~ n c o r r e s p o n d i n g to {(-1} i[~]

I [~] \ [~] is a skew r-hook of le~-len~th i}

is zero on all classes e x c e p t those c o n t a i m i n ~ an r-cycle. Proof:

Suppose that [~] is a d i a g r a m a p p e a r i n g in [V]([r] - [r-l,l] + Jr-2,12]

Then, by L e m m a 21.5,

- ... ± [ir]).

Ill \ [9] is a union of m d i s j o i n t skew hooks and

its c o e f f i c i e n t is m-I r-x x=l (c-x) (-i) As before,

this is

(-i) r-c if m = i, and zero if m ~ i.

Therefore

[~]([r] - [r-l,l] + Jr-2,12] - ... ± [ir]) = Z {(-l)i[l]

J Ill \ [~]

But, by definition, all of

~n

is a skew r-hook of leg length i}.

X ~ X (r)-(r-l'l)+

except the s u b g r o u p

here, e x c e p t on ~p

~(n-r,r)"

"''±

(lr) ÷

However,

(p an r-cycle), by T h e o r e m 21.4.

~n

is zero on

it is zero even

84

Remark: wraps 21.8

The proof shows that"the o p e r a t o r

x

x

(i)

x

'~{hen 9 = _

XX

x

x

(3,2)

x

and r = 3

_

XX..

+

x x x

XX

XX

shows the ways of w r a p p i n g skew 3-hooks on to [3,2]. c h a r a c t e r X (6'2) of

- X (4~) - X (3'22'I)

+ X (3'2'I~)

The g e n e r a l i s e d

is zero on all classes

~8 e x c e p t those c o n t a i n i n g a 3-cycle. (ii)

of

...-+ [ir]"

skew r-hooks on to the rim of a diagram". EXAMPLES

+

[r]" - [r-l,l]'+

For n a 4, X (n) + X (n-2'2)

- x(n-2'12)is

zero on all classes

~ n e x c e p t those c o n t a i n i n g a 2-cycle. These e x a m p l e s s h o w that X (6'2) + X (3'2'13)

= X (42) + X (3'22'I)

as

a 3 - m o d u l a r character, since this e q u a t i o n holds on 3 - r e g u l a r classes, and X (n-2'12) = X (n-2'2) + X (n) as a 2 - m o d u l a r character. At once, it follows that X (n-2'12) ~n"

Also,

block of 21.9

X (n-2,2) 3 and x(n) are in the same 2-block of X (6'2) , X ~3'2'I ), X (42) and X (3'22'I) are in the same 3-

~8'

since

THEOREM

Let Z a A X

A

= 0 be a n o n - t r i v i a l

characters on p - r e g u l a r classes.

relation between

Then a A is n o n - z e r o

for

some p-

s i n g u l a r A t and if aA is n o n - z e r o for just one p - s i n g u l a r A t then all the characters w i t h n o n - z e r o c o e f f i c i e n t s are in the same p-block. Proof:

If the only n o n - z e r o c o e f f i c i e n t s b e l o n g to p - r e g u l a r partitions,

consider the last p a r t i t i o n p whose c o e f f i c i e n t ap is non-zero. c h a r a c t e r X p contains a m o d u l a r i r r e d u c i b l e c h a r a c t e r ~ tc the factor D ~ of S ~. By C o r o l l a r y

12.2,

other o r d i n a r y c h a r a c t e r in our relation,

~

The

corresponding

is not a c o n s t i t u e n t of any

and this contradicts

the fact

that the m o d u l a r i r r e d u c i b l e c h a r a c t e r s of a grouparelinearly~dependent. If the p a r t i t i o n s w i t h n o n - z e r o coefficients p-block,

lie in more than one

then there are two n o n - t r i v i a l s u b r e l a t i o n s of the given one,

and each s u b r e l a t i o n must involve a p - s i n g u l a r partition, have just proved.

by w h a t we

The T h e o r e m now follows.

A l t h o u g h it is fairly easy to prove that all relations b e t w e e n the o r d i n a r y characters of

~n'

r e g a r d e d as p - m o d u l a r characters,

come from

a p p l y i n g T h e o r e m 21.7, there seems to be no way of c o m p l e t e l y d e t e r m i n ing the p-block s t r u c t u r e of

~n

along these lines.

85 21.10 EXAMPLE It is an easy e x e r c i s e to prove from the M u r n a g h a n N a k a y a m a Rule that w h e n n = 2 m is even X (n) _ x(n-l, I) + X (n-2,2)

_ ... ± X (m,m)

is zero on all classes of ~ n c o n t a i n i n g an odd cycle. Hence x(n) , x(n-l,l) ,..., X (m,m) are all in the same 2-block of ~2m' by T h e o r e m 21.9. This is a c o n v e n i e n t point at w h i c h to state 21.11

THEOREM

("The N a k a y a m a C o n j e c t u r e " ) .

same p-block of of {ir2r...}

~n

S ~ and S 1 are in the

if and only if there is a (finite)

permutation

such that for all i

~i - i ~ ~ i a - ic

modulo p.

We do not prove the N a k a y a m a C o n j e c t u r e here - the i n t e r e s t e d reader is r e f e r r e d to M e i e r and T a p p e

[17] w h e r e the latest proof and refer-

ences to all e a r l i e r ones appear.

It seems to the author that the

value of this T h e o r e m has b e e n overrated; not essential)

it is c e r t a i n l y useful

w h e n trying to find the d e c o m p o s i t i o n m a t r i x of

(but

~ n for

a p a r t i c u l a r small n, but there are few general theorems in w h i c h it is helpful.

In fact, there is just one case of the N a k a y a m a C o n j e c t u r e

B e e d e d for a T h e o r e m in this book, 21.12 of ~ n Proof:

LEMMA

and we prove this now:

If n is odd¢ S (n) and S (n-l'l)

are in d i f f e r e n t 2-blocks

" Let ~ =

(i 2)(3 4)... (n-2,n-l).

is the c o n j u g a c y class of

~n

X (n-l'l) (~) = O, by Lemma 6.9. I ~I

General theory

x(n)(w) X (n) (i)

~

(see Curtis

S (n) and S (n-l'l)

Then

c o n t a i n i n g ~.

I 6~I

But

is odd, w h e r e

6W

X (n) (~) = 1 and

Therefore,

I ~I

~(n-l,l)(w) X (n-l'l) (i)

mod 2.

and Reiner [2], 85.12)

now tells us that

are in d i f f e r e n t 2-blocks.

The proof we have given for the M u r n a g h a n - N a k a y a m a Rule has been desiqned to d e m o n s t r a t e the way in which

skew-hooks come into play.

The Rule can also be deduced from the D e t e r m i n a n t a l Form,

and we conclude

this section with an outline of the method. 21.12

LEMMA

Suppose ~ a t

~p ~ ~ n where

p e r m u t a t i o n of the r e m a i n i n g n-r numbers. p a r t i t i o n of n.

~ is an r-cvcle and ~ is a Let

(Dl,~2,...,~n)

be a

Then

x[Ul][~2]'''[Un](z0)

= n~ x [ U l ] [ ~ ] . . . [ ~ i _ l ] [ ~ i _ r ] [ U i + l ] . . . [ ~ n ] ( ~ ) " i=l

86

Proof: X[HI]'''[Hn](zD) = the n u m b e r of H-tabloids fixed by ~p n = i~l (the n u m b e r of u - t a b l o i d s fixed by z in w h i c h all the numbers moved

by p lie in the ith row) t

since

a H-tabloid

is fixed by

p if and

only if each orbit of O is c o n t a i n e d in a single row of the tabloid. n = i~l (the n u m b e r of ( H l , ~ , . . . , U i _ l , H i - r , U i + l , . . . , ~ n ) - t a b l o i d s fixed n = i~l x [ U l ] [ U 2 ] ' ' ' [ ~ i - l ] [ ~ i ' r ] [ H i + l ] ' ' ' [ H n ] ( z ) , As usual, 21.14

[k] is taken

EXAMPLE

p is a 5-cycle

to be zero

(cf. Example

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

X(5'~'4) (~p) : the c h a r a c t e r

of

=

=

[0]

[i]

[2]

[3]

[4]

[5]

[2]

[3]

[4]

[3]

[4]

[5]

[2]

[3]

[4]

[0]

[I]

(X (3,3p2)

[2]

[7]]

[3]

[4]

[5]

should

[6] [-i] [3]

[7] [0] [4]

[5] [2] [-2]

[6] [3] [-i]

[7] [4] [03

the above

[4]

evaluated

[5]

[6]

{7]

[3]

[4]

[5]

[-3 ] [-2]

[-I]

+

example,

w~th

Rule

where

8 numbers.

Then

at zp, by the

Determinantal

Form

at ~, by Lemma

21.13

at

by the D e t e r m i n a n t a l

the M u r n a g h a n - N a k a y a m a

have no d i f f i c u l t y

[3]

= O.

that np ~ ~ 1 3

of the r e m a i n i n g

[6]

[5] [-~] [2]

- X (5,3,0)) (~),

By i n s p e c t i n g to prove

+

Suppose

[5] [?]

=

if k < O, and xO(z)

21.q(i)).

by ~) to show.

as we w i s h e d

Form.

the r e a d e r will

see w h a t

from the D e t e r m i n a n t a l

the details.

is r e q u i r e d

Form,

and

87

22

BINOMIAL

COEFFICIENTS

In the n e x t sentations

of

~n

problems

which

divides

certain

collected 22.1

couple

of s e c t i o n s ,

over

arise

a field

depend

binomial

in t h i s

of

upon

we

finite

Suppose

0 -< n i < p a n d n r ~ O.

Then

= max

(ii) (iii)

Op(n) £p(n)

= n o + n I + ... + n r = r + i.

define

22.2

{iln j = 0

rational

9p(O),

LE~,tMA

repre-

Many

of the

p.

or not

relevant

to the

the p r i m e Lemmas

p

are

number

but we

~_p(n')

for

n/m,

let Op(O)

=

+ ... + n r p r w h e r e ,

for e a c h

i,

let

9p(n)

a positive

whether

a n d the

n = n O +nlP

(i)

not

characteristic

deciding

coefficients,

put our mind

section.

DEFINITION

For

shall

(n-

j < i}

let ~p(n/m) = ip(O)

Op(n))/(p

= ~p(n)

- 9p(m).

W e do

= O.

- 1).

Proof: T h e r e s u l t is t r u e f o r n = O, so we m a y a p p l y i n d u c t i o n . If r p r _ l n = p , ti~en ~p{ ( ) '} = ( p r - l - r ~ + r ) / ( n - l ) , b~f i n d u c t i o n . B u t ~ (pr, ) = r+u Assume, (x)

{(Dr-l) '} =

( p r - l ) / ( D - 1 ) , a n d the

therefore,

that O < n-p r < r+l r+l r for O < x < p - p ,

result .

is t r u e in this Vp (pr+ Since x)

_ pr

case. =

P ~p{n(n-1)... qherefore

~p(n')

= 9p(pr,) =

(pr

by i n d u c t i o n ,

and this

22.3

Assume

L~[,~4A

Proof:

We m a y

(pr+ i) } = Vp{ (n-p r) '} •

_

+ ~p{(n 1 +

n

pr

_

Op(n)

is the r e q u i r e d

a -> b > O.

apply

-

- pr) :}

induction

Then

+

l)/(p-l),

result. ~ p ( b ) < ip(a)

o n a, s i n c e

the

- ~p(m).

result

is t r u e

for

a = i. If P I b,

let D'

a - a° modulo

p.

= b/p

Using

a n d a' =

the

~ p ( b ) = {~p(b)

=

Vp(b)

< ~p(a')

= ~p(b')

Vp(b,)

(a-ao)/p , where

Lemma,

+ Op(a-b)

= {~p(b') a' a' But ~p(b,)

last

- Op(a)}/(p

+ Op(a'-b')

0 < a O < p and

we have - i)

- ~p(a')}/(p

- l)

•

~p(b') ' by i n d u c t i o n ,

+ l, so ~ p ( b ) < %p(a)

and

- Vp(b),

~ p (a) = ~p(a')

in this

case.

+ l and

88

Now

suppose

that

Vp(b)

= O.

the

result

p ( b _al ) < £p(a)

is true

(b)

a+b-i ~-

-

a (b_l) ,

a + Vp(b_l ) .

Vp(b ) = Vp(a-b+l) Because

Since

for b = i, we may

- Vp(b-l).

~lence,

unless

assume

that

Vp (a-b+l)

> O,

b > i, and

V p ( b ) < £p(a). But

if v

(a-b+l) P

by -

the

first

9p(b)

22.4

in this

(b)

case

Assume

As

of the proof.

- Vp(a-b+l), Therefore,

< Zp(a)

that

r ... + arP" r ° + b l P ' + ... + bz "p

(0 -< a l• < p )

b~b

(O -< b I• < p ) .

. ..

ar (br)

modulo

if a i <___bi for some a polynomial

over

p.

In p a r t i c u l a r r p d i v i d e s

the

field

of p e l e m e n t s ,

we h a v e

(xpr+l)ar the

result.

22.5 COROLLARY A s s u m e a a b a 1. T h e n all the b i n o m i a l (~) a-1 ,a-b+l '(b-i ) r'''q~ 1 ) are d i v i s i b l e by p if and o n l y if

Proof:

By c o n s i d e r i n g coefficients a-b+l, 1 J'

Then

the

last

coefficients

~ -I m o d pZp(b)

binomial

Pascal's

if a n d o n l y

Triangle,

p divides

if p d i v i d e s

,a-b+l, .a-b+l [ 2 ;'''''{ b

sentence

(~)

i.

(x+l)ao(xP+l)al ... b c o e f f i c i e n t s of x , we o b t a i n

a-b

= Zp(a)

also.

(x+l) a =

Comparing

Vp(b)

a = a ° + alo~ +

- (a°) bo (al) bl

if and o n l y Proof:

then

paragraph

LE~R4A

Then

> 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
23.2

(~' b e i n g the par-

~ECt7 (sgn ~)~.

Let D t

w~Rt

LE~VuV~ Let the ~round field be Qf and t be a B-tableau. (i)

(ii)

Then

The ~reatest common divisor of the c o e f f i c i e n t s of the tabloia~

i n v o l v e d in {t}Ktp t and

in S ~

to ~, and < , >' b e i n g the b i l i n e a r form on M~').

is ~ ' .

{t}KtPt< t = H (hook lengths in [~])

{t}
!

Proof: (i) By definition, tion ~varies. But

g~

= g.c.d.

< et, ,et, ~ >' as the permuta-

!

sgn ~ < e t , , e t , w

>

= sgn W < {t'},{t'}Kt,~Kt,

>

= 7 {sgn~ s g n ~ s g n ~ J~, T ¢ Ct, , ~ ~ ~ ~ Rt,

}

= T {sgn w JT ( Ct,, ~ x -I 7 -1 ¢ Ct,, ~ ( Rt, }

g0

= 7 {sgn ~ I T E R t , w T-I - i = < {t},{t}Ktp t - i

E Rt, ~ ¢ C t }

>

= < {t}~, {t}<tp t > and result

(i) follows.

(ii) Corollary To evaluate remarks

4.7 shows that {t}
at the end of section

The right ideal a complementary

4).

We have

PtKt ~ ~n of

Q ~n

(See the

Pt
(which is isomorphic

right ideal U, by Maschke's

Multiplication ~n"

for some c ¢ ~.

c, it is best to consider the group algebra Q ~n"

on the left by Pt
to S u) has

Theorem. a linear transformation

Taking a basis

for PtKt Q ~n'

followed by a basis of U, this

linear t r a n s f o r m a t i o n

is r e p r e s e n t e d

by the matrix

dim S ~

of

0

0 On the other hand,

taking the natural

the linear transformation diagonal,

is r e p r e s e n t e d

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

the product

{wI~ E

~ n } for ~ n ,

by a matrix with l's down the occurs with coefficient

1 in

PtKt .

A comparison the dimension

of traces

gives c dim S p = n~

of S ~, c = H(hook

lengths

Since {t}Ktp t ~ = {t~}Kt~ Pt~' llary 8.10 show that we may give: 23.3

basis

DEFINITION

Suppose

By the Hook Formula

for

in [p]).

the first part of the Lenuna and Coro-

that F is the field of p elements.

Let 0 be

the n o n - z e r o element of H o m F ~ n ( M P , S p) given by e : {t} ÷ where 1gP'

(I, gP

{t}
P

this means that the image of {t} is obtained

{t}KtP t

23.4

from the vector

in S ~ by reducing all the tabloid coefficients

modulo p.

THEOREM (i)

If Im 0 c S p

e~uivalentl~

if Ker 0 ~ S ~I ~ then S p is reducible.

(ii)

If Im 8 = S ~

equivalently

if Ker 8 = S P l r

HomF~n(SPrSP)

~ F r then S ~ is irreducible.

and if

91

Proof:

If F = ~,

the

{t}~

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

$ defined

= ~p'{t}<tp t

sends

{t}< t to a n o n - z e r o

multiple

of i t s e l f ,

fore

dim Ker

, and by

the S u b m o d u l e

By L e m m a Therefore, The

Ker

Ker

first

submodule

and

~ = d i m S ~I~

8.14,

8 ~ S p±, w h e n

0 ~ S ~I part

0 = S ~±, (ii)

23.5

if and o n l y

of S ~ in this

result

THEOREM

and o n l y

we work

of the T h e o r e m

If Ker

by

over

23.2(ii).

Theorem,

the

if

Im @ c S ~.

is n o w

trivial,

Ker

There-

~ = S~

.

field

of p e l e m e n t s .

since

Im 8 is a p r o p e r

case.

then

8 gives

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

follows

from Lemma

Suppose

that

if p d i v i d e s

by Lemma

the

between

M ~ / S ul and S ~,

23.1. Then

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

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

if

integer !

{H Proof:

The

if and o n l y unique

if K e r

8 ~ S ~.

= =

by L e m m a

23.2

and o n l y

if p d i v i d e s

(ii).

EXAMPLES

(~ is p - r e g u l a r is in a b l o c k (ii) does

not

~(~ook where

since

if

Ker

that

~ is p - r e g u l a r , (by T h e o r e m

S ~ is r e d u c i b l e M P / S ~± has

4.9).

a

Therefore,

0 = S~

(H(hook Since the

lengths g~'

in

S ~ is a c y c l i c integer

module,

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

lengths gP'

in

divide

~(hook

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

p and

g~'.

(cf. T h e

~' are

Thus

[~]).

not

This

Hook

p-regular,

instance,

is

if

[~]).

lengths just

the

in

[p]),

then

case w h e r e

Formula).

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

For

{t}
H(hook

and)

O

[~])

p

If p does

in

show

(!p, p {t}
of d e f e c t

divide

13.17

then

from Corollary

if a n d

only

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

10.5,

If p =

{t}Ktp t =

(3,2)

and

t = 1 2 3 , Wen 45

1~~4

direct

p

if p d i v i d e s of ~ =

((p-l) x)

1 < x < p.

(iii)

that

But,

(i)

If b o t h

lengths

[~])}/g~

(S u + S P ± ) / S ~i

if a n d o n l y

{t}< t 8

in

and C o r o l l a r y

submodule

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

23.6

lengths

last T h e o r e m

minimal

But

(hook

computation

shows

92

The duct if

g.c.d,

of

the

char

Example 23.7

F = 2 or

3.

5.2,

when

the of

the

and

of

When

ducible.

that

is

~ is

so is

is

g~'

the

= -F(-4)

But if

vector

the

and

called

pro-

only F in

- F(-5).

partition!

S ~ is

= 4.

reducible

and

irreducible

let

if

and

S ~ be only

define d if

holds: ~ =

(n)

or

(i n)

{ n and

p =

(iii)

p

{ n and

p

we

4, S#

{t}0

a hook

Then

p

Thus,

so

3 , {t}8

(ii)

S (n)

is

24,

F = 2,

F =

p elements.

followin~

Since

[~J

char

char

(i)

Proof:

coefficients

in

Suppose

field

the

edge

lengths

THEOREM

over one

of

hook

and

S (In)

may

assume

(n-l,1)

or

(2,1 n-2)

# 2.

have

dimension

that

~ =

i,

(x,lY)

they with

are

certainly

x > 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 } <

(x - i) ~ b y

@ be

-

lengths

1 -- , { t } K t P t ( l g~ Let

(12)

of

(i

(y+l))

23.3. =

=

(x+y){t}K t

.

Then

(x+y){t}Kt,

p elements.

This

shows

that

93

But Hom F ~ 13.17.

Using

(S~,S ~) ~ F if p ~ 2 or if ~ = ~emma

a n d p ~ 2 or ~ = Next

23.1,

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

(n-l,l)

suppose

that {t)(l

(also w h e n

p I n.

Then

-

-

(12)

(y+x)

~ =

(n-l,1),

in the

by C o r o l l a r y

cases

where

(2,1 n-2) , by T h e o r e m

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

E Ker

(y+x - i) ... (y+2)

p ~ n

8.15) •

0

1

2 Let

t* = (y+l)

Since

x > 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 <

ana

U p ( ( a r + 1 + I)P r+l

tion

gives

x+l

< x+l

(ar+ 1 + l)p r+l + ar+2 pr+2

+

a r ~ 0 ~ as).

... + asps

+

... + as ps ) > 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


as w e w i s h e d

I is p - r e g u l a r

applications

the Corollary

(cf. E x a m p l e

to SP,

to t h e d u a l

D 1 is a c o m p o s i t i o n

is a f a c t o r

= bp(~i

Altogether,

If w e

{t}@T~i-l,v

of S ~ is a c o m p o s i t i o n

factors

COROLLARY

k,k+l,...

belongs

the h y p o t h e s i s

D (4'12)

24.9

{t}
of S p e c h t m o d u l e s

EXAMPLE

in r o w s

of S 1 w h e n

are very many

factors

all

i-i

of t h e T h e o r e m ,

Under

factor

but we

sition

to

thus,

submodule

hypothesis,

There one,

(i-l) 's in r o w k.

k,k+l,...,

~ Nms s=k

M I / S ~± is i s o m o r p h i c

unique

to

to i - 1 i n rows

T 2 has

i~l l k + m - l ' it f o l l o w s t h a t s o m e c o l u m n of T 2 c o n t m=l numbers. T h e r e f o r e , T 2 is a n n i h i l a t e d b y
a n d 0 ~ v ~ ~i - i; Since

i-i E m=l

- vk +

or e q u a l

identical

that

changed

equal

Similarly,

at l e a s t

Nik numbers

- vk n u m b e r s

to m i n r o w s k , k + l , . . . to T.

reverse

semistandard

Our hypothesis

@T b e l o n g s

~-tableau and

to H o m F ~ n ( M B , S

~)

the

T of t y p e

first

and K e r

part

p has

of T h e o r e m

@T ~ S~l

105

By d i m e n s i o n s ,

M P / S ~± ~ S ~.

so ~ is c e r t a i n l y When

p = 2,

of the a b o v e of the

diagram

following

To d e s c r i b e if we can the each

node

of ~ m u s t

result

to v e r i f y

two d i f f e r e n t

contains

Conjecture

special

from

case

through

23.1.

the h y p o t h e s i s

statement

of T h e o r e m some

that

no c o l u m n

numbers;

24.6,

number

to the end of the

a multiple

decreasing,

from Lemma

cf.

the

23.17•

[p] by m o v i n g [~]

strictly

that

[~]2

ith r o w of

is m o v e d

be

follows

to the

the C a r t e r

[I]

now

is e q u i v a l e n t

another

obtain

end of the

parts

The

it is s t r a i g h t f o r w a r d

Corollary

2-power

comments

The

p-regular.

of plp(d)

we w r i t e

p ~ 1

d > 0 of n o d e s

from

(i-l)th

r o w of

[~] and

spaces•

(See E x a m p l e

24.11). 24.10

COROLLARY

partitions

Let

p(1) If 1 < a < < r - b Proof:

We m a y

restrict

char

F = p and

~(i),~(2) ,...,~(r)

our

k ~(2) ~-i

and

(3) k~2

... k - r ~ 2

I = ~(b) , ~ = ~ (a)

suppose

that

attention

and

a = 1 and b = r,

to the

piP (di)

sequence

let T be

the

~(a)

to v e r i f y

. (i) Ni_l, 1 = ~i-i

so T h e o r e m

24.6(ii)

~(r)-tableau

EXAMPLE .....

Suppose 4

.

the

.

.

~(b)

~(k-j+l)

= di

we may

~ ~(k-j+2)

By c o n s t r u c t i o n ,

+ d

of type

for all

+ 1

i

for

j ~ 2,

p(1)

in T h e o r e m

semistandard).

if j = i, and 0

(i) - Pi - di+l

+ d

i

if j ~ 2.

{ -i rood p £ p ( d i ) '

result.

.

.

3

.

.

.

.

2

.

-~

Q

•

•

Q

•

•

I

Q

•

÷ •

X

Hom F~II(S

otherwise

~ O.

char F = 3

-~

Therefore,

since ÷...÷

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

(r) (i) - ~i = ~i-i gives

~(r)

(i) (I) ~i-i - ~i - di+l N (r) . _ p(r) ij = ~i+j-± i+j

i-i m =~l (~j+m-l(r) _ s=j~Nms)

Now,

24.11

and

corresponding

(It is s i m p l e

Also,

divides

• (i) (r) = ~i - Pi+l

Let Nil

24.6

(proper)

H e m F ~ n ( S I , S p)

then

Let d. be the n u m b e r of n o d e s m o v e d in 3 defining d = O if j > 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 <- i < j-l, by C o r o l l a r y S (n-js'Js) Applying

24.16

again,

every

is a f a c t o r of S (n-js'js)

B u t D (n-j'j) 12.2,

is not a f a c t o r of

so D (n-j'j)

f a c t o r S (n-js'js)

is a f a c t o r of

is a f a c t o r of

108

S (n-js-l'js-l) D (n-j'j)

or of one

is a f a c t o r

Jo = m, we h a v e 24.17

When Next,

of

{S (n-i'i) IO < i -< j-l}

of s ( n - j s - l ' J s - i ) .

Continuing

= i, D (n-j'j)

consider

case w h e r e

m-i

the

is a f a c t o r

n - m-i

mod

r-i

b

=

ao+

alP

+

...+

ar_IP

n = ao+

alp

+

...+

ar_iP

= 0 if m = pr

r fp(n-l,m-l)

= i.

Returning 24.18 some

Thus,

Similarly, to the

If m -> 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

<

Then

r

n contains

1 -< j -< m

s(n-m'm) and D ( n - j ' J ) %

say m = pr.

to

of S (n-m~m)

(O -< a I. + brP

fp(n-l,m)

fp(n-l,m)

j with

multiplicity

argument

plp(m)

r-i where

this

proved

fp(n-2j,m-j)

so

Therefore,

then

there

is a f a c t o r factor

is

of

D (n-l)

with

+ fp(n-l,m-l). consider

inequality

first

the

fp(n-l,m)

case w h e r e

m is a p o w e r

+ fp(n-l,m-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_<13.

114

25

YOUNG'S We

the

turn

action

poned

ORTHOGONAL now

theory

reference

stage

of

~n

to the

Since

~n

of S U

.

in o r d e r can

finding

the m a t r i c e s

module

to e m p h a s i z e

(and we b e l i e v e

the

which

represent

S U.

This

has b e e n

fact

that

the

should)

post-

represen-

be p r e s e n t e d

without

matrices.

is g e n e r a t e d

by the

to d e t e r m i n e

Consider

of

on the S p e c h t

representing

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

to the p r o b l e m

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

to a late

tation

FORM

first

the

transpositions

action

the b a s i s

(x-l,x)

of these

for

i < x ~ ~,

transposition

of s t a n d a r d

on a

polytabloids

e t.

Here we have 25.1

(i)

If x-1

and x are in the

same

column

(ii)

If x-i

and x are

same

row of t,

+ a linear

combination

(by c o m b i n i n g (iii)

8.3

by

applying

25.2

(ii),

the

EXAMPLE

order

and the

If t(x-l,x)

In case

in the

of s t a n d a r d

the

relevant

Garnir

relations.

If U =

(3,2)

24

o

o

0

o

1

O

O

O

(1 2) <-->

0

0 -i

O

0

O

O -i

1

O

-i

O

1

10

(3 4) <->

-i

O

O

O

O

-

1

O

O

O

-

0

1

O

O

O O

O 0

O 1

1 O

oO

In m a n y ways, is the b e s t w a y for e x a m p l e , cases

into

involves

lems

can be rest

and the

an u n p l e a s a n t

avoided

of this

when

section

t* m a y be

1 2 4

1 2 3

25

35

45

0

O

O

O

O

1

O

O

1

O

1 -1

O

O

where

over

the

be d e v o t e d

in the

O

1

O

0

O

O

10

1

O

O

O

O

1

O

O

0

1

0 -i

O

1

as this

represent

Ilowever,

~of

out

is called, permutations;

we m u s t

and x are

It turns

to the

O

O

x-i

field

then

O

O

which

field.

one,

calculation.

we w o r k

1

representation,

of the

second

calculated

O1OO l

1 3 4

34

the m a t r i c e s

= et ~ {t}

u-tableau

1 2 5

natural

will

{t*}

8.9).

the s t a n d a r d

(4 5) <-->

it is i n d e p e n d e n t

et(x-l,x)

= -e t.

= et(x_l,x)

tableaux

(2 3) <-->

of d e s c r i b i n g

account,

row,

the

Young's

to p r o v e

take

et(x-l,x)

et, w i t h

et(x-l,x)

standard

and we

tl,t2,t3,t4,t 5 = 1 3 5

-1

used then

then

then

polytabloids

technique

is s t a n d a r d ,

of t,

that

real

case w h e r e

take

in the these

numbers, the

three same

proband

~round

115

field

is JR.

given

by d e f i n i t i o n

Let

viation

t I < t 2 <...<

e i for the

Since

we

t d be

3.10. standard

are w o r k i n g

e 2 , . . . , e d an o r t h o n o r m a l Schmidt normal To

fix

that w e

notation,

lization

we h a v e

by e l , . . . , e j o v e r form

<

and t h a t >

{tj+ I} is i n v o l v e d

This

being

=

(±f)/(<

uniquely.

25.3

The

THEOREM

standard

the s t a n d a r d ~

, ~iven same

YOUNG'S

basis

the

~12 + P22 = 1

x-i

e1,

Gram-

to the n e w o r t h o -

orthogona-

time,

It does

Young's

x are in the

a linear

the

combinathat

take

a positive

of S ~

coefficient

depends

total

on the o r d e r

we p r o v e of S ~

of the t o t a l the

in fj+l"

constructed

order we

order

contains

choose

on

the p a r t i a l

FORM

= Plfr and

in

~n,

then

for

all

r

+ P2fs

pl (= Pl(X,r))

and x is in the

equals (k,£)th

(i-k+

£_j)-i

if x-i

position

of t r,

ts equal

to tr(X-l,x)

is in

and

P2 > O. not matter same

Orthogonal

same

be

fact

Now,

we prove

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

with

1.3) .

the

to the

combination

(see

fl,f2,...,fd

is i n d e p e n d e n t

3.11

position

we m a y

spanned

relative

linear

f would

However,

basis

by d e f i n i t i o n

and x are in the

cases.

from the

o f the s p a c e

contradicting

.

that

t s = tr(x-l,x) (i,j)th

Remark:

abbre-

"

provided

fr(x-l,x) where

the

permutations.

Gram-Schmidt

1 < _ i < _ j

fl,f2,...,fd

orthonormal

ORTHOGONAL

(x-l,x)

8.9,

el,e2,...,e d

tableaux,

At the

If

respect

are o r t h o n o r m a l

{tj+ I} has

the n e w b a s i s basis

order

using

fl,...,fj

(othe~;ise of

f'f >)%

fj+l

Of course,

the

use

construct S~

is a n o n - z e r o

Therefore,

so t h a t

of the o r i g i n a l

25.4

in f

chosen

determines

from

of the

a basis

there

the p r o o f

> = 0 for 1 ~ i ~ j.)

sign

reader

< ei, f > = O for

with

fj+l the

of

representing

fl,...,fj

Then

t i o n of e l , . . . , e j by < ei,f

we m a y

It is w i t h

matrices

the

in the o r d e r

shall

eti

reals,

fl,f2,...,fd

constructed

I~ ,

f of e l , . . . , e j + 1 tabloid

we

process.

Suppose

bilinear

the

"nice"

remind

~-tableaux,

possible,

process.

get

we

standard

polytabloid over

basis

orthogonalization basis

the

Wherever

that

there

row or c o l u m n Form

row o r c o l u m n

says

is no

of t r, s i n c e

that

fr(x-l,x)

of t r, r e s p e c t i v e l y .

when

P2 = 0 in these = ±fr

if x-I

and

116

Before preliminary 25.5 x-i

LEMMA is

lower

Proof: less

Suppose than

Recall

than

miu(t)

on the p r o o f s

of 25.3

and 25.4,

we

require

a

If

{t}

a2th

that b 2 < a 2 . U s i n g 3.14,

row

two p - t a b l e a u x , then

first

u rows

and

{t}(x-l,x)

t h a t miu(t)

that

4 {t*}(x-l,x).

is the n u m b e r

of t.

we

alth

row

and x be

Since

in the b l t h

and x be in the b 2 t h

deduce

miu(t*(x-l,x)),

except

or m a x

< u < a I.

(bl,a2)

~ {t*}

3.11

to i in the

any

{t}

of e n t r i e s ~ {t*}

,

for all i and u.

be in the

be in the

t and t* are

x in t*.

or e q u a l

x-i

that

from definition

< miu(t*) Let

x-i

embarking Lemma.

f r o m miu(t)

perhaps

< miu(t*)

for i = x-i

row of t.

row of t*.

We

are

Let given

that miu(t(x-l,x))

and e i t h e r

b I < u < m i n ( a l , b 2)

F o r b I < u < m i n ( a l , b 2) , mx_l,u(t(x-l,x))

= mx,u(t),

since blth

< mx,u(t*),

(bl,a 2)

mx_l,u(t(x-l,x))

since

Assume (Both Step X

<

1

We

{t}

since

4 {t*} b 2 < a2 < u

_< m i u ( t * ( x - l , x ) ) do n o t h a v e

equality,

and Y o u n g ' s

Orthogonal

both

results

true

true w h e n

The m a t r i c e s

which

we

are

n = O). claim

in all

since

~ - i

proof

represent

oases.

Thus

{t} ~ {t*}.

Form:

for all The

.

Specht

now proceeds

(x-l,x)

are

modules in

correct

3 steps. for

n.

We the

that

and x is in the

u < b 2 < a2 .

25.3

are v a c u o u s l y

row

b I -< u < a 1

+ i, s i n c e

miu(t(x-l,x))

of T h e o r e m

, since

mx_l,u(t*(x-l,x)),

{t(x-l,x) } ~ { t * ( x - l , x ) } . Proofs

alth

~ {t*}

+ i, slnce

-< mx_2, u (t*)

Therefore,

in the

t and b I < u < a 1

-< u < a I,

= mx_2,u(t)

=

is

{t}

mx_l,u(t*(x-l,x)

=

For max

x-i

row of

take

our notation

l~n_l-mOdule

spanned

and n is in the rith, the p r o o f

we

gave

for the p r o o f by t h o s e

r2th,...,or

for M a s c h k e ' s

Vi = U1 • U2 • where

U i is the

~n_l-mOdule

et's rith

of T h e o r e m where

shows

so t h a t V i is

t is a s t a n d a r d

row of t.

Theorem

9.3,

Since

~-tableau,

V 1 c V 2 c ...,

that

... • U i, spanned

by

those

ft

!

s where

n is in the

117

r th row of t.

(Recall

that

oar

total

order

on t a b l o i d s

puts

all

those

l

with

n in the In the

@i mapping

rlth

row b e f o r e

all

proof

of T h e o r e m

9.3 w e

V i onto

li W h O S e SIR

@ Ui_ 1 and V i = U 1 ~ . . . ~

Define

where

the

second

irreducible

original

c such

that

Because

both

bilinear

one,

<

,

>

< u,v

>

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<

... + ar re

Therefore,

is

>

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.)

<< {t2

} <<

representing

o r d e r on t a b l o i d s ,

We

..<<

t(i±l)~"

if b o t h

at the b e g i n n i n g of p o l y t a b l o i d s

ei

standard

(n-l,n).

containing

are s t a n d a r d .

fix o u r n o t a t i o n

of { l , 2 , . . . , d }

then ti~(n-l,n)=

hypothesis

the p r o o f of S t e p 2.

of the m a t r i c e s

a new t o t a l

+ brer

Our induction

concludes

...<{t d} are the d i f f e r e n t and

choice

fr is a l i n e a r c o m b i n a t i o n

and {t(n-l,n) } are a d j a c e n t v i e w of L e m m a

our minimal

+ "'" + b r - l f r - i

now t h a t

with

Take

appear elsewhere

B u t m x _ l , l ( t k) = z + 1 > z =

thus pro~L~dthat in the e x p r e s s i o n fr = b l f l

of S t e p

f3 m u s t

that b k is n o n - z e r o .

and this c o n t r a d i c t s

We h a v e

+ brer

(x-l,x)

= -blfl(X-l,x)

line.

"'" + b r _ i f r _ l

by s a y i n g

tabloids

ordered

{td } is the new order. and if b o t h

4 , in w h i c h

{t}

(This is p o s s i b l e that

by d e f i n i t i o n Thus,

tiw and ti~(n-l,n)

in

{~} < {t 2} < 3.10,

z is a are s t a n d a r d

120

We p l a n tr~(n-l,n)

to e v a l u a t e

is

standard,

fr~(n-l,n).

then

Let

G denote

the

Let

X denote

the s p a c e

Let Y = X + e

r~ or not b o t h

on w h e t h e r Since neither

our new

or b o t h

tr~(n-l,n)

group

]RG t

spanned

and

r~

for the m o m e n t ,

that

if

= t(r+l)z.

{i, (n-l,n) }

(so t h a t

total

Assuage,

t

order

r~

by e l z , e 2 ~ , . . . , e ( r _ l ) ~

d i m Y = d i m X + 2 or (n-l,n)

are

contains

e t and et(n_l,n)

~ , for e v e r y

belong

to X

i, d e p e n d i n g

standard.)

(using

standard

25.1).

t,

Hence

both

X and Y are G - i n v a r i a n t . By S t e p

2,

flz,...,f(r_l)z

is an o r t h o n o r m a l

fl~ .... ' f r z ' f ( r + l ) ~

is an o r t h o n o r m a l

basis

d i m Y = d i m X + i).

The

gonal the

complement

space

for Y

space

spanned

by

to X in Y , a n d

because

our i n n e r

spanned

by

frn and

f(r+l) z

bas±s

fr~ and

for X and

(Omit

f(r+l)~

f(r+l)~ product

is G - i n v a r i a n t

if

is the o r t h o is G - i n v a r i a n t ,

(Omit

f(r+l)z

if

d i m Y = X + i) . Now, cient

of

f

= an e l e m e n t of X + b e , w h e r e b > O ( s i n c e the c o e f f i r~ r {try} in fr~ is c h o s e n to be p o s i t i v e ) . T h e r e f o r e , w h e n n-I

and n b e l o n g

to the f

where

I

£ =

these

rz

same

(n-l,n)

and n are in the

same

row

and n are

same

column

have

cases,

and

just p r o v e d comparing

completes

On the

r~

-1 if n - 1

B u t we

this

of trz,

of X + eb e

+l if n-1

fr~(n-l,n) and

row or c o l u m n = an e l e m e n t

other

the

in the

that

fr~(n-l,n)

coefficients

of trz of t

r~

is a m u l t i p l e

of erz,

we

see

of

frz

in

that

= Ef ~ case w h e r e

hand,

when

t

both

(n-l,n) is n o t s t a n d a r d . r~ tr~ and t r ~ ( n - l , n ) (= t(r+l)~)

are

standard, fr~(n-l,n) Since

the s p a c e

= an e l e m e n t spanned

fr~(n-l,n) where that

Pl and

P2 are

P2 is s t r i c t l y

real

by

= Plfrw

and

Pl + P22 = 1

with

P2 > O.

(b > O)

is G - i n v a r i a n t ,

the

coefficient

of

{t(r+l)~}

shows

Now

< fr~(n-l,n),fr~(n-l,n) so

f(r+l)~

+ P2f(r+l) n

numbers,

positive.

of X + b e ( r + l ) ~

fr~ and

>

=

< fr~,fr~

> = 1

Also

fr~ = Plfr~ (n-l'n)

+ P 2 f ( r + l ) z (n-l'n) '

whence f(r+l)~(n-l,nJ It remains, t h e r e f o r e ,

= 0~fr~to s h o w

Plf(r+l)~ that

Pl may be

calculated

as in the

121

statement

of Y o u n g ' s

tr~(n-l,n)

= t(r+l~..

group

5 3 Since

Also,

trz

cases

to c o n s i d e r

n-i

n-2,

n-i

(ii)

Some

are

(iii)

Some

two

(iv) of t

are

in the

be d o n e

in the

so n-i

~ase

using

same

is l o w e r

and n a p p e a r

under some

discussion,

properties

row or c o l u ~ n

than

of the

of trz,

n in trz

where

There

n > 3. are

4

in trn thus:

n,

two n u m b e r s

in the

Form

will

and n are n o t

In-2 I n-i

b u t no

This

~ tr~(n-l,n),

(i)

no two

Orthogonal

same

from

column

two n u m b e r s in the

same

N o two n u m b e r s

{n-2,n-l,n}

are in the same

row,

but

are

column,

of trn

from

{n-2,n-l,n}

row of t from

in the

same

rz

{n-2,n-l,n}

are in the

same

row or colu~nn

r~ We

tackle

comparatively

case

(ii)

trivial.

Case

(ii)

L e t H be

n).

Since

n-i

is

first;

Finally

the g r o u p

lower

than

case we

(iii)

is s i m i l a r

deal w i t h

generated n in t

rz

by , t

the h a r d

gl =

r~

has

and case case

(n-2,n-l) the

(i) is

(iv). and

g2 =

(n-l,

form:

/or /

n

n- ln-1

/ In the n).

The

results action

first

space so

case,

spanned

far s h o w

of H on

this

by

let t = try, ft,ftgl

that,

with

space

is

and

respect

given

and in the

second

ftg 2 is H - i n v a r i a n t . to the b a s i s

by

o gl =

where =

(n-2,n-l)<-~

~i is known,

-(the

have

axial

finished Now,

a2-~i

0

O

1

0

from Step

distance if we

trace

g2 =

(n-l,n) ~

axial

distance

f r o m n-2

can p r o v e

glg 2 = - ~ I T I

Therefore

fact,

1

0

f r o m n-i shall

our

, the

io0o ]

to n-I in t) + i. W e -i -i t h a t ~i = 1 + T1 . - ~i + TI"

In

ft,ftgl,ftg2

[ T2 i. The

let t = t r ~ ( n - l ,

T1 J to n in t therefore

122

Itrace The

glg21

character

s lOl~II

table of

(i 3 )

Case

(3)

i

i

X (2'I) (i 3 ) X

2

0

-i

1

-i

1

representation and

trace

Itracel

of d i m e n s i o n

3 having

trace

of o r d e r

3 is X (3) + X (2'I)

glg 2 = O, g i v i n g

than n'l,

some h in H.

~i = ~i~i

and n-i is h i g h e r

Taking

for ftlRH,

~ ½ + ½ + 1 : 2.

s 2 on e l e m e n t s

(iv) Let H, gl and g2 be as in Case

is h i g h e r

basis

(2,1)

i

The only

+ I~iI

is

X (3)

transpositions Therefore,

~3

+ I~iI

+ ~i

(ii).

1 on the

' as required.

~{e may

assume

than n in t, and that

tr

that n-2 = th for

ft 'ftg] ' f t g 2 ' f t q ~.g ].' f. t.g 2.g l.g ~ ' f t ~ 2 g l ~ 2 g l

gl and g2 are r e p r e s e n t e d

as a

by I

gl =

-v I

v2

~2

Vl

(n-2,n-l) <->

-~i

w2

e2

-~i

71

72

72

-nl

~2 -$i

B2

e2 g2 = (n-l,n)

<-> -YI

Y2

Y2

Y1

62 (Omitted entries are

Values

81

zero) .

Here we k n o w that each of V l , e l , Z l , e 2 , 6 2 , Y 2 is non-zero. The of 91,el and Wl are k n o w n and 9 1 1 + 711 = ell ,

from Step

i.

be no more

~e w a n t

efficient

~i = ~i'

B1 = el

and

Y1 = 91"

this

than e q u a t i n g

g2g I, u s i n g

the fact that glg 2 has o r d e r

3 (cf. T h r a l l

(4,1), (5,2)

and

(3,1)

w a y of p r o v i n g entries

e2 e2 el 9 1 - ~ i

in the r e l e v a n t

matrices

There [23]). give

~2 el ~2 - ~i e2 Y1 ~2 = O

-"~2 82 ~i 61 + 72 ~i 62 Y 1 -

seems

(glg2)2 w i t h

71 72 81 62 = O

The

to

123

2

and

- el ~)i (~i ~2 + e~ (~i ~2 - e2 %'1 ~2 e 22 = 1 -

Substituting give the r e q u i r e d This

result:

finishes

Step

e~

~.~

-c~2 ~)i

and el I = ~iI + ~i I ,

a I = HI,

these

rapidly

81 = e I and Y1 = ~i"

3 and c o m p l e t e s

the p r o o f of Y o u n g ' s

Orthogonal

Form.

EXAMPLE

25.6

the g r a p h s

Here

is the o r t h o n o r m a l

used in E x a m p l e

basis

of S (3'2)

in terms of

5.2:

4

2 fl =

&,

_ ~ ~ .

4 ¥

Z

=e I

ti=135 24

J3 't

2

¢'3

f2

=

Z

= -e I + 2e 2

t2 = 1 2 5 3 4

= -e I + 2e 3

t3 = 1 3 4

& 4

2 /3 f3 =

2 5 4

-I

4

I

~ f4 =

= e I - 2e 2 - 2e 3 + 4e 4 t4 = 1 2 4 3 5

3 /2 f5 =

= 2e I - e 2 - e 3 - e 4 + 3e 5 t5 = 1 2 3 4 5

124

For clarity, we have chosen the graphs so that the edges have integer coefficients. the graphs in G i.

are orthogonal,

The numbers

(For example,

Corollary

Writing

multiplying

each

< G3,G 3 > = 1 2, so

{t i} is the last tabloid fi ensure

(2/3)-IG3

8.12 has been used to write

tabloids. Since Theorem 25.3. and

and that

(= GI,G2,..., G 5, say) It is easy to check that

out in full the matrices

the graphs

basis,

-i 1 -i

in terms of p o l y

in f3' illustrating

representing

to the orthonormal

> = 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_
be w r i t t e n

as

cF) (aij

(The r e a s o n for this p e r v e r s e act on W ( 1 ) ® W (I) by

notation

E [_g = Z i,j-
will

emerge

aijgikgj£ ~

later.)

Let G Ld(F)

(g E G Ld(F)) ,

as usual. For the m o m e n t , invariant

subspaces

assume

char F = O.

of W ( 1 ) 8 W (I) , n a m e l y

{ kl +

~ . i_

.I 1 . < . i < j < d }

{][

~

[ 1 < i < j ~ d}

There

are two n a t u r a l

those s p a n n e d

G Ld(F)-

by

and by

These exterior

are c a l l e d

power

respectively.

the s y m m e t r i c

Si n c e

p a r t of W ( 1 ) ® W (1))

char F = O

W ( 1 ) @ W (I) = Write

p a r t of W ( 1 ) 8 W (I) and the s e c o n d

of W (I) (or the s k e w - s y m m e t r i c

( s y m m e t r i c part)

S

(2nd e x t e r i o r power).

this as W ( 1 ) ® W(I)

Less w e l l k n o w n

=- W ( 2 ) S W (12)

is t h a t

W (I) ® W (I) ~

W (3) ~ 2W (2,1)

@ W (I~)

126

for some subspaces which

W (3)

(called the 3rd symmetric power),

there are two copies) AIBo

W (2'I)

(of

and W (13) (called the 3rd exterior power)

W(1) ~ W(1) 8 W(1)

® W(1)

~ W(4) ~ 3W(3,1)@

2W(2,2)~

3W(2,12 )

W (I~) "and so on".

Further W(2)8 W(2)-~ W (4) (9 W(3'I)~ W(2, 2)

Most of the work needed to prove these results has already been done,

since they are similar to those for the symmetric

the last example with S(2)~ S(2)+ ~4 ~ S(4) char F = O). Consider

again W(1)® W (I) .

skew-symmetric

parts when F is arbitrary

adjust our notation, nomials of degree

(compare

How do we deal with the symmetric (allowing char F = 2)?

by letting W (2) be the space of homogeneous

2 in commuting variables

i ~

group

~ S(3'I)~ S (2'2) , when

i, 2,...,-_d .

and

We poly-

We write

for the monomial !

so that i ~ = j i

and W (2) is spanned by {~-~

We keep our previous now

notation

for W (I) ® W (I) and for W (12) , and

(W(1)~ W ( 1 ) ) / W (12) =~ W (2) as vector spaces, i_- = ~

~i,0

since

modulo W(12)

Another way of looking at this is to define mation

Ii -< i s j < d}.

: W(1)® W(1)+

the linear transfor-

W (2) by

[ (12 ) Then ker 41,O = W

If we let G Ld(F)

act on W (2

in the natural way,

then ~i,O turns out to be a G Ld(F)-homomorphism:

= g

-~ ~ k,ZZ gikgjZ --[ 91 O k,Z

It is the generalization

gikgj£

k ~ = i

of W (2) , described

g .

in the way above,

the kth symmetric power of W (I) which we take as our b u i l d i n g block the r e p r e s e n t a t i o n 26.1

DEFINITION

to for

theory of G Ld(F). The kth s[mmetric power of W (I) is the vector space

W (k) of homogeneous

polynomials

i, 2,...,d, with coefficients i I i2...i k

of degree k in commuting variables

from F.

We write

for the monomial ~i ~2

"'" ik

127

and we

let the G Ld(F)

action

ili2...i k g = where

the

s u m is o v e r

g =

(gij).

the

subspace

The

reader who

symmetrized between

on W (k) be d e f i n e d

Z gilJlgi2J2

all s u f f i c e s

is m o r e

vectors0

this

may

and w(k)

find

... gikJ k

jl,J2,...,jk

familiar

S y m m k ( W ( 1 ) ) of W ( 1 ) @

by

with

it u s 4 f u l

between

the k t h

...® W (I)

JlJ2"''Jk

symmetric

(k times)

to k n o w

1 and d,

that

power

spanned the

and

by

as

certain

connection

is:

w(k)* = S y m m k (W (i)*) where

* denotes

the p r o c e s s

Corresponding we

consider

preliminary be

clear

with

that

Let

t X

it is u s e f u l entries

the

r the m u m b e r

in the

rth

in the

before

we

time

in terms x n be

array

representation There

come

to d i s c u s s

vector

being,

of the

obtained

Let

of m's

rth row of

in the let

{tX}

~n' more

spanned

by t a b l o i d s any

of G Ld(F)). positive (of type

by m a k i n g

t I X - t2X

denote

however,

to f o r g e t

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

(i s i ~ n).

r o w of t2X , and

spaces

of n and t is a ~ - t a b l e a u

of i n t e g e r s

of

a little

It should,

it is b e s t

action

theory

is s t i l l

to this.

a sequence

If Z is a p a r t i t i o n denote

duals.

...@ W (Zn) .

(For the

X = x I x 2 ...

Then

the

if and o n l y

tlX e q u a l s the

substituif for all

the n u m b e r

~-class

(in))

containing

of m's tX.

{t} ÷ {t}~ = {t~}

is c l e a r l y onto

though,

i ÷ x i in t

m and

W(ZI)@

interpretation

integers.

tions

to M Z = S O'Z

space

work,

repeated

intended

let

the

of t a k i n g

the

a well-defined

map

set of z - t a b l o i d s

from

the

of type

set

of z - t a b l o i d s

~, w h e r e

of type

the p a r t i t i o n

(i n)

~ is d e f i n e d

by ~i = the n u m b e r (As in some partitions by

the

26.2

of o u r e a r l i e r of n.)

Extend

of

terms

work,

we

X to be

of X e q u a l

do n o t

require

a linear

map

EXAMPLES

(i)

If X = 1 1 2, is s p a n n e d

S(2,1) , (2,1)~

then

by

is s p a n n e d

~i- -1

by

and

If X = 1 1 l, t h e n S ° , (2,1)~

is s p a n n e d

S(2,1) , (2,1)~

= O.

by 11 1

Y7 1

1 1 2 m

U and

on S 0'~,

z-tabloids.

S ° , (2,1)~

(ii)

to i.

21 1 m

~ to be p r o p e r the

space

spanned

128

Certain

linear transformations

S O'~

17.

spaces mations

in s e c t i o n

Define

defined

on the v e c t o r

linear transfor-

on s O ' ~ x by {t}X ~ i , v = { t } ~ i , v

(It is c l e a r 26.3

~i,v were

the c o r r e s p o n d i n g

that this is w e l l d e f i n e d . )

THEOREM

partition,

~ "

Suppose

and ~ , p

that X is a s e q u e n c e o f

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

type

~, I is a p r o p e r

as in 15.5.

Then (i)

d i m SIX = the n u m b e r

(ii) S P ~ ' ~ X (iii)

Proof.

~c-l,~

= SP''~Rc

S p~'~ X n ker ~ c - l , ~

In 17.12, we p r o v e d

= sP~Ac'P

~ "

11 #~

we d e d u c e

s~"~ x ~c-l,~ s~Ac

e~

= O

.

= s~'~1~c ~

last t a b l o i d s ,

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

of the S p e c h t module, o b v i o u s l y

where

is the set of s e m i s t a n d a r d

strictly

~c_l,~c

'p X ~ ~ = O. c I,U c

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

this i n e q u a l i t y

~A c , p

that

standard basis ~o(l,~)

of type

~

= etI~cPRC and

X to t h e s e e q u a t i o n s ,

and

l-tableaux

that

e~W, ~j t ~c-l,p~

Applying

of semi s t a n d a r d

is s t r i c t

contains

for some

sP~Ac'Px

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

partitionn ~ of n,

respectively

Then

f a c to r s

isomorphic

there

is a s e r i e s to S ° X

of o p e r a t i o n s

of s u b s p a c e s

(cf. C o r o l l a r y

~.

If

S p4~'p X n k e r 9c-i

(using 15.12).

let a O be the m u l t i p l i c i t y

S~ '~ .

of type

for some p a i r of p a r t i b i o n s

O,v and a s e q u e n c e

f r o m O,v to l,l or p~,~,

l-tableaux

l, or if

of the

dim S ~ X >_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££

~ ~

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o

945

t--' t--' 0"1

1936 54 1728

F--' l--a p--j :--J ~.-.i

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F.J

t-J l--J F~p-,

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I--I ~--' ,u J

1428 143 1728

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945 Co

3564 131

297 p,J O~

3564 1936 891 1431 1013

",,..1

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2673

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150

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£L&I 96&OI

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v v v v v ~ , v v v ~

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12 64 143 417 ~28

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66 220 1299 1275 2287 12 495 792 5082 66 924 792 220 495 1065 4212

I-' }-,

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l-J

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64 1938

1428 10296 143 1938 1065 8568 417 7371 428 7371 8568 1299 3367 4212

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l-J

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10296 1275 2287 5082

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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