Characteristic-free Littlewood-Richardson type decompositions of arbitrary skew Schur complexes

Table of contents :
Introduction
CHAPTER I SCHUR COMPLEXES
I.1 Shape matrices
1.2 Schur modules. Weyl modules and
Schur complexes
I.3 The standard basis theorems
CHAPTER II PIERI FORMULAS FOR SCHUR COMPLEXES
II.1 Pieri formulas
II.2 The ideas behind skew Pieri formulas
II.3 Outlines of the decompositions of skew
Schur complexes
CHAPTER III LITTLEWOOD—RICHARDSON RULE
III.1 Preliminary notions in Combinatorics
III.2 Another description of
Littlewood—Richardson rule
III.3 Classical definitions of Schur modules and
Schur complexes
CHAPTER IV DECOMPOSITIONS OF SKEW SCHUR COMPLEXES
IV.1 The U+(u1.R)-invariants associated with the
standard LR-tableaux
IV.2 Universal filtrations of skew Schur complexes
IV.3 Consequences
Bibliography
11
21
30
3O
35
37
45
45
49
55
59
59
66
71
74

Citation preview

CHARACTERISTIC-FREE LITTLEWOOD-RICHARDSON TYPE DECOMPOSITIONS OF ARBITRARY SKE KO, HYOUNG JUNE ProQuest Dissertations and Theses; 1987; ProQuest

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8715745

Ko, Hyoung June

CHARACTERISTlC-FREE LITTLEWOOD-RICHARDSON TYPE DECOMPOSITIONS OF ARBITRARY SKEW SCHUR COMPLEXES

Brandeis University

PHD.

1987

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CHARACTERISTIC-FREE LITTLEWOOD-RICHARDSON TYPE DECOMPOSITIONS OF

ARBITRARY SKEW SCHUR COMPLEXES

A Dissertation

Presented The Faculty of

to

the Graduate School

of Arts and Sciences

Brandeis University Department of Mathematics

of

In Partial Fulfillment the Requirements for the Degree Doctor of Philosophy

by HYOUNG JUNE KO March 1987

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This dissertation. directed and approved by the candidate's Committee. has been accepted and approved by the Graduate Faculty of Brandeis University in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

/\

Dean.

raduate SEhool

Arts

and Scienc s

“g?! /7, /7f’7 iss

rtation

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To my mother And in memory of my father

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ACKNOWLEDGEMENT

I would

like

to express my sincere gratitude

to my

dissertation mentor. Professor David A. Buchsbaum. for his constant encouragement. guidance and kindness. I would also like to thank Professor Gerald Schwarz and Professor Jerzy Weyman for many helpful

discussions.

My

thanks go to Maggie Beucler for her excellent job of word processing.

Finally,

I would

Eunsil and my daughter,

like

Seohee.

to

thank my wife,

for all

their love and

understanding.

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TABLE OF CONTENTS

Introduction

CHAPTER I

SCHUR COMPLEXES

I.1

Shape matrices

1.2

Schur modules.

Weyl modules and

Schur complexes

I.3

The standard basis

CHAPTER II

11

theorems

21

PIERI FORMULAS FOR SCHUR COMPLEXES

30

II.1

Pieri formulas

3O

II.2

The ideas behind skew Pieri formulas

35

II.3

Outlines of

the decompositions of

skew

Schur complexes

CHAPTER III

37

LITTLEWOOD—RICHARDSON RULE

III.1

Preliminary notions

III.2

Another description of

in Combinatorics

Littlewood—Richardson rule III.3

Classical

IV.1

45 49

definitions of Schur modules and

Schur complexes

CHAPTER IV

45

55

DECOMPOSITIONS OF SKEW SCHUR COMPLEXES

59

The U+(u1.R)-invariants associated with the standard LR-tableaux

IV.2

Universal

IV.3

Consequences

Bibliography

filtrations of

59 skew Schur complexes

66 71

74

-.—..— _.

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INTRODUCTION

The study of finite free resolutions [3.4.9.10].

modular representations [11]. and algebraic geometry over

invariant theory [13.16.27]

field of positive

characteristic [18] has led to the characteristic-free representation theory of

the general

linear group.

Many

authors constructed the characteristic—free representations of

K.

the general

Akin, D.A. Buchsbaum.

J.

linear group.

Weyman and others [12.17.27]

have developed a general and fundamental Z-forms of

group.

rational

Moreover.

Among them,

representations of

theory of

the general

linear

this important development admitted of

a natural generalization to Schur complexes.

whose

usefulness is abundant [1.2,3.4.7.15.20.25.26].

instance.

the

For

Schur complexes play central roles in the

resolutions of determintal varieties

[3.4.7.8.9,10.20.25,26].

Consequently this forces us to

further study Schur complexes. complexes

is

looking for

One way

to study Schur

the complex—theoretic versions

of classical character relations for the general

linear

group.

The purpose of this thesis is to present a method

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for finding the characteristic-free Littlewood-Richardson

type decompositions of Schur. Wyle modules and Schur complexes of arbitrary skew shapes. specialize our questions

Let

¢

:

G -e F

We shall

first

in terms of Schur modules.

be any morphism of

finitely

generated free modules over a commutative ring Given any pair of partions

p Q A

.

is

there any natural

isomorphism between the skew Schur module.

and the direct sum integers“

C2”

say

LA/uF ,

3 C2” LvF , where the non-negative

are the Littlewood-Richardson

coefficients?

It

very convenient

ideals (e.g., modules.

R

turned out

that

this

isomorphism

is

for calculating resolutions of generic

Pfaffian.

Plucker,

determinantal) and

The characteristic-zero statement was first

stated in terms of formal characters - Schur functions of

the general

[23].

linear group by Littlewood-Richardson

Recently the complete proof of the

characteristic-zero statement was done combinatorially by means

of

the

'jeu de

taquin'

which underlies

formalism of Schur functions [5.24.29.30].

the

Also the

Pieri formulas and the Littlewood—Richardson rule for Schur modules were proved in [4].

However.

the

characteristic-free situation is somewhat complicated. The reason is

that

the direct sum

2 C3” LvF

is natural

D

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only in characteristic zero. cannot exist and we need

Hence such an isomorphism

to start

looking for a

filtration natural in every characteristic. and having the associated graded module such an isomorphism exists. is

isomorphic

to

2 C3” LvF . D it will

2 0:” LvF

(Clearly,

follow that

as modules.)

if

LA/uF

Theoretically

D

[1.15], we know that such a filtration exists. but we don't know what

it

is explicitly.

As special

cases,

such

filtrations are known for the Pieri formulas [1] and the Littlewood-Richardson rule [6]. With these examples

more general Chapter

modules.

in mind.

we will now describe

in

terms the main substances of this thesis. I contains

the definitions of Schur and Weyl

as well as Schur complexes.

and some of

their

important properties which are utilized in the main body

of the thesis. Chapter II deals with the analog for Schur complexes

of the Pieri formulas for symmetric functions (or Schubert cycles).

any morphism of commutative ring

filtrations of

More precisely.

let

¢

=

G —» F

be

finitely generated free modules over a R

.

Then we construct an explicit

o 8 Sp¢

and

LA¢ 0 Apo

so

that

their

associated graded complexes are isomorphic to direct sums

of Schur complexes.

Elaborating on the ideas presented

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in the proofs of skew Pieri formulas we will set up a general framework for filtering the Schur complexes of arbitrary skew shapes.

partitions

More precisely.

given any pair of

u = (ul.---.nq) Q A = (A1.°~-.kq)

,

let

u w

=

0 —e R 1

the

be a zero map.

filtrations of

the

Not only does knowing what

'Schur complexes'

LA(W.¢:|u|.|7\/u) 2 LAN»: lLIul)

are. help in

finding the desired filtration of skew Schur complex

LA/u¢

. but also Schur complexes have the straightening

law [4.7].

For the complex

the universal

LA(¢.W:IA/ul,lul),

we denote

filtration by

ND = d,\(¢.w: |7\/u|- lul) 2 Ag» 8 Amy OE} 02v

for

u g A

and

partition

2

ID]

the

=

IA/ul

sequences

.

Let

T

A - u =

be the smallest

(Al-u1,---.Aq-uq)

in the lexicographic order of sequences of natural

numbers and

[TI =

[A/ul

.

Then we define a surjective

GL(G)xGL(F)—equ1variant map complexes by means of the natural

J 2

NT —+ Lk/u¢

the straightening

filtration inherited from

law.

of As

(0) =

N

Next we define

C N O

T

C N 1

T

C

°°~

C N

2

T

= N n

T

.

to be the image of the restriction J

J

1

i

NT —4 LA/u¢

t

N

Ti

—» L

Alu¢

to

NTi

.

Ji

has

that of

LA(¢.¢:|7\/u|.lu|). we denote it by T

NT

X. 1

of the chain map

Then the natural

chain maps

induce surjective

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GL(G)xGL(F)-equivariant maps

Ji :

NT /NT i

of

complexes

for

i

= 1,'-',n

.

—» Xi/Xi_ i-l

1

Before we continue our

discussion it is necessary to bring some combinatorial machinery into

the picture.

In chapter

III,

we prove

the useful

the Littlewood-Richardson coefficients defined as

the number

of

standard

TabK/“({1.°".p1})

of content

AssT associated to

T

T

CA up

tableaux

;

(formed by

listing

CA uv

T

in

is

the entries of

starting from the

is a word of Yamanouchi.

well-known combinatorial

.

such that the sequence

from bottom to top in each column.

left-most column)

description of

facts we prove

Using the

that

the sequence

AssT may be replaced by the sequence dssT associated to T

(formed by listing

right

the entries of

T

from

left

to

in each row starting with the bottom row and

continuing to

the

top).

In 111.3 we briefly describe

the

classical definitions of Schur modules and Schur complexes.

and the decomposition of Schur complexes over

the rational numbers

Q

(see.

e.g..

[25]).

We begin chapter IV by constructing the universal free module

R3”

corresponding to the

A

Littlewood-Richardson coefficients

CH”

generated by

C(T)

the

U+ (u1.R)-invariants

standard LR-tableaux of

Denoting by

(L

u/v

R

shape

“1 ) U+(u1.R) N u

It

h/v

the

,

is

where T’s are

and content

-

u

.

U (#1.R)-1nvar1ant +

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submodule of of content

_

LA/vR u

in

7

integers

R = m .

go back to

spanned by all standard tableaux LA/vR

. we then prove that over the +

DA

“fl” = (LA/DR

#1 U (ill-R)~ )

H .

Now let us

the filtrations of Schur complexes of

arbitrary skew shapes.

In IV.2 we prove

that

surjective

GL(G)xGL(F)-equivariant maps

—9 Xi/Xi_1

of complexes induce surjective

GL(G)xGL(F)-equivariant maps

Ji

:

ji

the

1

LT ¢ 0 R21 i

Xi/Xi_1

of complexes

Ni/Ni-l

(1 g i S n)

.

—» i

The proof of the

an

existence of

such maps

J;

is proceeded by showing

that

the straightening formula expresses any standard basis element

in

LR/u¢

as R-linear combinations of

standard basis elements

in the

LT ¢ 8 R:T i

show

that

the Schur complex

LA/u¢

.

the We

then

i

has a universal

free

filtration whose associated graded complex is

2 . vCA. Iv|=wui where

RA uu

Leticia u no".

is the trivial complex.

i.e.. O —+ R

Finally we discuss the immediate consequences of above decompoositions.

Moreover.

A

uv

the

the above results

in nicely with the connection between the

—»'O

tie

'jeu de taquin'

in Combinatorics [5.29.30] and the straightening law in

Algebra [4.12.16].

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CHAPTER I

SCHUR COMPLEXES

In

this chapter we will

review some of

the basic

facts and give some of the basic notations that will be used throughout.

For proof and details we refer the

reader to the papers [1.4.7] of the bibliography.

I.1.

Shape matrices

A partition is a non-decreasing sequence

A = (A1.°'°.Aq)

of non-negative integers. A1 2 A2 2 ...

We

say

that

length.

the number of

£(A)

.

its nonzero

. by

'IAI

.

sequences

We will denote by non-negative

finite number of nonzero

m”

terms.

A = (A1,--‘.Aq)

sequence

(A1.'°°.Aq.0.°°°)

and

is

A

A .

the set of all containing

in

m

infinite

only a

Given any finite

m”

it as a

by extension with

Thus we will not distinguish between

(A1.°°°.Aq.O.-°-)

or more

and is denoted

we may think of in

its

of non-negative

integers

sequence

zeros.

A

is the sum of the terms of

of

terms

The weight of a partition

generally any finite sequence

integers,

2 Aq

(A1,--°,Aq)

co

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A relative sequence is a pair in

mm

i 2 1

such that

.

u g A

meaning

(A.u) that

We shall use the notation

relative sequences.

If both

then the relative sequence partition.

A

A/u

”i

A/u

and

of sequences

u

S Ai

for all

to represent are partitions

will be called a skew

The skew partion A/

u

=

A

( 1

,ooo'k

/

q) (u1.°--.uq)

may be described graphically by its diagram

is the set of all ordered pairs

(i.j)

AA/u

,

which

of integers

satisfying the inequalities 1Si$e(A)=q,pi+lgj$A simultaneously.

That

is.

AA,“ = {(i.j) e Nxmll S i S q. ”1+1 S j S Ai} The shape matrix of

A = (aij)

is an q1 matrix

defined by the rule

{1 a.. 13

As an example take the

A/u

shape matrix of

if

0

ui+1 g j 3 xi

otherwise

A = (4.3.2) A/u

and

u = (2.1)

.

Then

is

O 0 1 1 0 1 1 O 1 1 O O and

the diagram of

A/u

is

where each dot represents an ordered pair (i.j).

It is often convenient to replace the dots by

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squares,

in which case the diagram is

1

[ In general we define a shape matrix to be a finite matrix with zeros and ones as entries. sequence

A/u

we can associate

shape matrix with

partitions.

matrix

However.

A = (aij)

simply

Ih/ul

If

Alp

a diagram and a skew

we will usually not distinquish

and its diagram

defined to be

to

the similar rules used for

between a relative sequence

A = (aij)

Given any relative

Alp .

its shape matrix

AA/u .

The weight of a shape

of a relative sequence

IAI—Iul

h/u

and is denoted by

[AI

is .

or

.

A = (A1.A

2|

~-.) 6 m”

is a partition then its

conjugate or transpose is defined to be the parition N

~

A;

N

A = (h1.k2.'"')

where

which are greater

A =

(aij)

At

Aj

is the number of terms of

than or equal

to

j

is a shape matrix we define

A = (aij)

of

Similarly if

the

transpose

in the usual way by taking

2

A

.

A

I”

I

.. — a.. 13 31

Let

A = (aij)

the rwo sequence

be an

sxt

shape matrix.

aA = (a1.'°°.as)

of

A

We define

by

t

ai =

E

aij

for

i = 1,°'°.s

and the column sequence

i=1 bA

to be

aAt

.

Clearly

bA = (b1,-'°,bt)

where

s b. J

=

2 a.. i=1 1‘]

for

j

= l.°°°.t

.

That

is.

a

is

just

A

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the sequence of row sums of

of column sums of

A

and

bA

is the sequence

A

Finally two shape matrices are said to be equivalent if

one can be

of

its

transformed into

the other by permutations

rows and columns.

-10-

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1.2.

Schur modules,

Weyl modules and Schur complexes

From now on. we denote by (with identity). and by

R

F . G

a commutative ring

finitely generated free

R—modules unless otherwise specified.

ApF

,

SPF

.

and

DpF

divided powers of is a sequence

F

Also we denote by

the p—th exterior,

symmetric.

respectively.

a = (a1,°.°,as)

If

and

no

in

m

.

a A F _ A

we use the following notations: a

1

a

F @R

8R A

s

F

SaF = Sa F @R 1

---

8R Sa F s

DaF = DalF 0R

'0'

0R DaSF

For any shape matrix

A

we use

AA(_)

. SA(—)

. DA(—)

(or A~(_) , s~(_) . n~(_)) to denote AaA(_) . sa (_) , A A A A b DaA(_) (or A A(_) . sbA(_) . nbA(_)). It will be recalled that the exterior algebra symmetric algebra

DF = 2 DPF

SF = E SPF

are all

algebras.

Thus,

S D

ai F a

if

-——# A

F —-—# S F ———» D

1

For any module

F

.

ail F a

8 ~-- 8 A

F 8 '0' 0 S

i1

a.

F 8 ~-- 0 D

+ ait

.

it will

ait F a

.

F .

it

a.

F

it

shape matrix

We are going

°--

the diagonalizations

11

sxt

co-asscciative Hopf

ai = ai1 + aiz +

1

a.

and divided power algebra

co-commutative,

not cause confusion about

A

.

AF = 2 ApF .

A = (aij)

to define

and a free

two maps:

-11-

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dA(F) =

AAF -——e s~F A

dA(F)

as follows. A

a1 F

:

DAF ———» AXF

Diagonalizing each

@ "- 0 A

a5 F

to

A

ail F

A

ai F

in

@ °°° 8 A

ait F

A F = A

=

. we get a map

a

a.

a1 F

a

A A F ——A-» (A 11F a ... a A 1tF)0---®(A 8‘$1 F a --- A StF) By rearranging terms along the columns of

A';

we have an

a.

a.

9’

a

II? CD

isomorphism

(A 11F o ... a A 1tF) 9 ~-- 0 (A 51F o ... o A CL

3.

a.

a

(A 11F @ ... a A 51F) @ --- e (A 1tF @ ... a A StF) ai.

As

a.. = 0 15

or

1 , A

JF = s a.. F .

Thus

13 a a a h a (A 11F 9 -~- @ A 51F) o --- o (A 1‘F 0 ~-- 8 A StF) =

(3a

11

F @ ... a sa

F) 8

0"

9

(S

s1

F 0

00-

8 S

a1t

F)

ast

Finally use multiplication in the symmetric algebra

to map each

Sa

13

F 8 °-- 8 Sa

.F SJ

to

S

b

F

SF

and we obtain

J

the map

(sa

F @ ... o sa 11

F) @ --- 6 (Sa 51

3

b1

F o --- o s

bc

F = s

The whole thing amounts the composition

map

dA(F)

m 0 6 0

to be

identification.

F @ ... 0 Sa 1t

bA

F = SNF

A

to saying that A

F) -E—+ st

.

the map

Similarly.

dA(F)

one defines a

the composition of multiplication.

rearrangement and diagonalization maps:

-12_

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is

A

aA

-—:L——» (Da F o ... @ Da F) e --- @ (Da F @ ... Da F) 11 9

1t

51

st

I

E

(D

a11

F G --- 8 Da

51

a

F) 0 00° Q (Da

a

1t

F 8 °°° 8 D

a

ast

F)

a

= (A 11F 0 --~ 0 A 51F) e ... a (A 1tF @ ~-- 0 A StF) .

b

b

b

E—» A 1F Q -~- 8 A tF = A AF =.A~F A

The Schur module on

LAF

is the image of the map .

The Weyl module on

to the shape matrix

is the image of

is a skew partition

A/u

.

is denoted by

We shall now give

simply If

L

A

(1.-~o.1) . so that p ”I

.

LAF

and

dA(F)

KAF

are

If the shape matrix

A/uF (KA/u F)

A

(Weyl)

module

.

u = 0 .

If

LAF (KAF)

some simple examples.

then the matrix

A = (1.---.1)

associated

the map

then the Schur

then we denote this module by

A = (p)

It

A

GL(F)-modu1es in a natural way.

A

.

K F

It should be observed that

of shape

dA(F)

corresponding to

LAF = APF

and

then its shape matrix

If we

A

KAF

take

is U

It is denoted by

A

F

to

m

is denoted by

A

associated

“U

the shape matrix

F

ll

Definition 2.1.

A

P

-13-

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A

I—IOOIH

corresponding to

is

.

so that

LAF = SPF

and

_ APF KAF —

Remark 2.2. (1) F

If

R

is a field of characteristic zero. and

is an n-dimensional R-vector

set of distinct

the Schur modules

A

then a complete

irreducible homogeneous polynomial

representations of

partitions

space.

GL(F)

{LAF}

(A1.A2.-'-)

of degree

where

of

d

A

d

is described by

runs over all

with

< n = dim F

(2) over

Schur module and Weyl module are isomorphic

the rationals.

integers.

general

far from isomorphic over

(K

A/u

F)*

and

L~ NF,6 k/u

are isomorphic.

In classical representation theory of the linear group.

the Schur module

LAF

described either as a certain submodule of

(cf.

the

However it is proved in [4] that the

GL(F)—modules

(3)

but

1.2.1).

can be

S~F A

or as a certain quotient module of

AAF

(cf. 1.3.1).

We shall now describe the Schur complexes and show how the Schur complexes generalize both Schur modules and

-14-

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Weyl modules.

After that. we focus on the study of Schur

complexes.

Let

¢

:

G —» F

be any morphism of

generated free R-modules. algebra and

so

on the map

We define the symmetric ¢

the exterior algebra

Hopf algebra

DGéAF

tensor product

,

over

A¢

.

and

A¢

the Hopf algebra

on the map

é

denotes

The element

corresponding to the map natural isomorphism

to be

where

R

HomR(G.F) E G*®F

1—1

¢

to be

the

under the acts on both

*

c¢

D.G®A1F ——e D.1—1 G®AJ+1F , where c ¢ e sc*éAF 1 c

G®Sq+1F . where

to be

S¢

In particular.

A GQSJF -—+ A

Sp¢

the

'twisted'

c¢

Define

AGGSF

c¢ € G*®F

¢ 6 HomR(G.F)

making them into complexes. i

finitely

c¢ 6 AG @SF , and

the complex

c

c

c

c¢

c

C

o a ApG —3+---—$» Aicesp_iF —$»--——1» cos p—l F ——» SPF a o and define

Apo

0

to be

the complex

C

.

c

o » DPG —1»--~—1e DiGQAp_1F —$+---—$» G®Ap_1F —$e APP e o. It is clear that

s¢ =

can also be checked

2

s ¢

p20

P

and

A¢ =

2

AP¢ .

It

p20

that multiplication and

comultiplication maps

in

so

and

A¢

are compatible

with the boundary maps.

Remark 2.3.

is a zero map

The reader

0 ——9 F

symmetric algebra

SF

should keep in mind

. on

then F

so

is

just

and

A¢

is

that

if

¢

the usual the exterior

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algebra G —» 0 and

AF .

then

A¢

that

on

is

F . s¢

is

Now we to be

the

is

the

trivial

AC

DC

complex

on

on

G

G

.

Note

0 —% R —» O

sequences of

shape matrix and

sxt

be a

A

and

row and column sums of

aA A

,

bA

as we

We can then define a map of chain complexes.

the Schur map.

=

AA¢ -* SX¢

the composition a

a

a. 11¢0090@A

AA¢=AA¢—’(A

a. 1t¢)@ooo@(A

H?

Ii?

a

a

¢)®°°-®(A 1t¢ ®---@A

(S

¢)0--°@(S

—» s the

a. 51¢80'08A

(A 11¢®-'°®A asl ¢®--~®S all

where

is the map

the exterior algebra

dA(¢) to be

¢

o -» c —» F -» o

let

did before. called

is

if

the divided power algebra

80¢ = Aoo

31¢ = A1¢

Similarly.

b1

3‘51

¢o---es

b:

bA

terms.

st

¢)

¢) ¢)

a'st

¢ = s~¢ . A

first map is diagonalization.

isomorphism rearranging

a

¢®~--@S a1t

¢ = 3

St

the

the second is

third is

the

the

a.. isomorphism identifying

A

13¢

with

(for

¢

Sa

aij

=

ii

Notice that

0 or 1). and the last map is multiplication. the Schur map

G

and

dA(¢)

F

dA(¢)

are used.

does not depend upon

Since each of

is a map of complexes,

¢

.

for only

the maps comprising

the Schur map

dA(¢)

is a

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natural

Hence

transformation and a morphism of complexes.

its

image is a complex and we make

the following

definition.

Definition 2.4.

The image of

dA(¢)

is called the Schur complex on

shape matrix partition

A .

When

.

we write

h/u

should be noted

that

A

¢

. denoted by

associated to

LA¢ . the

is the shape matrix of a skew LA/u¢

instead of-

the Schur complexes

LA¢

LA¢

.

It

are

complexes of GL(G)xGL(F)—modules ("GL(¢)-complexes".

for

short).

Notice that complexes over

LA(—) R .

is a functor from maps to

If we restrict our attention to the

maps of the form

0 —e F .

Schur module

.

LAF

then we recover the usual

Although

traditionally

the above

terminology has been used in connection with shapes given

by skew partitions there should be no harm in extending it to include general shapes.

Similarly.

attention to maps of the form

C -» O

if we restrict

we obtain the Weyl

module 'KAG When

the shape matrix

Schur complex This

I

is because

LA¢ if

Sb ¢ = (O —» R —4 0) A

A

is

the zero matrix,

is the trivial complex A

is

the zero matrix,

and the Schur map

the

0 —» R —+ O then

dA(¢)

a A A¢ =

is the

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identity.

It will be convenient for us

shape matrix

A

such that

LA(-)

to also have a

is the zero functor.

For this purpose we will use the empty matrix If'we have a relative sequence

k/u

such that

then we will assign the empty matrix matrix for

A/u

.

Note

that

If

(cf.

for negative

1.1)

taking

integers

two shape matrices

)

as

u $ A

,

the shape

Apo

and

Sp¢

to

p

A1

then the functors

(

)

these conventions are

consistent with the custom of be zeros

A = (

and

LA (

A2

)

are equivalent

and

1 ‘

LA (

2 ‘

)

are

naturally equivalent (see [4]). Suppose now that

¢1$¢2

of

two maps

we mean that

¢

¢i

=

:

G —+ F

Gi —+ Fi

is

,

i

the direct sum

= 1,2

G = G1$G2 , F = F1$F2 . and

(¢1(g1).¢2(g2)) .

.

By this

¢(g1.g2) =

For any non-negative integer

p

we

have the direct sum decomposition

(1)

AP¢ =

2

Aa¢1 o Ab¢2

a+b=p

of chain complexes.

If

P = (p1.-‘°.p ) is a sequence q of non—negative integers of length q . then (1) immediately yields a natural direct

sum decomposition of

P1 the chain complexes

(2)

AP¢ = A

Pq ¢®---®A

¢

as

follows:

AP¢ = E Aa¢1 Q Ab¢2 .

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where the sum is taken over all sequences

a = (a1,-°-.aq) integers

such that

integers

a

and

AP(¢1.¢2:a.b) a

,

b

. b = (b1,-°°,bq)

of

b

= pi

for

such that

to be

length

lb] = b .

ai+bi

q

2 Aa¢1®A

i

a+b =

b¢2

satisfying

It follows from (2)

of non—negative =

1.°°°.q

IPI

Fix

. and define

over all a+b = P

.

,

sequences Ial

= a

.

and

that there is a direct sum

decomposition

A m (¢) = a+b=P E A r (¢ 1 .¢ 2 :a.b) The above discussion and definitions may be repeated with

S

in place of If

A

A

is a shape matrix then we can apply

discussion to

the row and column sequences

obtain natural direct

of

the above A

to

sum decompositions

A A (\/u|.|u|)

Now for any sequence

[bl =

IA/nl

0

satisfying

. We define a subcomplex

LA(¢,¢:IA/ul.lu|)

ND

D g A

and

of

by the following formula

Nu = dk(¢.¢:IA/n|.lu|)

3

Aaf Q AA/aW

USA

lal=lMuI 02v

Then it follows from Corollary 1.3.5 that the complex LA(¢.W:IA/ul.lul)

{NDIv C K.Ivl =

has a natural filtration

Ik/ul}

whose associated graded complex

is isomorphic to 2 v Q LA/uw of

complexes where

v Q A

such that

the sum is

ID] =

lA/ul

taken over all

and

Ik/vl =

partitions

lul

.

As

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LA/u¢ want

is a GL(¢)—subcomp1ex of to know what

Lk/u¢

Lk(¢.w:IA/ul.lul) . "straightening

like inside

which we now give below.

Let

B = {x1 < '--

< xr}

ordered set.

For any pair of partitions

T e TabA/D(B)

.

we define

Tp q

#{(i.j) e AMDII 5 i g p

and

be a totally D g A

.

T(i.j) e {x1.-°-.xq}}.

T'

g T

if

Té'q 2 Tp.q

for all

we say th a t

T'

< T

' if

T'

T’p.q > Tp.q

S T

an d

p.q : for

p.q

Notice that T'

and

to be

we say that

some

. we

This can be done by means of

law".

Definition 3.1.

looks

LA(¢.¢;IuI.IA/fll)

= T

.

that

is,

of all

tableaux.

T'

.

¢ T

T'

S T

and

T S T'

do not imply

we just have a pseudo-order

However.

if

T'

< T

.

in the set

then certainly

The pseudo-order defined above

is consistent

with the lexicographic order of the row—standard tableaux:

and

T

if

T'

< T

in the pseudo-order.

are row—standard.

lexicographic order. lexicographic order

tableaux by means of

then

T'

< T

T‘

also in the

(Remark that we have a in the set of all

the order of

B

that

the pseudo-order still works for

set

B(¢)

(cf.

and both

row—standard

.) the

Using the

fact

totally ordered

1.3), we can state the straightening law:

-39-

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Proposition 3.2.

[4]

Let

T e TabA/v(B(¢))

row—standard mod C but not standard mod G.

exist

row-standard mod G tableaux

Ti

.

be Then there

with

T1

< T

.

such that

Z T - E aiZTi € im(uA/v(¢)) where

ai

,

e Z

Now knowing that Schur complexes have the straightening law.

one has to check what happens when the straightening

law is applied to all standard basis elements in

[RuQLA/uq) g LA(¢>.¢;I)\/ul. luI) .

First of all. let

2 = c(1)-2100(2)'z20---@c(q)°zq

111

¢®-'-0A

uq

A -u A -u w)®(A 1 1¢®°~°0A q q¢)

that up to sign.

Z'

If we let

be a standard basis

mfleAk/M g AuwQAA/u¢ =

element in

(A

LA/u¢

T

Z is equal

.

Then we know

to

= zl°c(1)@z2-c(2)®'-~®z -c(q) q be the smallest partition 2 the sequence

A-p = (Al-u1,---.hq-uq)

in the lexicographic order on

sequences of natural numbers and

dk(¢.¢;Ik/nl.lui)(2')

ITI =

is clearly in

IA/ul

N7

.

(see 1.3 for

the discussions of this matter). We will now show that

the isomorphism

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(**)

Ak(¢.¢:IR/u|.lul) E Ak(w.¢:|u|.lk/u|)

induces a map the

J

straightening

:

NT —9 LA/fl¢ law.

Notice

of complexes by means of that

the

straightening

is a GL(¢)xGL(¢)-equivariant operator on

and preserves the content [4].

law

TabA(B(¢$¢))

Recalling that

mqk/u¢

is a direct summand of an R-free GL(w)xGL(¢)-complex

AA(¢.¢:IuI.lA/ul)

. we apply the straightening law

(relative to "w < ¢")

02A

to the image of a basis element in

Aa¢aAk/a¢ g AA(¢.W:IA/ul.lul)

via the

|a|=|x/ul 027

isomorphism (**) and then prOJect it onto Therefore we have an induced natural

RMQAK/u¢

surjective

GL(¢)-equivariant map

J =

NT —» L

A/u¢

of complexes.

(Remark that the Schur

complexes have the straightening law and so their boundaries are comparable with the law.)

We can now

state formally

Proposition 3.3.

a) The straightening formula expresses a standard

basis element of

LA/p¢

as a linear combination of all

standard basis elements of

the subcomplex

N1

of

LA(¢.w:|R/u|.lu|)

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b)

The isomorphism

Ax(w.¢:lul.IA/ul)

Ax(¢,w;IA/ul.lul) E

induces a natural surjective

GL(¢)-equivariant map

J ‘

NT ‘* LA/u¢

of complexes by means of

Notice

that

the bounds of

the straightening law.

the above proposition tells us at

the filtration quotients of

there exists any filtration). LA(¢.w:IA/ul.lul)

(if

As the complex

has the natural filtration

{NDID g A. lvl = IA/ul} , NT

has also a natural

filtration inherited from that of that

LA/u¢

least

LA(¢.W.IA/ul.lul)

the associated graded complex is

2 DEA

Lv¢ 9 L

isomorphic

so

to

A/u¢

|v|=|A/u| v21

We denote

that very filtration of

(O)=N

To

CN

Tl

CN

NT

by

C"°CN

T2

Tn

=N

For the convenience to simplify the notation.

Ni

for

NT

(0 S i S n)

.

Next we define

T

we write

Xi

to be

i

the

J :

image of

the restriction

NT -+ L A/u¢

filtration complexes.

to

N.1

of a natural map

(0 g i g n)

{xilo < i g n} increasing as

Ji

of i

.

LA/u¢

Thus we have a

by chain

increases.

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Example 3.4.

1

Let

1.2 e B(¢)

2 a

and

a.b.c 6 B(¢)

a b c

1 b c

=

a 1

2

a b 1

-

.

Then

a c

a 1 2

+

1

a 1 2

straighten

L

a

1

c

h~o—-J m

W m

‘-—'———--————-) m

N

(3.3.1)/(2’1)¢

b

N

(3.1)

(2.1.1)

0n the other hand.

a b c

a

1

1

2

= unstraighten ,

1

2 a

1 a b

1

+

c

2 a

1

1 b c

—

2 a

1 a c

a

b

and

a b

1

— a 1 2

a c +

1

1

a 1 2

=

-

2 a

1 a b

1 +

2 a

1 a c

unstraighten c

Thus

b

J(2.1'1)

c

a b 1 a c 1 - a 1 2 + a 1 2 0 b

X(2'1’1)/X(3'1) X(2’1'1)/X(3'1)

because .

1 2 a = 1 b c a

a b c J(3’1)[? 1 2]

In general,

we have

b

.

1n

= 0

in

the following

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proposition.

Proposition 3.5.

The map

J 1

NT —9 LAIu¢

induces the

surjective GL(¢)-equivariant maps

J.1 : of complexes

Ni/Ni_ 1

—-) X./X. 1

1-

1

(1 g i S n)

The proof of

the above proposition follows from the n

definition.

Now we claim that

2

xi/Xi—l

is

1:1 isomorphic

to

2 CAD v

.

The proof of our claim

D

involves a combinatorial argument and

the

invariant

theory to describe the Littlewood-Richardson coefficients C

A uv

In the next

several

sections we will develop

the

necessary machinery.

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CHAPTER III LITTLEWOOD-RICHARDSON RULE

In

this chapter we provide a useful description of

the Littlewood—Richardson coefficients.

Then we briefly

review the classical definition of Schur complexes

in

characteristic zero.

III.1.

Preliminary notion in Combinatorics

The purpose of

this section is

to review some

preliminary notions in Combinatorics.

a = (a1,-°°.an)

Yamanouchi.

of positive integers is a word of

or a Y—word.

number of times (a1.°°',ak)

i

1 2 2 1 2 3)

Y-word.

If

integers.

if for each

the

appears in the subsequences

for every positive integer is a Y—word:

a = (a1.°-°.an)

(1

i

1 2 2 2 1 3)

.

i+1

E.g.=

is apt a

is any sequence of positive

its content is defined to be the sequence

v = (v1,v2.°-°)

.

appears

It follows

in a

.

where

only if the content of

each

k = 1,-'-.n .

is not smaller than the number of times

appears in there, (1

A finite sequence

k = 1,-°°,n .

vi = the numbers of times i that

a

(a1,---.ak)

is a Y-word

if and

is a parition for

A finite sequence

(xi

.--°.xi 1

) n

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from an ordered set Y—word if

B = {x1 g °°° S xn}

(i1,°°°,in)

is called a

forms a Y-word of numbers.

of Yamonouchi are also called

Words

lattice permutations

in

the

literature.

Lemma 1.1.

[4]

There is a bijection between the set of

Y—words of content of

shape

v

v

0

and define content

D

a(T)_ .

by

1.°-°.n

a

v

,

if

.

Each

T(j.k)

.

Let

is a

n

(ak .-°°,ak ) 1 vi

with entries equal to

B(a)(i,j) = kj

.

It is clear that

i

a

An easy induction argument on

takes Y-words

the reverse.

to standard tableaux and

giving

be the .

Define

v

and n

given

B

are

shows that

that

a

does

the desired bijection.

As an illustration.

(1.1.2.1,3,2)

of

n

a = (a1,°'-,a )

let

ai = j

(a1,°'-.a )

to be the row-standard tableau of shape

inverses. B

be a row-standard tableau of

to be the sequence

Conversely.

subsequence of

tableaux

from the set

appears as an entry

sequence of content

5(a)

T

with distinct entries

i e {1.---,n}

standard

n = Ivl

To see this let

shape

the set of

with distinct entries

{1.2.---,n} . where

Proof.

and

B

takes

the Y-word

to the tableau

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knoaw

Suppose that its transpose

T

T

2 4|

is a tableau of shape

is the tableau of

v

= T(j.i)

.

standard

tableau with distinct entries,

It is clear that

T = T

v

.

Then

given by

and if

T

T(i.j) is a N

Definition 1.2. content

v

Y-word

.

Let

We define

a(B(a)

)

ak

Then a tableau (p1,u2,---)

times.

2

occurs

Let

T

- AssT

to be

the

a = a .

a = (a1.--°.an)

. where

ai

Let

T

ak = a

u g A

in

u2

and

It is

is the

k < 1

be a pair of partitions.

Tabx/D({l,2.'-°.n})

whenever

1

occurs

times,

and

so

has content

in it exactly

“1

on.

be a standard tableau in .

associated with is

a

so that

TabA/D({1.2.°°°,n})

'dssT

transpose

u

T

be a Y-word of

.

such that

Definition 1.3.

p =

its

of content

easy to see that number of

a = (a1.°°°.an)

then so is

We define two sequences

T

the sequence of

entries of

T

starting with

AssT

and

as follows: T

formed by

listing

the

from bottom to top in each column, the

left-most column and continuing

to

the right.

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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

- dssT

is the sequence of

entries of

T

T

formed by listing the

from left to right in each row.

starting

with the bottom row and continuing to the top. With

these auxiliary notations.

we can now define

A

the Littlewood-Richardson coefficients

Definition 0:”

is

1.4.

The Littlewood-Richardson coefficient

the number of

standard tableaux

TabA/v({1.2.°'°.u1})

of content

a Y-word.

to see

It

is easy

017:” = 0 unless

CM”

n

that

|7\| = lul+lvl

T

in

such that CA

= CA

w

W

and p,u gx

,

AssT

is

and

(cf. [22],

[24])-

A (standard)

tableau

T

Littlewood-Richardson ("LR". is a Y-word.

is called a (standard)

for short)-tableau if

Thus the coefficient

standard LR—tableaux of content

TabA/v({1.2,°°-.u1}) that

AssT

definition.

.

Z

CZ”

AssT

is the number of

in

In the next section we will prove

may be replaced by

dssT

in the above

This allows us a little bit more freedom to

compute the coefficients

A

Cu”

in practice.

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III.2.

Another description of Littlewood—Richardson

rule

We begin this section by recalling that a sequence

a = (a1.---,an)

is a Y—word if and only if the content

of

is a partition for each

(a1.'°°,ak)

Proposition 2.1. T

Given a pair of partitions

be a standard LR-tableau of

than one row.

the

If

top row of

Egggi.

R = l,°°°,n

T1

T

.

If we let

is

the

then

shape

A/v

the content

T

,

AssT1

is a Y-word.

k = 1.-°°,n .

B(AssT)

and

let

which has more

.

clearly enough to show that the content of

discussion in III.1.

.

tableau obtained by erasing

AssT1 = (a1.a2.'-°.an)

is a partition for each

v E A

then it is (a1,-°-,ak)

From the

is a tableau of shape of

the entries

in the

top row of

T

correspond to the outside boxes in the rows of the

diagram of listed from and a

that

ak

B(AssT) a1

.

up to

If we let ak

in

a' AssT

are not in the top row of

is a partition because

the sequences

AssT

(a1.'-'.ak)

to be a subsequence (remark that T).

the content of

is a Y-word.

Notice

can be obtained from

by erasing the terms in the top row of that

T .

a partition.

shape of a partition has also

a‘

Now observe

the diagram obtained by erasing an extremal box

the diagram of

a1

the

in

shape of

(A box is extremal if it is an outside

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corner. it.)

i.e..

If we

if it has no neighbor to its right or below

let

b1

°‘°

be

the entries

b1 < b2 < '°-

sequence

.

bi

a'

then

is extremal.

Thus

top row of

T

and

are involved in the B(a')

corresponding

to

the diagram obtained by erasing

B(a')

the shape of a partition.

corresponding to Using

we next keep erasing

corresponding to

the

< bi

the box in

that extremal box in

above.

be

bi

'has

the same argument as

the boxes

in

B(a')

bi_1.°°-.b1 . one after another.

to

obtain the sequence of the diagrams of shapes of partitions.

This

B((a1.-°-.ak)) follows

shows

has

from Lemma

(a1.--°.ak)

that

the diagram of

the shape of a partition. III.1.1

that

It then

the content of

is a partition.

Theorem 2.2.

Given a pair of partitions

be a standard LR-tableau of

shape

A/v

.

D g A . Then

let

dssT

T is

a Y—word.

Proof.

We shall proceed by induction on the number

r(k/D)

of

rows of

dssT = AssT

r(A/v) = 1

the diagram

AMD

.

observing

that

is a Y—word in the trivial case

.

T's with rows

Assuming the theorem is true for all such

< r(k/v) - 1 ,

standard LR-tableau

T

with

let us prove it for a r(R/v)

rows.

From

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Proposition III.2.1.

(a1.--°.an)

AssT1

is a Y—word and so

dssT1

1:

is a Y—word by our induction hypothesis.

Using the same notation as in Proposition III.2.1.

dssT = (a1,°~°.an.b1.-°‘.be)

.

Therefore it is clearly

enough to show that the content of

is a partition for each are

i = 1.°°-.8

the entries corresponding to

rows of

B(AssT)

the diagram of

fi(AssT)

be

up to

bi+1

then we get the diagram of From the proof of

diagram of

Hence

that

a Y-word implies

.

be.b e_1’.. -,b

i+l

one after another).

the shape of

and

the

is a partition.

for any standard

AssT

so

is a

tableau

T

.

D Q A

.

a Y-word.

dssT

tableau obtained by erasing

T

then

adssT1

the boxes of

Given a pair of partitions

and let

in the

is a Y-word.

the ,

we erase

< be

B((a1.'-°.an.b1,°°°.bi)).

be a standard tableau of shape

than one column,

b1 < °°°

(a1,---.an.b1.'°-.bi) dssT

Next we prove

T

.

B((a1.-°-.an.b1.--°.bi))

Proposition 2.3.

As

Proposition III.2.1.

that the content of partition.

.

the outside boxes

corresponding to the terms

(starting from

dssT

(a1.---.an.b1,---,bi)

k/v

let

which has more

be a Y—word.

If

T1

is

the right-most column of

is a Y-word.

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Bragg.

If we let

adssT1 = (c1.~°'.cn)

clearly enough to show that

is a partition for each

partition.

k = 1.---.n

the content of

right-most column of

T

.

(remark that

c1

right-most column of

correspond

partition because sequence

and T).

dssT

(c1.°--.ck)

From the

T . and the entries in the to

ck

the outside boxes

B(dssT)

to be a subsequence listed from

dssT

(c1.°°°,ck)

is a tableau of shape of a

in the columns of the diagram of

c'

then it is

the content of

fi(dssT)

discussion in 111.1.

.

01

.

If we let

up to

ck

are not in the

the content of

is a Y-word.

c'

is a

Notice that the

can be obtained from

c'

erasing the terms in the right-most column of observe

that

in

by

T .

Now

the diagram obtained by erasing an extremal

box in the diagram of

shape of a partition has also

the

If we

let

be

the right-most

n.

shape of a partition.

on.

d1 2 column of

T

and

the entries

are involved in the sequence corresponding to

d

obtained by erasing

corresponding to repeat

d

1+1

di+1

0

.

g di+2 g

3 d2

then the box in

is extremal.

Thus

that extremal box in

i+1

'0-

B(c')

the diagram B(c')

has the shape of a partition.

the same argument with

d i+2'

.0

°.de

.

We

one after

another. and then we obtain the sequence of the diagrams

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of shapes of partitions.

B((c1.'°‘.ck))

This shows that the diagram of

has the shape of a partition.

It then

follows from Lemma III.1.1 that the content of (cl.°°-.ck)

is a partition.

Theorem 2.4.

Given a pair of partitions

be a standard tableau of shape Y-word.

Then

AssT

A/v

v Q A .

, and let

let

dssT

We shall proceed by induction on the number

C(h/u)

of columns of the diagram

C(A/v) = l

Ak/v

.

observing that

is a Y-word in the trivial case .

Assuming the thoerem is true for all such

T's with columns

S c(h/v)

standard tableau with III.2.3.

slssT1

- 1

c(h/v)

.

let us prove

columns.

is a Y-word and so

notation as in Proposition III.3.3. (c1.-°°.cn,de,°°°.d1)

.

d1 S d2 g ... S de

== (c1.‘°-.cn)

Using the same

AssT =

(cl,°°'.cn.de.°--.di+1)

i = o.1,~-~.e-1

are

.

the entries corresponding

d1.d2,°°°,di

B(dssT)

is

As

outside boxes in the columns of the diagram of

we erase the boxes of

for a

Therefore it is clearly enough

to show that the content of a partition for each

it

From Proposition

AssT1

is a Y-word by our induction hypothesis.

terms

be a

is a Y-word.

2322;.

AssT = dssT

T

to

the

B(dssT)

corresponding to the

(starting from

d1

up to

di

.

one

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.

after another). and then we get the diagram of B((c1.'0'.cn.de,--°,di+1)) Proposition III.3.3.

Hence

From the proof of

the shape of the diagram of

5((C1’""°n'de'°..'di+l))

content of

.

is a partition.

(c1,-'-.cn.de.--°.di+1)

AssT

is a partition.

is a Y-word.

0n the basis of

the above

theorems,

the Littlewood—Richardson coefficients alternative

nu

is

we can now state

in the following

form.

Corollary 2.5.

CA

so that the

The Littlewood-Richardson coefficient

the number of

TabA/D({1.-'°.u1})

standard tableaux

of content

n

T

such that

in

AssT

is a

word of Yamonouchi.

We end

the above

but

this section by mentioning

that

the proofs of

theorems can be done directly by definition.

they are not neat.

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III.3.

Classical definitions 0f Schur modules and Schur complexes

This section contains a brief

summary of

results

from the classical representation theory of the general linear group.

group

Over the rationals

GL(n.Q)

Q .

the algebraic

is linearly reductive. which means that

every polynomial

representation of

completely reducible.

GL(n.Q)

is

First. we will briefly describe

the classical definition of Schur modules. which give the irreducible homogeneous polynomial

representations of

GL(n.Q) Let

V

be a Q-vector space of dimension n.

is a partition of

subspace of

V

0d

d

.

then the Schur module

If

LAV

A

is

the

defined by

exVed Here

QEEd]

eA

is the Young idempotent in the group ring

of the symmetric group

corresponding

{LAVIIAI = d .

to

the partition

A1 3 n}

2d A

on .

d

symbols

The spaces

form a full set of inequivalent

irreducible homogeneous polynomial representations of GL(V)

of degree d.

Notice that in our situation Schur

modules and Weyl modules are isomorphic.

i.e..

LV=K~V 7‘ A There is a well-known equivalence between the category of En-modules and the category of homogeneous

polynomial GL(V)-modules of degree n (n = dim V)

.

It is

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given by

two covariant functors

¢ .

W

inverse

to each

other

43 E —modules n

polynomial GL V -modules of degree n

('—

w

¢

. . 15 Simply

V8n

Hom2 (V n

8n

.

via permutations.

diagonal matrices

GL(V)-module

M

. . With the action of

)

To define

Tn

embedded in

W

2n

on

consider the GL(V)

.

Then given a

we set

MM) = {xeMl(t1,---,tn)x=t1--~tnx for (t1,---,tn)€Tn} : w(M)

is a Zn—module with the action induced by the

embedding

2n

into

In particular. associated

GL(V)

we call

SA = ¢(LAV)

to a partition

{skllhl = n}

via the permutation matrices.

A

.

the Specht module

The Specht modules

form a full set of inequivalent irreducible

representations of

2n .

We now consider the natural action of

{1,°°-.n} elements

.

This makes

fixing

we denote by

zn-i

n-i+1.--°,n

N T 2n

2n

a subgroup of .

For any

on

2n

En_i-module

the induced module.

p g A

[AI = n .

are partitions with

lul

= m

N

With this

notation we can define the skew Schur modules where

of all

LA/uv ,

and

as

L A/u v = Hon.E (s".(s"ev@“‘"‘)12n) n

Here

(SHQVQn-m)12n

is the module induced from

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n

EmXEn-m _ module

5 0V

@n-m

over

.

2n .

In particular.

there is a well-known decomposition .

A _ 3 Cu” LUV

LA ,pV where

A

the

CM”

coefficients.

L

A/u

are

the Littlewood-Richardson

Thus we see that for

u = 0

.

V = L V A Next we review the definition of Schur complexes

introduced for characteristic zero by Nielsen [25]. Schur complexes can also be defined Schur modules. action of

2n

For a complex on

Ffin

by

F

in

the same manner as

we can define

I‘

‘

x _1

U r =

2 i