On Unipotent Specht Modules of

Finite General Linear Groups

Von der Fakult at Mathematik und Physik der Universit at Stuttgart zur

Erlangung der Wurde eines Doktors der Naturwissenschaften (Dr. rer. nat.)

genehmigte Abhandlung

Vorgelegt von

Marco Brandt

aus Boblingen

Hauptberichter: Prof. Dr. R. Dipper

Mitberichter: Prof. G. D. James

Prof. Dr. W. Kimmerle

Tag der mundlic hen Prufung: 12. Februar 2004

Institut fur Algebra und Zahlentheorie der Universit at Stuttgart

2004Contents

Introduction iv

1 Basics 1

1.1 The setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Compositions and Partitions . . . . . . . . . . . . . . . . . . . . . 1

1.3 -tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Gaussian polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 - ags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Finite groups with a BN-pair . . . . . . . . . . . . . . . . . . . . 12

2 Facts about Specht modules S 14

2.1 The permutation module M . . . . . . . . . . . . . . . . . . . . 14

2.2 The Specht module S . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 The branching theorem . . . . . . . . . . . . . . . . . . . . . . . . 21

(n m,m)3 The Specht modules S 24

(n m,m)3.1 The permutation module M . . . . . . . . . . . . . . . . . 24

(n m,m)3.2 Basic properties of S . . . . . . . . . . . . . . . . . . . . . 26

3.3 The polynomials p (q) . . . . . . . . . . . . . . . . . . . . . . . . 29t

3.4 Calculation of the polynomials p (q) . . . . . . . . . . . . . . . . . 37t

3.5 The idempotents e . . . . . . . . . . . . . . . . . . . . . . . . . . 43L

3.6 Construction of the elements b . . . . . . . . . . . . . . . . . . . 46L

3.7 IsB a basis of S ? . . . . . . . . . . . . . . . . . . . . . . . . . 64

(2,2,2)4 The Specht module S 80

4.1 Basic de nitions and properties . . . . . . . . . . . . . . . . . . . 80

4.2 Using the branching theorem . . . . . . . . . . . . . . . . . . . . . 83

5 German summary 96

A Notation 106

B Some polynomials p (q) 109t

Bibliography 111Introduction

Many outstanding problems in representation theory can be solved with a proper

understanding of the irreducible unipotent modules for the nite general linear

group GL (q) [5, 6].n

In [11], Gordon James investigated these irreducible unipotent modules: For each

partition of n there is a Specht module S for GL (q), de ned over a eld Fn

in terms of the intersection of the kernels of certain homomorphisms. If F is a

eld of characteristic zero, then S is irreducible and {S | is a partition of

n} is a complete set of pairwise non-isomorphic irreducible unipotent modules

for GL (q). If the characteristic of F is coprime to q, then, in general, S hasn

a unique top composition factor D and the D ’s are the irreducible unipotent

modules for GL (q).n

For each Specht module S , a generating element e is known but, in general,

no explicit basis for S as a vector space over F has been found. In [7], Richard

Dipper and Gordon James make signi cant progress towards the construction of

a basis of S for a two part partition . My thesis is based on this paper and

further develops and improves the techniques introduced there.

Chapter 1 sets the scene and gives an overview of the fundamental de nitions

and propositions in the area of compositions, partitions, -tableaux and Gaus-

sian polynomials. Furthermore, we introduce - ags and a manageable notation

for these chains of vector subspaces. We conclude the chapter by delivering a

short insight into the theory of nite groups with a BN-pair.

In chapter 2 we de ne M as vector space over F with basis . The canonical

operation of GL (q) on turns M into an FGL (q)-module. As we can assignn n

a -tableau to each - ag and we have a total ordering on the set of -tableaux,

P

wede ne,foranelement v = c X ∈M ,last(v)asthelast-tableauwhichX

X∈

can be assigned to a - ag X occurring in this sum with nonzero coe cient c .X

Motivated by the fact that the unipotent Specht module S is a submodule of

M ,wecarefullyexamineM andtheoperationofGL (q)onM .Nextwede nen

S as the intersection of the kernels of certain homomorphisms and present our

main tool for understanding the structure of S , namely the branching theorem.

Sinead Lyle proves in her thesis [14] that, for every element v ∈ S , last(v) is a

standard -tableau. This leads us to the de nition of a standard basis of S , i.e.

a basisB = {b | i ∈I} of S , which is de ned independently of the concretei

choice of the eld F, together with a set of polynomials{p (q)|t∈Std()} sucht

that p (q) =|{b∈B | last(b) =t}| and p (1) = 1 holds for everyt∈Std().t t

Finding a standard basis ofS for a two part partition = (n m,m) is the goalv

ofchapter3.Westartwiththeintroductionofashortnotationofthe(n m,m)-

rk ags. Then we de ne, for every t∈Std((n m,m)), a subsetM (q) of (n m,m)t

rkand set p (q) := |M (q)|. By a recursive approach we develop an algorithm tot t

calculatep (q) and prove that allp (q) are polynomials overq. The main theoremt t

(n m,m) rkof this chapter is the existence of a setB ={b | L∈M (q),t∈Std()}L t

(n m,m)of linearly independent vectors in S . The proof of this theorem is con-

rkstructive and lists, for every t ∈ Std() and every L ∈ M (q), the operationst

(n m,m)necessary to obtain the element b from the generator e of S . Un-L (n m,m)

(n m,m) (n m,m)fortunately we can’t prove thatB is a generating system of S and

therefore it remains only a conjecture. But with the help of GAP [9], the idea

from [7] to divide Std((n m,m)) in some special intervals and the branching

theorem,wecollectalotofevidenceforthisconjecture.Weformulatetwofurther

(n m,m) (n m,m)conjectures and nally prove that B is a standard basis of S with

corresponding polynomials {p (q) |t∈Std((n m,m))}, if 1m 11.t

(2,2,2)In chapter 4 we deal with the Specht module S . The third part in the parti-

tion (2,2,2) signi cantly complicates the task. But again the branching theorem

(2,2,2)turnsouttobeverypowerfulandhelpsustoconstructastandardbasisofS .Acknowledgments

Many people have supported, encouraged and helped me during the time I spent

working on this thesis. I wish to express my gratitude to all of them.

First of all, I would like to thank my supervisor Prof. Dr. Richard Dipper. He

has been a great source of motivation and I am grateful to him for having

introduced me to the fascinating research area of representation theory of the

nitegenerallineargroupandforguidingmyresearchworkthatledtothisthesis.

Furthermore, I would like to thank my co-supervisors Prof. Gordon James and

Prof. Dr. Wolfgang Kimmerle for reading this thesis.

Many thanks to my colleagues and friends at the ”Abteilung fur Darstellungsthe-

orie” and the ”Fachschaft Mathematik” who have made me feel very comfortable

at the University of Stuttgart.

I would also like to thank Vanessa Miemietz for proof-reading this thesis.

For nancial support I am grateful to the Deutsche Forschungsgemeinschaft

(DFG) and my grandparents.

Finally, I would like to thank my parents for their encouragement and their

invaluable support over the last years which allowed me to fully concentrate on

my research and thus signi cantly contributed to the successful completion of

this thesis.Chapter 1

Basics

1.1 The setting

Throughout this thesis n is a natural number, p a prime, q a power of p and F a

th eld whose characteristic is coprime to p and which contains a primitive p root

of unity. GF(q) denotes the nite eld of q elements, GF(q) its multiplicative

groupandGL (q)thegroupofinvertiblennmatricesoverGF(q).Themonoidn

of ab matrices over GF(q) is referred to asM (q).a,b

LetX beaset.ThenwedenotebyS thegroupofpermutationsonX.MoreoverX

S :=S is the symmetric group on n numbers.n {1,2,...,n}

We embed S into GL (q) by assigning to a permutation the appropriaten n

permutation matrix P = (p )∈GL (q), whereij n

(

1 for j =i

p :=ij

0 otherwise.

A permutation of the form (i,j) is called transposition and a permutation of the

form (i,i+1) is a basic transposition.

If a and b are vectors over GF(q) of the same length l, we have the canonical

scalar product

l

X

ha,bi := ab ∈GF(q).i i

i=1

Furthermore, we x, once and for all, a non-trivial group homomorphism

: (GF(q),+)→F .

Thus, is a linear F-character of the group (GF(q),+).

1.2 Compositions and Partitions

In this section we introduce the fundamental de nitions of compositions and

partitions. Thereby we follow [12] and [15].2 Basics

1.2.1 De nition:

1.) = ( , , ,...) is a composition of n, if , , ,... are non-negative1 2 3 1 2 3

integers with

∞X

|| := =n.i

i=1

The non-zero are called the parts of . The last part is denoted by (hi h

standing for ”height”).

2.) A partition of n is a composition of n for which

....1 2 3

In the notation of compositions we often suppress the zeros at the end.

For example

(2,0,3,1,4,0,0,...) = (2,0,3,1,4).

Inpartitionsthesequenceofentriesisunique,becausetheymustdecrease.There-

fore we can indicate repeated parts by a superscript.

For example

3(4,2,2,2,1,0,0,...) = (4,2 ,1).

1.2.2 De nition: If is a composition of n, then the diagram [] is the set

{(i,j) | i,j ∈ Z,1 i,1 j }. If (i,j) ∈ [], then (i,j) is called a nodei

thof []. The k row (respectively, column) of a diagram consists of those nodes

whose rst (respectively, second) coordinate is k.

We shall draw diagrams as in the following example.

1.2.3 Example:

For = (4,2,3,1) we have [] = .

01.2.4 De nition: Suppose is a composition. If (j 1) equals the numberj

0 0 0 0of nodes in column j of [], then = ( , , ,...) is obviously a partition. It1 2 3

is called the conjugate partition of .

1.2.5 Example:

0For = (3,2,1,3) with [] = we get = (4,3,2).

We de ne a partial order on the set of compositions of n.

1.2.6 De nition: If and are compositions of n, we write D if

j j

X X

0 0 for all j 1i i

i=1 i=1

and C if E but =.

61.3 -tableaux 3

1.2.7 Example:

1.) We have the following order on the partitions of 3:

D D .

2.) It is possible to have E and E without =:

D D .

3.) D is only a partial order. For example, we can’t order the elements

and .

1.2.8 De nition: Let = ( , ,..., ) be a partition of n and (i,j)∈ [].1 2 h

Then the (i,j)-hook of [] consists of the (i,j)-node along with the j nodesi

0to the right of it and the i nodes below it.j

The length of the (i,j)-hook is

0h = + +1 i j.ij i j

If we replace the (i,j)-node of [] by the numberh for each node, we obtain theij

hook graph.

1.2.9 Example: Let = (4,2,1). Then

6 4 2 1

3 1

1

is the hook graph of .

1.3 -tableaux

We continue with the introduction of -tableaux and related de nitions. Our

main reference is [12].

1.3.1 De nition: Let be a composition of n. A -tableau is one of the n!

arrays of integers obtained by replacing each node in [] by one of the integers

1,2,3,...,n, allowing no repeats.

1.3.2 Example: Let = (1,2,4,1).

Then

1 1

2 3 2 5

t = andt =1 2

4 5 6 7 3 6 7 8

8 4

are -tableaux.4 Basics

The symmetric groupS acts on the set of-tableaux by permuting the integersn

1,2,3,...,n. For example

t (3,5,6,7,8,4) =t .1 2

1.3.3 De nition: If is a composition andt a -tableau, then

the row i oft is the set {x∈ N | there exists a node in row i of [] which,

int, is replaced with x} and

the column j oft is the set {x∈ N | there exists a node in column j of []

which, int, is replaced with x}.

1.3.4 Example: In example 1.3.2 the set {3,5} is column 2 of t and the set1

{3,6,7,8} is row 3 oft .2

1.3.5 De nition: Suppose is a composition of n andt a -tableau. We de ne

functions

row :{1,2,...,n}→{1,2,...}t

b7→i

if b belongs to row i oft and

col :{1,2,...,n}→{1,2,... n}t

b7→j

if b belongs to column j oft.

1.3.6 De nition: Let be a composition andt a -tableau. Thent is

1.) row-standard, if the numbers increase along the rows oft. We write T ()rs

for the set of row-standard -tableaux.

2.) column-standard, if is a partition and the numbers increase down the

columns oft.

3.) standard, ift is row-standard and column-standard. We denote the set of

standard -tableaux by Std().

1.3.7 De nition: Let be a composition of n. The -tableau, where the nodes

are replaced with the numbers 1,2,...,n in order along

1.) the rows is called the initial tableau and is denoted byt .

2.) the columns is denoted byt .

1.3.8 Example: Let = (1,2,4,1) andt ,t as in example 1.3.2. Then1 2

t =t andt =t .1 2