On unipotent Specht modules of finite general linear groups [Elektronische Ressource] / vorgelegt von Marco Brandt

On unipotent Specht modules of finite general linear groups [Elektronische Ressource] / vorgelegt von Marco Brandt

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On Unipotent Specht Modules ofFinite General Linear GroupsVon der Fakult at Mathematik und Physik der Universit at Stuttgart zurErlangung der Wurde eines Doktors der Naturwissenschaften (Dr. rer. nat.)genehmigte AbhandlungVorgelegt vonMarco Brandtaus BoblingenHauptberichter: Prof. Dr. R. DipperMitberichter: Prof. G. D. JamesProf. Dr. W. KimmerleTag der mundlic hen Prufung: 12. Februar 2004Institut fur Algebra und Zahlentheorie der Universit at Stuttgart2004ContentsIntroduction iv1 Basics 11.1 The setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Compositions and Partitions . . . . . . . . . . . . . . . . . . . . . 11.3 -tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Gaussian polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 - ags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Finite groups with a BN-pair . . . . . . . . . . . . . . . . . . . . 122 Facts about Specht modules S 142.1 The permutation module M . . . . . . . . . . . . . . . . . . . . 142.2 The Specht module S . . . . . . . . . . . . . . . . . . . . . . . . 202.3 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 . . . . . . . . . . . . . . . . . . . . . 263.3 The polynomials p (q) . . . . . . . . . . . . . . . . . . . . . . .

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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