Table Of ContentModular Representations of the
Symmetric Group: Structure of Specht
Modules
by
Craig J. Dodge
March 5, 2012
A dissertation submitted to the
Faculty of the Graduate School of
the University at Buffalo, State University of New York
in partial fulfillment of the requirements for the
degree of
Doctor of Philosophy
Department of Mathematics
Dissertation Committee:
Professor David Hemmer, Advisor
Professor Bernard Badzioch
Professor Yiqiang Li
Abstract
In this paper we will study the structure of Specht modules over a field of finite characteristic. Our
primary concern will be the existence and construction of homomorphisms between certain Specht
modules. We use the well understood radical filtrations of Specht modules in Rouquier blocks to
find Specht modules, whose homomorphism space will be of arbitrary dimension. We will study
semistandard homomorphisms and techniques developed by Fayers and Martin to find a previously
unknown infinite family of decomposable Specht modules in characteristic two.
ii
Acknowledgement
I would like to thank my advisor David J. Hemmer for his suggestions and guidance throughout
my entire graduate career. I owe a great deal to the collaboration of Mathew Fayers, which was a
fantastic experience and vital to this thesis. I would also like to thank my fian´cee Christy whose
love and support has made this challenge so much easier.
iii
Contents
Abstract ii
Acknowledgement iii
1 Introduction 1
1.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 FΣ -Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
n
1.3 Littlewood-Richardson Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Blocks and Cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Representing Partitions on the Abacus . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Rouquier Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Large Dimensional Homomorphism Spaces 13
2.1 Radical Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Repeated Composition Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Radical Layers of S(γ) and S((cid:15)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Specht Module Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Decomposable Specht Modules 22
3.1 Semistandard Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Homomorphism from S(n−3,3) to S(n−5,3,1,1) . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Homomorphism from S(n−5,3,1,1) to S(n−3,3) . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Composition of Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Bibliography 39
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Chapter 1
Introduction
Our goal is to understand the modular representation theory of the symmetric group. If F is an
algebraically closed field, and Σ denotes the permutation group on n elements, then our goal is
n
equivalent to understanding modules of the group algebra FΣ . We will begin by reviewing some
n
of the well known facts about symmetric group representations. Partitions of n have a natural
connection to the modules of FΣ , so it is necessary to first introduce some notation for partitions.
n
1.1 Partitions
We say λ = (λ ,λ ,...) is a composition of n if λ ,λ ,... is a sequence of non-negative integers such
1 2 1 2
(cid:88)
that λ = n. We say λ as a partition of n, and write λ (cid:96) n, if λ is a composition of n such
i
i
that the sequence λ ,λ ,... is non-increasing. If λ = (λ ,...,λ ) it is understood that λ = 0 for all
1 2 1 s i
i > s. We will often refer to partitions of an unspecified integer, therefore in some cases we will
(cid:88)
say that λ = (λ ,...,λ ) is a partition and |λ| = λ . Let Λ be the set of all partitions of any
1 s i
i
positive integer.
Young Diagrams
It will be useful to represent partitions by Young diagrams. If λ = (λ ,...,λ ) then we will define
1 s
the diagram [λ] := {(i,j)|i,j ∈ Z, 1 ≤ i ≤ s, 1 ≤ j ≤ λ } to be the Young diagram of λ. We will
i
call an element (i,j) ∈ [λ] a node of [λ]. It is useful to represent [λ] with a diagram of boxes, one
boxforeachnode,adjustedtothetopandleft. Forexampleletλ (cid:96) 16suchthatλ = (6,4,2,2,1,1).
The Young diagram [λ] would be illustrated as follows:
1
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4)
(3,1) (3,2)
.
(4,1) (4,2)
(5,1)
(6,1)
In the above figure each box is filled in with the node that box corresponds to. Often it will be
convenient to refer to a partition by its Young diagram, so we may not distinguish between λ and
[λ].
For a given diagram [λ], we will distinguish certain nodes. Define (i,λ ) to be removable if
i
λ > λ . Define (i,λ +1) to be an addable node if λ > λ . These labels are appropriate,
i i+1 i i−1 i
since a removable node, (i,j) ∈ [λ], is a node such that [λ]−{(i,j)} is a diagram for a partition
of |λ|−1. Similarly an addable node, (i,j) ∈ [λ], is a node such that [λ]∪{(i,j)} is a partition of
|λ|+1.
Fix a positive integer e, we will say that a partition λ is e-singular if there exists i ∈ Z such
that λ = λ = ... = λ . If λ is not e-singular, then λ is e-regular. Define λ to be e-restricted
i+1 i+2 i+e
if there exists no i ∈ Z such that λ −λ > e. Let [λ(cid:48)] = {(i,j) ∈ Z2|(j,i) ∈ [λ]}, that is the
i i+1
tableau obtained by interchanging the rows and columns of [λ]. The partition λ(cid:48) is referred to as
the conjugate of λ. It is easy to see that λ is e-regular if and only if λ(cid:48) is e-restricted.
Ordering of Partitions
Define Λ+(n) := {λ|λ (cid:96) n}. We will now define a total order on Λ+(n).
Definition 1.1.1. If λ,µ ∈ Λ+(n), let k = min{i ∈ Z|λ (cid:54)= µ }, then we write λ > µ if and only if
i i
λ > µ .
k k
This total order is often referred to as the dictionary or lexicographic order on the set of
partitions. We can also define a useful partial order on Λ+(n), called the dominance order.
Definition 1.1.2. If λ,µ ∈ Λ+(n), we write λ(cid:2)µ if and only if
s s
(cid:88) (cid:88)
λ ≤ µ
i i
i=1 i=1
for all s ≥ 1. In this case we say that µ dominates λ.
Thelexicographictotalorderisarefinementofthedominanceorder, thatisifλ(cid:2)µthenλ ≤ µ.
λ-Tabloids
If λ (cid:96) n, we define a λ-tableau as a bijection from [λ] to the set {1,2,...,n}. We can represent a
λ-tableau with a diagram by replacing each node with the integer it is assigned by the bijection.
2
For example,
1 2 3 4 5 6
7 8 9 10
1112
1314
15
16 (1.1.1)
is a diagram of a (6,4,2,2,1,1)-tableau. Any permutation of these sixteen integers also produces
a (6,4,2,2,1,1)-tableau. We define a λ-tableau t to be standard if it is strictly increasing along its
rowsandcolumns. Example(1.1.1)isastandard(6,4,2,2,1,1)-tableau. Itwilloftenbeconvenient
to refer to a λ-tableau by its diagram, so we will use both descriptions interchangeably.
If λ (cid:96) n, then we have a natural action of Σ on the set of λ-tableaux. We can define this
n
action in the following manner, if σ ∈ Σ and φ : [λ] → {1,2,...,n} is a bijection, then we define
n
σφ := σ◦φ. We can easily describe the action of σ on a diagram of a λ-tableau. If t is a λ-tableau
then σt is the λ-tableau obtained by replacing the entry equal to i with σ(i) for each box of t.
We can now use this group action to define an equivalence relation on the set of λ-tableaux.
Let t be a λ-tableau. Define R := {σ ∈ Σ |∀i, i and σi are in the same row of t} to be the row-
t n
stabilizer of of t. These are exactly the elements of Σ that fix the rows of t as sets. Note that if
n
λ has d non-zero parts, then
∼
R = Σ ×Σ ×...×Σ .
t λ1 λ2 λd
We can similarly define C to be the column-stabilizer of a λ-tableau t.
t
Let t and s be λ-tableaux. We will say t is row-equivalent to s if and only if there exists π ∈ R
t
such that πt = s. If t is row-equivalent to s, we write t ∼ s. Since ∼ is an equivalence
row row
relation, we will denote {t} as the equivalence class of t under the relation of ∼ . We refer to {t}
row
as a λ-tabloid. We shall represent tabloids with diagrams similar to the tableau, but we will not
distinguish the position of the entry in each row. For example if
1 2
t = 3 ,
then we represent {t} by
1 2 2 1
{t} = = .
3 3
p-Regularization
We wish to describe the process of p-regularization of a partition. Let λ (cid:96) n and [λ] be the
associated Young diagram. Fix some prime p. We define the kth ladder of N×N, L , to be the
k
following subset:
L := {(i,j)|i+(p−1)j = k+p−1}.
k
The kth ladder of [λ] is the intersection of L with [λ]. We say the regularization of λ, denoted λreg,
k
is the partition associated to the Young diagram obtained by moving all the nodes of [λ] as far up
3
each ladder as they can go. For example let p = 3. If λ = (43,24,12) then λreg = (5,42,32,2,1).
The following diagram illustrates both λ and λreg. Each node has been filled in with the ladder
that node is in.
1 3 5 7
2 4 6 8 1 3 5 7 9
3 5 7 9 2 4 6 8
4 6 3 5 7 9
λ = 5 7 λreg = 4 6 8 .
6 8 5 7 9
7 9 6 8
8 7
9
It should be clear that λreg is a p-regular partition. We can similarly define a process called p-
restriction to obtain a p-restricted partition from λ, which we call λr. To do this simply define
λr := ((λ(cid:48))reg)(cid:48).
1.2 FΣ -Modules
n
To each partition λ (cid:96) n, we associate a Young subgroup of Σ , Σ . If λ has d non-zero parts, we
n λ
define
Σ := Σ ×Σ ×...×Σ .
λ {1,2,...,λ1} {λ1+1,...,λ1+λ2} {n−λd+1,...,n}
DefineMλ := IndΣn1asaZΣ -module, where1representsthetrivialmodule. Wecallthismodule
Z Σ n
λ
thepermutationmoduleassociatedwithλ. Wehaveanotherdefinitionforthepermutationmodule
Mλ using the λ-tabloids.
Z
MZλ := (cid:110)(cid:88)a{t}{t}(cid:12)(cid:12)a{t} ∈ Z, {t} is a λ-tabloid(cid:111).
Since both are transitive actions with the same point stabilizer, it is not difficult to see, Σ acting
n
on the set of λ-tabloids is equivalent to acting on the set of cosets of Σ . This definition gives a
λ
well defined inner product, (cid:104), (cid:105), on Mλ, defined by
Z
{t}
(cid:104){t},{s}(cid:105) = δ .
{s}
We can now define a very important submodule of the permutation module, Mλ, which will be
Z
of great interest to us. Fix a λ-tableau t. Let C be the column stabilizer of t. Define
t
(cid:88)
κ := sgn(σ)σ ∈ ZΣ .
t n
σ∈Ct
Define e := κ {t} ∈ Mλ. We call e the polytabloid associated with t. Define
t t Z t
SZλ := SpanZ(cid:8)et(cid:12)(cid:12)t is a λ-tableau(cid:9) ⊆ MZλ.
4
Thus for each λ (cid:96) n and field F, we associate a FΣn-module, Sλ := F ⊗ZSZλ, which we call the
Specht module associated with λ. It is true but not obvious that dimSλ is independent of the field
[14, Corollary 8.5]. It is also true that Sλ has a basis {e |t is a standard} regardless of the ground
t
field.
Over the complex number then Sλ is an irreducible module, and {Sλ|λ (cid:96) n}, is a complete set
of non-isomorphic CΣ -modules. When F is a field of characteristic p > 0, Sλ is not irreducible in
n
general. However, James proved if Sλ⊥ is the submodule of Mλ which is perpendicular to Sλ with
respect to the inner product on Mλ, then we can describe the simple FΣ -modules in the following
n
way: if λ is p-regular, then Sλ/(Sλ ∩Sλ⊥) is the unique irreducible quotient of Sλ [14, Theorem
11.1]. We will label this module Dλ. It is well understood that the set, {Dλ|λ is p-regular}, forms
a complete list of non-isomorphic simple FΣ modules [14, Theorem 11.5].
n
We also find that if λ is p-restricted then Sλ has a unique irreducible submodule, which we
denote D . Similar to the p-regular case, we have {D |λ is p-restricted} is a complete set of non-
λ λ
isomorphic FΣ -modules. Moreover we can define a bijection between these two lists of simple
n
modules, but first we must make a few more observations.
Let sgn be the one dimensional module of FΣ , where Σ acts by multiplication by the sign of
n n
the permutation. James proved that
Sλ⊗sgn ∼= (Sλ(cid:48))∗, (1.2.1)
where (Sλ)∗ is the dual of Sλ [14, Theorem 8.15]. In characteristic zero the Specht modules are
self-dual so this statement simplifies to Sλ ⊗sgn ∼= Sλ(cid:48). In characteristic p this is no longer the
case. Still we have Dλ ⊗sgn is irreducible. We establish a bijection between p-regular partitions
called the Mullineaux map. This map is defined as the bijection which has the property
Dm(λ) ∼= Dλ⊗sgn.
A conjecture for the explicit combinatorial definition was made by Mullineux [19], which was later
proven correct by Ford and Kleshchev [9, Theorem 4]. From this we can recognize the relation
between the two indexings of simple modules:
Dλ ∼= D ⊗sgn ∼= D
λ(cid:48) (m(λ))(cid:48)
D ∼= Dλ(cid:48) ⊗sgn ∼= D(m(λ(cid:48))).
λ
The irreducible FΣ modules are not very well understood and not even a general formula for their
n
dimension is known. Since the simple modules are so closely related to the Specht modules, it is
clear that understanding the structure of the Specht modules is necessary to understanding the
modular representations of the symmetric group. We have a very important structure theorem of
James which will help understand the composition factors of a Specht module.
Theorem 1.2.1. [13, Theorem A] Let F be a field of characteristic p. Suppose λ,µ (cid:96) n such that
µ is p-regular, then [Sλ : Dλreg] = 1 and [Sλ : Dµ] (cid:54)= 0 implies µ(cid:4)λreg.
5
WewillbeparticularlyconcernedwithunderstandinghomomorphismsbetweenSpechtmodules.
This theorem will be important to us in order to find a homomorphism from Sµ to Sλ, if µ is p-
regular. This theorem tells us that for there to be any chance of such a non-zero homomorphism to
exist, we will require µ(cid:4)λreg. In the next section we will discuss another condition on partitions
which we will need to understand in order to construct non-trivial homomorphisms between two
Specht modules.
1.3 Littlewood-Richardson Rule
Foreachλ (cid:96) nwehaveanirreduciblemoduleSλ ofCΣ . Eachelementσ ∈ Σ actsbyaninvertible
n n
linear transformation on Sλ. Since this action preserves the group operation, we get a well defined
homomorphism φ : Σ → GL(Sλ). Now we can define a class function ϕ : Σ → C by
λ n λ n
ϕ (σ) = Tr(φ (σ)),
λ λ
where Tr is the trace function. The set of {ϕ |λ (cid:96) n} is the set of irreducible characters of Σ and
λ n
is actually a basis for the set of class functions on Σ .
n
Notice Sν ⊗ Sµ can be thought of as an C(Σ ×Σ )-module by the action
F |ν| |µ|
(σ,τ)◦(v⊗w) = (σ◦v)⊗(τ ◦w).
If n = |ν|+|µ|, then induction gives us a CΣ -module. Let Φ be the class function induced by
n ν,µ
the CΣ -module IndΣn (Sν ⊗ Sµ). Since the irreducible characters form a basis for the class
n Σ ×Σ F
|ν| |µ|
functions we may define Littlewood-Richardson coefficients, c(λ;ν,µ), as follows:
(cid:88)
Φ = c(λ;ν,µ)ϕ .
ν,µ λ
λ(cid:96)n
There exists a rule has for computing the Littlewood-Richardson coefficients using tableaux.
We say the set [λ \ µ] := {(i,j) ∈ Z2|(i,j) ∈ [λ] \ [µ]} is a skew diagram of shape λ \ µ. Let
ν = (ν ,ν ,...,ν ). If t is a box diagram of λ\µ, with each node assigned a positive integer less
1 2 s
than or equal to s such that the number of entries equal to k is ν ; then we say t is a λ\µ tableau
k
of weight ν. We say such a skew diagram is semistandard if it is non-decreasing along each row
and strictly increasing along each column. For example the diagram below is a semistandard skew
tableau of shape[(5,3,1,1)\(2,1)] of weight (2,2,1,1,1).
1 1 2
2 3
4
5
In order to state the rule for computing Littlewood-Richardson coefficients, we must first define
a ballot sequence.
6
