The Rook-Brauer Algebra

67 Pages·2012·0.554 MB·English

by  Elise G. delMas

Macalester College
DigitalCommons@Macalester College
Honors Projects Mathematics, Statistics, and Computer Science
5-1-2012
The Rook-Brauer Algebra
Elise G. delMas
Macalester College, [email protected]
Recommended Citation
delMas, Elise G., "The Rook-Brauer Algebra" (2012).Honors Projects.Paper 26.
http://digitalcommons.macalester.edu/mathcs_honors/26
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Honors Project
The Rook-Brauer Algebra
Advisor:
Author:
Halverson
Dr. Tom ,
delMas
Elise
Macalester College
Readers:
Flath
Dr. Dan ,
Macalester College
Garrett
Dr. Kristina ,
St. Olaf College
May 3, 2012
2
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1 Preliminaries 11
1.1 Algebraic Structures . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.1 Groups and Monoids . . . . . . . . . . . . . . . . . . . 11
1.1.2 Associative Algebras . . . . . . . . . . . . . . . . . . . 12
1.1.3 Group Algebras . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Representations and Modules . . . . . . . . . . . . . . . . . . 12
1.2.1 Representations . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3 Irreducible Submodules and Decomposition . . . . . . 14
1.2.4 Algebra Representations and Modules . . . . . . . . . 15
1.2.5 Tensor Product Spaces and Modules . . . . . . . . . . 16
2 The Rook Brauer Algebra RB (x) 17
k
2.1 The Symmetric Group, the Rook Monoid, and the Brauer
Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 The Symmetric Group . . . . . . . . . . . . . . . . . . 17
2.1.2 The Rook Monoid . . . . . . . . . . . . . . . . . . . . 18
2.1.3 The Brauer Algebra . . . . . . . . . . . . . . . . . . . 19
2.2 The Rook-Brauer Algebra . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Multiplication . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 Presentation on generators and relations . . . . . . . . 22
3 Double Centralizer Theory 25
3
4 CONTENTS
3.1 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Examples and Applications . . . . . . . . . . . . . . . . . . . 27
3.2.1 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
k
3.2.2 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
k
3.2.3 B (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
k
3.2.4 RB (n+1) . . . . . . . . . . . . . . . . . . . . . . . . 29
k
4 Action of RB (n+1) on Tensor Space 35
k
4.1 How RB (n+1) Acts on V k . . . . . . . . . . . . . . . . . . 35
k ⊗
4.2 Actions of Generators . . . . . . . . . . . . . . . . . . . . . . 37
4.3 π is a Representation . . . . . . . . . . . . . . . . . . . . . . 38
k
4.4 π is Faithful . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
k
5 The Orthogonal Group O(n) 45
5.1 Definition and Action on Tensor Space . . . . . . . . . . . . . 45
5.2 Commuting with RB (n+1) . . . . . . . . . . . . . . . . . . 45
k
6 Combinatorics and the Bratteli Diagram 51
6.1 A Bijection Between RB (n+1) Diagrams and Paths on . 52
k
B
6.1.1 Insertion Sequences Become Tableaux . . . . . . . . . 53
6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7 Future Work 59
7.1 Seminormal Representations . . . . . . . . . . . . . . . . . . . 59
7.2 Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.2.1 Action of p . . . . . . . . . . . . . . . . . . . . . . . . 61
i
7.2.2 Action of t . . . . . . . . . . . . . . . . . . . . . . . . 61
i
7.2.3 Action of s . . . . . . . . . . . . . . . . . . . . . . . . 61
i
Abstract
We introduce an associative algebra RB (x) that has a basis of
k
rook-Brauer diagrams. These diagrams correspond to partial
matchings on 2k vertices. The dimension of RB (x) is
k
k 2k (2(cid:96) 1)!!. The algebra RB (x) contains the group al-
(cid:96)=0 2(cid:96) − k
gebra of the symmetric group, the Brauer algebra, and the rook
(cid:80) (cid:0) (cid:1)
monoid algebra as subalgebras. We show that RB (x) is gener-
k
ated by special diagrams s ,t (1 i < k) and p (1 j k),
i i j
≤ ≤ ≤
where the s are the simple transpositions that generated the
i
symmetric group S , the t are the “contraction maps” which
k i
generate the Brauer algebra B (x), the p are the “projection
k i
maps” that generate the rook monoid R . We prove that for
k
a positive integer n, the algebra RB (n + 1) is the centralizer
k
algebra of the orthogonal group O(n) acting on the k-fold ten-
sor power of the sum of its 1-dimensional trivial module and
n-dimensional defining module.
5
6 CONTENTS
Introduction
ThispaperfindsthecentralizeralgebraoftheorthogonalgroupoverC,O(n),
acting on the k-fold tensor space V k where V is the (n+1)-dimensional
⊗
module Cn C. The introduction describes the motivation for finding this
⊕
centralizer and the work that has already been done in this area to support
that motivation.
We begin with the general linear group of invertible matrices over C,
GL(n), and the GL(n)-module V = Cn with standard basis v ,v ,...,v .
1 2 n
{ }
This module is the irreducible GL(n)-module labeled by the partition (1),
so V = V(1). We are concerned with the tensor product GL(n)-module V k
⊗
whose basis is the set of simple tensors
v v i 1,...,n .
{ i1 ⊗···⊗ ik | j ∈ { } }
The general linear group acts diagonally on elements of V k, and the cen-
⊗
tralizer algebra of the action of GL(n) on this tensor space is the set of
all GL(n)-module homomorphisms from V k to itself. We refer to this as
⊗
End (V k). It is known that this centralizer algebra is isomorphic to
GL(n) ⊗
the group algebra of the symmetric group CS .
k
The next part of the story is to look at the subgroup of orthogonal ma-
trices O(n) GL(n) acting on this same module V k. The natural question
⊗
⊆
to ask is, what is the centralizer algebra for O(n) acting on V k? Since
⊗
O(n) is contained in GL(n), its centralizer could possibly be much larger
than CS . In 1937, Richard Brauer succeeded in describing the centralizer
k
with an algebra called the Brauer algebra, B (n). This algebra over C has
k
a basis of Brauer diagrams on 2k vertices, which is equivalent to the set of
all possible partitions of a set of 2k elements into blocks of 2.
7
8 CONTENTS
V k
⊗
→ →−
−
GL(n) CS
k
⊆ ⊆
O(n) B (n)
k
Since the centralizer has been described for the actions of both GL(n)
and O(n) on the module V k, we now wish to explore the centralizer of
⊗
GL(n) and O(n) acting on a slightly different module. Now let V = V(1) V
∅
⊕
whereV isthetrivial1-dimensionalsubmoduleofGL(n). GL(n)actsonthe
∅
(n+1)k-dimensional module V k diagonally. The centralizer for this action
⊗
has been described by the rook monoid algebra CR . In this paper we take
k
the next natural step and describe the centralizer of O(n) GL(n) acting
⊆
onthe(n+1)k-dimensionaltensorspaceV k, whichwecalltherook-Brauer
⊗
algebra RB (n+1).
k
V k
⊗
→ →−
−
GL(n) CR
k
⊆ ⊆
O(n) RB (n+1)
k
This thesis is organized as follows.
Chapter 1 refreshes some important background information on rep-
•
resentations of groups and algebras, though some prior knowledge is
assumed.
In Chapter 2, we discuss the three important subalgebras of RB (x),
k
•
and we define the rook-Brauer algebra with a basis of rook-Brauer
diagrams on 2k vertices. The set of these diagrams on 2k vertices is
equivalenttothenumberofpartitionsofasetof2k elementsintoparts
of size 1 or 2.
Chapter 3 discusses double centralizer theory and the motivations be-
•
hind studying the rook-Brauer algebra.
InChapter4wedefineanactionofRB (n+1)onthetensorspaceV k
k ⊗
•
and prove that this action creates a representation of RB (n+1) and
k
CONTENTS 9
that this representation is faithful for n k. This shows that there is
≥
an injective linear transformation between RB (n+1) and End(V k),
k ⊗
the set of all endomorphisms of V k.
⊗
Chapter 5 presents the proof that the action of RB (n+1) commutes
k
•
with the action of O(n) on V k, which shows that in fact when n k,
⊗
≥
RB (n+1) End (V k) as a subalgebra.
k O(n) ⊗
⊆
Finally in Chapter 6 we use combinatorics on the Bratteli diagram of
•
V k toshowthatthedimensionofRB (n+1)isequaltothedimension
⊗ k
of End (V k).
O(n) ⊗
Chapter 7 discusses future work on this project, which includes con-
•
structing the irreducible representations of RB (n+1).
k
Acknowledgements
ThankyoutomyreadersDanFlathandKristinaGarrett,andtomyadvisor
Tom Halverson. This research was partially supported by National Science
Foundation Grant DMS-0800085.