Partition Algebras and Kronecker Coefficients [Master thesis]

102 Pages·2015·0.481 MB·English

by  Cameron Marcott

Partition Algebras and Kronecker
Coefficients
by
Cameron Marcott
A thesis
presented to the University of Waterloo
in fulfillment of the
thesis requirement for the degree of
Master of Mathematics
in
Combinatorics and Optimization
Waterloo, Ontario, Canada, 2015
(cid:13)c Cameron Marcott 2015

I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis,
including any required final revisions, as accepted by my examiners.
I understand that my thesis may be made electronically available to the public.
iii

Abstract
Classical Schur-Weyl duality relates the representation theory of the general linear group
to the representation theory of the symmetric group via their commuting actions on tensor
space. With the goal of studying Kronecker products of symmetric group representations,
the partition algebra is introduced as the commutator algebra of the diagonal action of the
symmetric group on tensor space. An analysis of the representation theory of the partition
offers results relating reduced Kronecker coefficients to Kronecker coefficients.
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Acknowledgements
I’d like to thank my supervisor Kevin Purbhoo for guiding me as I explored a large
variety mathematical topics over the past two years, and for offering valuable editorial
advice on this thesis. Also, I’d like to thank David Wagner and Jason Bell for agreeing to
serve as readers for this thesis.
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Table of Contents
List of Figures xi
1 Introduction 1
2 Representation Theory Background 3
2.1 Representations of Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Isotypic Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Induction and Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Bratteli Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Representation Theory of C[S ] . . . . . . . . . . . . . . . . . . . . . . . . 10
n
2.6 Kronecker Products of C[S ] Representations . . . . . . . . . . . . . . . . 17
n
3 Centralizer Algebras and Schur-Weyl Duality 19
3.1 Centralizer Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Schur-Weyl Duality and Seesaw Reciprocity . . . . . . . . . . . . . . . . . 21
3.3 A Centralizer for the Symmetric Group Algebra . . . . . . . . . . . . . . . 25
4 The Partition Algebra 29
4.1 Building the Partition Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Pretty Pictures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Special Elements, Ideals, and Quotients . . . . . . . . . . . . . . . . . . . . 34
4.4 A Trace for the Partition Algebra . . . . . . . . . . . . . . . . . . . . . . . 35
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5 Representation Theory of the Partition Algebra 37
5.1 Green’s Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6 Jones’ Basic Construction 45
6.1 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.2 Jones’ Construction and the Partition Algebra . . . . . . . . . . . . . . . . 50
7 Quasi-hereditary Algebras 59
7.1 Hand waving some hard math . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.2 Quasi-hereditary Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.3 P (n) as a Quasi-hereditary Algebra . . . . . . . . . . . . . . . . . . . . . 69
k
8 The n-pair Condition 73
9 Results about Kronecker Coefficients and Concluding Remarks 83
9.1 Results about Kronecker Coefficients . . . . . . . . . . . . . . . . . . . . . 83
9.2 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Bibliography 87
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