Combinatorics of $(\ell,0)$-JM partitions, $\ell$-cores, the ladder crystal and the finite Hecke algebra

TL;DR

This thesis explores the combinatorial structures underlying the representation theory of finite Hecke algebras, extending existing results on irreducibility criteria, crystal descriptions, and core bijections.

Contribution

It introduces new combinatorial descriptions for irreducibility of Specht modules and deepens understanding of core bijections related to affine symmetric groups.

Findings

Extended irreducibility criteria for Specht modules.

New crystal descriptions related to affine Lie algebra representations.

Deeper analysis of core bijections and their combinatorial properties.

Abstract

The following thesis contains results on the combinatorial representation theory of the finite Hecke algebra $H_n(q)$. In Chapter 2 simple combinatorial descriptions are given which determine when a Specht module corresponding to a partition $\lambda$ is irreducible. This is done by extending the results of James and Mathas. These descriptions depend on the crystal of the basic representation of the affine Lie algebra $\widehat{\mathfrak{sl}_\ell}$. In Chapter 3 these results are extended to determine which irreducible modules have a realization as a Specht module. To do this, a new condition of irreducibility due to Fayers is combined with a new description of the crystal from Chapter 2. In Chapter 4 a bijection of cores first described by myself and Monica Vazirani is studied in more depth. Various descriptions of it are given, relating to the quotient $\widetilde{S_\ell}/{S_\ell}$ and to the bijection given by Lapointe and Morse.