A bijection on core partitions and a parabolic quotient of the affine symmetric group

Let $\ell,k$ be fixed positive integers. In an earlier work, the first and third authors established a bijection between $\ell$-cores with first part equal to $k$ and $(\ell-1)$-cores with first part less than or equal to $k$. This paper gives several new interpretations of that bijection. The $\ell$-cores index minimal length coset representatives for $\widetilde{S_{\ell}} / S_{\ell}$ where $\widetilde{S_{\ell}}$ denotes the affine symmetric group and $S_{\ell}$ denotes the finite symmetric group. In this setting, the bijection has a beautiful geometric interpretation in terms of the root lattice of type $A_{\ell-1}$. We also show that the bijection has a natural description in terms of another correspondence due to Lapointe and Morse.Comment: 23 page

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

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