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

TL;DR

This paper explores a bijection between certain core partitions, providing new geometric and algebraic interpretations, and connecting it to the structure of affine symmetric groups and root lattices.

Contribution

It offers new interpretations of a known bijection, linking core partitions to affine symmetric group coset representatives and root lattice geometry.

Findings

Bijection relates $ ext{ell}$-cores and $( ext{ell}-1)$-cores with specific first part constraints

Geometric interpretation in terms of root lattice of type $A_{ ext{ell}-1}$

Connection to Lapointe and Morse's correspondence

Abstract

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.