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

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.

23 pages