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.
Categories: math.CO