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

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