Let $\ell,k$ be fixed positive integers. In an earlier work, the first andthird authors established a bijection between $\ell$-cores with first partequal 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 theaffine symmetric group and $S_{\ell}$ denotes the finite symmetric group. Inthis setting, the bijection has a beautiful geometric interpretation in termsof the root lattice of type $A_{\ell-1}$. We also show that the bijection has anatural description in terms of another correspondence due to Lapointe andMorse.
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