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

Let l, k be fixed positive integers. In [1], the first and third authors established a bijection between l-cores with first part equal to k and (l − 1)-cores with first part less than or equal to k. This paper gives several new interpretations of that bijection. The l-cores index minimal length coset representatives for f Sl/Sl where f Sl denotes the affine symmetric group and Sl denotes the finite symmetric group. In this setting, the bijection has a beautiful geometric interpretation in terms of the root lattice of type Al−1. We also show that the bijection has a natural description in terms of another correspondence due to Lapointe and Morse [8].