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

Let ℓ , k be fixed positive integers. In [C. Berg, M. Vazirani, ( ℓ , 0 ) -Carter partitions, a generating function, and their crystal theoretic interpretation, Electron. J. Combin. 15 (2008) R130], 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 S ℓ ˜ / S ℓ where S ℓ ˜ denotes the affine symmetric group and S ℓ denotes the finite symmetric group. In this setting, the bijection has a beautiful geometric interpretation in terms of the root lattice of type A ℓ − 1 . We also show that the bijection has a natural description in terms of another correspondence due to Lapointe and Morse [L. Lapointe, J. Morse, Tableaux on k + 1 -cores, reduced words for affine permutations, and k-Schur expansions, J. Combin. Theory Ser. A 112 (1) (2005) 44–81].