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

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

2008-04-08

arxiv.org/abs/0804.1380v1

Mathematics

Combinatorics

23 pages

Scientific paper

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.

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