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

2008/04/08 by Chris Berg, Berg, Chris, Brant Jones +3 · 1 citation
Mathematics · #05E10 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05E10

paper · pdf · doi:10.48550/arxiv.0804.1380

23 pages

arxiv created 2008/04/08 · arxiv updated 2009/12/01

Abstract

Let ℓ,k be fixed positive integers. In an earlier work, 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 \widetildeS_ℓ / S_ℓ where \widetildeS_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.

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