Tableaux on k + 1-crores, reduced words for affine permutations, and k-Schur expansions
Journal of Combinatorial Theory Series A
The enumeration of fully commutative affine permutations
European Journal of Combinatorics
A bijection between dominant Shi regions and core partitions
European Journal of Combinatorics
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Let @?,k be fixed positive integers. In [C. Berg, M. Vazirani, (@?,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 @?-cores with first part equal to k and (@?-1)-cores with first part less than or equal to k. This paper gives several new interpretations of that bijection. The @?-cores index minimal length coset representatives for S"@?@?/S"@? where S"@?@? denotes the affine symmetric group and S"@? denotes the finite symmetric group. In this setting, the bijection has a beautiful geometric interpretation in terms of the root lattice of type A"@?"-"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].