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In this talk, we'll share an explicit description of the corresponding "affine evacuation'' map on tabloids, and we show that the number of tabloids fixed by this map is equal to the evaluation of a certain Green's polynomial at q = -1. Along the way, we discover a combinatorial interpretation of the evaluation of the Kostka-Foulkes polynomials at q=-1.  These findings are based on joint work with Mike Chmutov, Dongkwan Kim, Joel Lewis, and Elena Yudovina.

The Robinson-Schensted correspondence is a bijection between permutations and pairs of standard Young tableaux of the same shape. Under this bijection, the reverse complement of a permutation corresponds to the evacuation of the two tableaux. The number of standard tableaux of shape λ which are fixed by evacuation is equal to fλ(-1), where fλ(q) is the q-analogue of the hook-length formula. Chmutov, Pylyavskyy, and Yudovina recently introduced a generalization of Robinson-Schensted which maps elements of the affine symmetric group to pairs of tabloids (=standard row-strict tableaux) of the same shape. There is a natural involution of the affine symmetric group that generalizes the reverse complement of permutations.


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