Mark Shimozono

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This is a combinatorial study of the Poincaré polynomials of isotypic components of a natural family of graded GL(n)-modules supported in the closure of a nilpotent conjugacy class. These polynomials generalize the Kostka-Foulkes polynomials and are q-analogues of Littlewood-Richardson coefficients. The coefficients of two-column Macdonald-Kostka(More)
We present a simple proof of the Littlewood-Richardson rule using a sign-reversing involution, and show that a similar involution provides a com-binatorial proof of the SXP algorithm of Chen, Garsia, and Remmel 2] which computes the Schur function expansion of the plethysm of a Schur function and a power sum symmetric function. The methods of this paper(More)
We study balanced labellings of diagrams representing the inversions in a permutation. These are known to be natural encodings of reduced decompositions of permutations w 2 n , and we show that they also give combinatorial descriptions of both the Stanley symmetric functions F w and and the Schubert polynomial S w associated with w. Furthermore, they lead(More)
We introduce “virtual” crystals of the affine types g = D (2) n+1, A (2) 2n and C (1) n by naturally extending embeddings of crystals of types Bn and Cn into crystals of type A2n−1. Conjecturally, these virtual crystals are the crystal bases of finite dimensional U ′ q (g)-modules associated with multiples of fundamental weights. We provide evidence and in(More)