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This is a combinatorial study of the Poincar e polynomials of iso-typic 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 coeecients. The coeecients of two-column Macdonald-Kostka polynomials(More)
We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n, C). Our main results are: • Pieri rules for the Schubert bases of H * (Gr) and H * (Gr), which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. • A new combinatorial definition for k-Schur(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)
The X = M conjecture of Hatayama et al. asserts the equality between the one-dimensional configuration sum X expressed as the generating function of crystal paths with energy statistics and the fermionic formula M for all affine Kac–Moody algebra. In this paper we prove the X = M conjecture for tensor products of Kirillov–Reshetikhin crystals B 1,s(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 A 2n−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(More)