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- Mark Shimozono, Jerzy Weyman
- Eur. J. Comb.
- 2000

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)

- Victor Reiner, Mark Shimozono
- J. Comb. Theory, Ser. A
- 1995

- THOMAS LAM, MARK SHIMOZONO
- 2007

Let G be a simple and simply-connected complex algebraic group, P ⊂ G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH∗(G/P ) of a flag variety is, up to localization, a quotient of the homology H∗(GrG) of the affine Grassmannian GrG of G. As a consequence, all three-point genus zero… (More)

- MARK SHIMOZONO
- 1998

Answering a question of Kuniba, Misra, Okado, Takagi, and Uchiyama, it is shown that certain Demazure characters of affine type A, coincide with the graded characters of coordinate rings of closures of conjugacy classes of nilpotent matrices.

Combinatorial objects called rigged configurations give rise to q-analogues of certain Littlewood-Richardson coefficients. The Kostka-Foulkes polynomials and twocolumn Macdonald-Kostka polynomials occur as special cases. Conjecturally these polynomials coincide with the Poincaré polynomials of isotypic components of certain graded GL(n)-modules supported in… (More)

Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the k-Schur functions in homology and affine Schur functions in cohomology. Our results rely on Kostant and Kumar’s nilHecke ring, work of Peterson on the homology of based loops on a compact group, and earlier work of… (More)

- Jeffrey B. Remmel, Mark Shimozono
- Discrete Mathematics
- 1998

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 give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are… (More)

- Sergey Fomin, Curtis Greene, Victor Reiner, Mark Shimozono
- Eur. J. Comb.
- 1997

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)

- MARK SHIMOZONO
- 2003

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)