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- MARK SHIMOZONO
- 1998

Answering a question of Kuniba, Misra, Okado, Takagi, and Uchi-yama, 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.

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

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)

- 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 polyno-mials 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… (More)

- MARK SHIMOZONO
- 1999

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)

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

- Mark Shimozono, Dennis E. White
- Electr. J. Comb.
- 2001

We describe the domino Schensted algorithm of Barbasch, Vogan, Garfinkle and van Leeuwen. We place this algorithm in the context of Haiman's mixed and left-right insertion algorithms and extend it to colored words. It follows easily from this description that total color of a colored word maps to the sum of the spins of a pair of 2-ribbon tableaux. Various… (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)

- 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 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)