Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties

Abstract

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. This entails a translation of the affine type A crystal theory into the language of tableaux following Nakayashiki and Yamada, for the case of tensor products of the classical crystals indexed by rectangular partitions. In particular the explicit action of the zero-th crystal raising operator on the above crystals is given, and its direct connection with the generalized cocyclage on Littlewood-Richardson tableaux is explained.

25 pages, AMS-LaTeX

References in corpus (3)

• Inhomogeneous lattice paths, generalized Kostka polynomials and A${}_{n-1}$ supernomials
• A generalization of the Kostka-Foulkes polynomials
• A cyclage poset structure for Littlewood-Richardson tableaux

Cited by in corpus (8)

• Inhomogeneous lattice paths, generalized Kostka polynomials and A${}_{n-1}$ supernomials
• Kirillov--Reshetikhin crystals for nonexceptional types
• Box ball system associated with antisymmetric tensor crystals
• RSK correspondence and classically irreducible Kirillov-Reshetikhin crystals
• Markov chains for promotion operators
• Rigged configuration bijection and proof of the $X=M$ conjecture for nonexceptional affine types
• Soliton cellular automaton associated with $D_n^{(1)}$-crystal $B^{2,s}$
• Similarity and Kirillov-Schilling-Shimozono bijection