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

TL;DR

This paper establishes a connection between Demazure characters of affine type A and graded characters of nilpotent matrix varieties, translating crystal theory into tableau language for tensor products of rectangles.

Contribution

It demonstrates that certain Demazure characters match graded characters of nilpotent varieties and provides explicit crystal actions in tableau terms for tensor products of rectangles.

Findings

Demazure characters coincide with graded characters of nilpotent varieties.

Explicit zero-th crystal raising operator action is described.

Connection with generalized cocyclage on Littlewood-Richardson tableaux is established.

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.