A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations

Abstract

The main purpose of this paper is to give a combinatorial realization of Kirillov-Reshetikhin (KR simply) crystals $B^{r,s}$ for type ${\text{E}}_n^{(1)}$ with a minuscule node $r$ and $s\ge 1$. To do this, we describe explicitly the crystal of the quantum nilpotent subalgebra associated with the translation by the negative of the $r$-th fundamental weight. Then the crystal can be extended as an affine crystal, in which a certain subcrystal characterized by the ${\epsilon }_r^{\ast }$-statistic is isomorphic to $B^{r,s}$ as an affine crystal, where ${\epsilon }_r^{\ast }$ is also realized precisely in terms of triple and quadruple paths.

35 pages, Abstract and Introduction are revised, rearranged Sections, revised several sentences, added several references, added Remark 5.4, added comment in Remark 6.5, typos corrected, minor corrections, v4; minor corrections, to appear in Journal of Algebra

References in corpus (1)

• Existence of Kirillov-Reshetikhin crystals for nonexceptional types

Cited by in corpus (1)

• Perfectness of Kirillov-Reshetikhin Crystals $B^{r,s}$ for types $E_6^{(1)}$ and $E_7^{(1)}$ with a minuscule node $r$