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 $\varepsilon_r^*$-statistic is isomorphic to $B^{r,s}$ as an affine crystal, where $\varepsilon_r^*$ is also realized precisely in terms of triple and quadruple paths.