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

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.