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

The 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 $s \ge 1$. It is obtained from the crystal of the quantum nilpotent subalgebra associated with the translation by the negative of the $r$-th fundamental weight, which can be viewed as the limit of KR crystals $B^{r, s}$ as $s \rightarrow \infty$. Then the KR crystal $B^{r, s}$ is characterized by a certain statistic given in terms of triple and quadruple paths, while it was known that the statistic for types $\text{A}_n^{(1)}$ and $\text{D}_n^{(1)}$ is given in terms of single and double paths, respectively.