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

TL;DR

This paper provides a combinatorial model for Kirillov-Reshetikhin crystals of type E, using quantum nilpotent subalgebras and path combinatorics, advancing understanding of affine crystal structures.

Contribution

It explicitly constructs the crystal of the quantum nilpotent subalgebra for type E and extends it to an affine crystal, linking it to KR crystals via path statistics.

Findings

Explicit combinatorial realization of KR crystals for type E

Extension of the crystal to an affine structure

Characterization of subcrystals via path statistics

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.