A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations
Il-Seung Jang
TL;DR
The paper provides a uniform combinatorial realization of KR crystals $B^{r,s}$ for $E_n^{(1)}$ with minuscule $r$ by embedding the crystal of a quantum nilpotent subalgebra into $B(\infty)$ and extending it to an affine crystal. The main construction identifies ${\bf B}^{\mathrm{J},s}$ with $B^{r,s}$, and encodes the $0$-arrow structure via a detailed analysis of $\varepsilon_r^*$ through triple and quadruple paths on planar arrangements $\Delta_n$, yielding a polytope-like model. The results hinge on simply braided reduced expressions for $w_0$ in types $E_6$ and $E_7$, explicit convex orders of positive roots, and a detailed trail-path correspondence that makes the $\varepsilon_r^*$ statistic computable in terms of combinatorial objects. This advances the uniform understanding of KR crystals across exceptional types and connects with previous tableau/polytope models, while establishing a clear pathway to potential generalizations beyond minuscule nodes. The work also situates the KR crystals within the broader framework of $U_q^-(w)$ and PBW crystals, using a robust signature-rule approach to crystal operators.
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.