Kirillov-Reshetikhin Dual Equivalence Graphs

TL;DR

This paper introduces Kirillov-Reshetikhin dual equivalence graphs (KR DEGs) on tensor products of crystals, proving their equivalence with certain stratification methods and generalizing previous Kazhdan-Lusztig graphs in affine type A.

Contribution

The paper defines KR DEGs on tensor products of Kirillov-Reshetikhin crystals and proves their equivalence with stratification methods, extending Kazhdan-Lusztig dual equivalence graphs.

Findings

KR DEGs are defined on 0-weight spaces of tensor products of crystals.

The authors prove the equivalence of two stratification methods involving KR DEGs.

KR DEGs generalize Kazhdan-Lusztig dual equivalence graphs in affine type A.

Abstract

Let $U$ be a tensor product of highest weight modules of $GL_n(\mathbb C)$ corresponding to multiples of fundamental weights (i.e. rectangles). We consider three ways to stratify $U^{\otimes k}$ into components: using isotypic components of the cyclic action on tensor factors, using a generalization of the charge statistic, and using certain generalizations of Assaf's dual equivalence graphs. We conjecture that all three ways coincide, and we prove that the latter two ways coincide. The Kirillov-Reshetikhin dual equivalence graphs (KR DEGs) we introduce for this purpose are defined on $0$-weight spaces of tensor products of Kirillov-Reshetikhin crystals. They generalize Kazhdan-Lusztig dual equivalence graphs (KL DEGs) that previously appeared in the study of Kazhdan-Lusztig cells in affine type A. While the tensor products of Kirillov-Reshetikhin crystals are connected as affine crystals, the KR DEGs in general are not.