On higher level Kirillov--Reshetikhin crystals, Demazure crystals, and related uniform models

TL;DR

This paper establishes a connection between tensor products of higher level Kirillov--Reshetikhin crystals and Demazure crystals, generalizing previous results and enabling new combinatorial interpretations of Q-system relations.

Contribution

It generalizes the isomorphism between tensor products of KR crystals and Demazure crystals to mixed levels without perfectness, and reduces complex crystals to simpler models for analysis.

Findings

Tensor products of nonexceptional KR crystals are isomorphic to Demazure crystals.

Connected components of tensor products with same maximal weight are isomorphic after removing certain arrows.

Provides a combinatorial interpretation of Q-system relations.

Abstract

We show that a tensor product of nonexceptional type Kirillov--Reshetikhin (KR) crystals is isomorphic to a direct sum of Demazure crystals; we do this in the mixed level case and without the perfectness assumption, thus generalizing a result of Naoi. We use this result to show that, given two tensor products of such KR crystals with the same maximal weight, after removing certain $0$-arrows, the two connected components containing the minimal/maximal elements are isomorphic. Based on the latter fact, we reduce a tensor product of higher level perfect KR crystals to one of single-column KR crystals, which allows us to use the uniform models available in the literature in the latter case. We also use our results to give a combinatorial interpretation of the Q-system relations. Our results are conjectured to extend to the exceptional types.