Categorifying the tensor product of the Kirillov-Reshetikhin crystal $B^{1,1}$ and a fundamental crystal

TL;DR

This paper uses KLR algebras to categorify a crystal isomorphism involving tensor products of Kirillov-Reshetikhin and fundamental crystals in affine type, linking modules to crystal nodes.

Contribution

It provides a categorification of a crystal isomorphism in affine type using KLR algebras, connecting modules to crystal nodes and operators.

Findings

Nodes of KR crystal correspond to trivial modules

Nodes of fundamental crystal correspond to simple modules

Crystal operators match socle of restriction behavior

Abstract

We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal isomorphism between a fundamental crystal and the tensor product of a Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine type. The nodes of the Kirillov-Reshetikhin crystal correspond to a family of "trivial" modules. The nodes of the fundamental crystal correspond to simple modules of the corresponding cyclotomic KLR algebra. The crystal operators correspond to socle of restriction and behave compatibly with the rule for tensor product of crystal graphs.