Categorifying the tensor product of a level 1 highest weight and perfect crystal in type A

TL;DR

This paper categorifies the tensor product of a level 1 highest weight crystal and a perfect crystal in type A affine using Khovanov-Lauda-Rouquier algebras, linking crystal theory with module categories.

Contribution

It provides a categorification of a crystal isomorphism in affine type A, connecting crystal combinatorics with module categories via KLR algebras.

Findings

Nodes of perfect crystal correspond to trivial modules

Nodes of highest weight crystal correspond to simple modules

Crystal operators match socle of restriction and tensor rules

Abstract

We use Khovanov-Lauda-Rouquier algebras to categorify a crystal isomorphism between a highest weight crystal and the tensor product of a perfect crystal and another highest weight crystal, all in level 1 type A affine. The nodes of the perfect crystal correspond to a family of trivial modules and the nodes of the highest weight crystal correspond to simple modules, which we may also parameterize by $\ell$-restricted partitions. In the case $\ell$ is a prime, one can reinterpret all the results for the symmetric group in characteristic $\ell$. The crystal operators correspond to socle of restriction and behave compatibly with the rule for tensor product of crystal graphs.