Crystal basis theory for a quantum symmetric pair $(\mathbf{U},\mathbf{U}^{\jmath})$

TL;DR

This paper develops a crystal basis theory for quantum symmetric pairs of type AIII, classifying irreducible modules and introducing the $ ext{ extjmath}$-crystal basis with favorable combinatorial features.

Contribution

It extends Kashiwara's crystal basis theory to quantum symmetric pairs, providing classification and combinatorial structure for their irreducible modules.

Findings

Classification of irreducible $ ext{ extjmath}$-modules

Introduction of $ ext{ extjmath}$-crystal basis with combinatorial properties

Establishment of basis at $p=q=0$ for modules

Abstract

We study the representation theory of a quantum symmetric pair $(\mathbf{U},\mathbf{U}^{\jmath})$ with two parameters $p,q$ of type AIII, by using highest weight theory and a variant of Kashiwara's crystal basis theory. Namely, we classify the irreducible $\mathbf{U}^{\jmath}$-modules in a suitable category and associate with each of them a basis at $p=q=0$, the $\jmath$-crystal basis. The $\jmath$-crystal bases have nice combinatorial properties as the ordinary crystal bases do.