# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1704.01277