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

Hideya Watanabe

Published 2017-04-05Version 1

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 basis of a finite-dimensional $\mathbf{U}$-module is thought of as a "localization" of the $\jmath$-canonical basis, which was introduced by Huanchen Bao and Weiqiang Wang in 2013. Also, the $\jmath$-crystal bases have nice combinatorial properties as the ordinary crystal bases do.