Hideya Watanabe — Page 2

arXiv2018

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

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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arXiv2017

Multiparameter quantum Schur duality of type B

We establish a Schur type duality between a coideal subalgebra of the quantum group of type A and the Hecke algebra of type B with 2 parameters. We identify the $\imath$-canonical basis on the tensor product of the natural representation with Lusztig's canonical basis of the type B Hecke algebra with unequal parameters associated to a weight function.

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