Quantum Howe duality and Schur duality of type AIII

Abstract

We establish a new connection between the iHowe duality of type AIII established by Luo-Xu and the iSchur duality established by Bao-Wang. We show that iweight $\overline{\rho}$ space in the iHowe duality is naturally isomorphic to the tensor space in the iSchur duality. Under this isomorphism, we show that the relative braid group action on this iweight space coincides with the action of the type B Hecke algebra in the iSchur duality. As a consequence, we derive from multiplicity-free decompositions that the iweight $\overline{\rho}$ spaces of irreducible modules over iquantum groups are irreducible modules over the type B Hecke algebra. Meanwhile, in the iHowe duality, we identify the relative braid group action from one side with the action of $K$-matrices and $R$-matrices from the other side.