Webs and finite dimensional Representation theory for quantum symmetric pairs

We consider the type AIII (quasi split) quantum symmetric pair subalgebras $U_q^{\mathrm{\prime }}$. We define a new Cartan subalgebra in one of the two subfamilies. Together with Letzter’s Cartan subalgebra in the other, we construct a triangular decomposition of $U_q^{\mathrm{\prime }}$ via the Letzter map. Then we define Verma modules as induced modules and prove that finite dimensional simple $U_q^{\mathrm{\prime }}$-modules are quotients of our Verma modules. These treatments are uniform in both subfamilies. In the second part we define a diagrammatic category $We{b}^B$ that controls polynomial representations of $U_q^{\mathrm{\prime }}$. We prove a multiplicity-free decomposition of the quantum exterior powers ${\bigwedge }^k\mathbb{V}$ as a $U_q^{\mathrm{\prime }}$-module, using a explicit eigenspace decomposition of certain dot morphisms in $We{b}^B$. As applications, we deduce the multiplicity-freeness of the involved anti-spherical Hecke module. For ${\bigwedge }^k\mathbb{V}$, we explicitly determine the highest weight vectors and their weights, thereby determining the isomorphism classes of their irreducible summands. Finally, we determine the kernel of the $We{b}^B$-action on certain finite dimensional representation category of $U_q^{\mathrm{\prime }}$.