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arXiv · 2610.08057

Webs and finite dimensional Representation theory for quantum symmetric pairs

Liao Wang
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Abstract

We consider the type AIII (quasi split) quantum symmetric pair subalgebras $U_q'$. 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 $U_q'$ via the Letzter map. Then we define Verma modules as induced modules and prove that finite dimensional simple $U_q'$-modules are quotients of our Verma modules. These treatments are uniform in both subfamilies. In the second part we define a diagrammatic category $Web^B$ that controls polynomial representations of $U_q'$. We prove a multiplicity-free decomposition of the quantum exterior powers $\bigwedge^k\mathbb{V}$ as a $U_q'$-module, using a explicit eigenspace decomposition of certain dot morphisms in $Web^B$. 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 $Web^B$-action on certain finite dimensional representation category of $U_q'$.

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BibTeXRIS

Liao Wang. 2026-10-06. Webs and finite dimensional Representation theory for quantum symmetric pairs. https://arxiv.org/abs/2610.08057

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