Webs and finite dimensional representation theory for quantum symmetric pairs
Abstract
We develop a highest weight framework and construct explicit finite dimensional irreducible modules for the type AIII (quasi split) quantum symmetric pair subalgebras Uq'(glnθ). In Part 1, we provide a uniform construction of Cartan subalgebras and Verma modules for both subfamilies. Our construction recovers Letzter's Cartan subalgebra in one subfamily and gives a new Cartan subalgebra in the other. We construct a weak triangular decomposition of Uq'(glnθ) via the Letzter map. Then we define Verma modules as induced modules and prove that finite dimensional simple Uq'(glnθ)-modules are quotients of our Verma modules. In Part 2, we define a diagrammatic category WebB that controls polynomial representations of Uq'(glnθ). We prove a multiplicity-free decomposition of the quantum exterior powers bigwedgek𝕍 as a Uq'(glnθ)-module, using a explicit eigenspace decomposition of certain dot morphisms in WebB. As applications, we deduce the multiplicity-freeness of the involved anti-spherical Hecke module. For bigwedgek𝕍, 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 WebB-action on certain finite dimensional representation category of Uq'(glnθ).
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BibTeX
@misc{wang2026webs,
title = {{Webs and finite dimensional representation theory for quantum symmetric pairs}},
author = {Liao Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08057},
eprint = {2610.08057},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08057},
}
RIS
TY - UNPB
TI - Webs and finite dimensional representation theory for quantum symmetric pairs
AU - Liao Wang
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.08057
UR - https://arxiv.org/abs/2610.08057
AB - We develop a highest weight framework and construct explicit finite dimensional irreducible modules for the type AIII (quasi split) quantum symmetric pair subalgebras U_q'(gl_n^θ). In Part 1, we provide a uniform construction of Cartan subalgebras and Verma modules for both subfamilies. Our construction recovers Letzter's Cartan subalgebra in one subfamily and gives a new Cartan subalgebra in the other. We construct a weak triangular decomposition of U_q'(gl_n^θ) via the Letzter map. Then we define Verma modules as induced modules and prove that finite dimensional simple U_q'(gl_n^θ)-modules are quotients of our Verma modules. In Part 2, we define a diagrammatic category Web^B that controls polynomial representations of U_q'(gl_n^θ). We prove a multiplicity-free decomposition of the quantum exterior powers bigwedge^k𝕍 as a U_q'(gl_n^θ)-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^k𝕍, 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'(gl_n^θ).
ER -