{
  "abstract": "We develop a highest weight framework and construct explicit finite dimensional irreducible modules for the type AIII (quasi split) quantum symmetric pair subalgebras $U_q'(\\mathfrak{gl}_n^θ)$. 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 $U_q'(\\mathfrak{gl}_n^θ)$ via the Letzter map. Then we define Verma modules as induced modules and prove that finite dimensional simple $U_q'(\\mathfrak{gl}_n^θ)$-modules are quotients of our Verma modules. In Part 2, we define a diagrammatic category $Web^B$ that controls polynomial representations of $U_q'(\\mathfrak{gl}_n^θ)$. We prove a multiplicity-free decomposition of the quantum exterior powers $\\bigwedge^k\\mathbb{V}$ as a $U_q'(\\mathfrak{gl}_n^θ)$-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 $\\bigwedge^k\\mathbb{V}$, 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 $U_q'(\\mathfrak{gl}_n^θ)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08057",
  "author_total": 1,
  "authors": [
    {
      "name": "Liao Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7hpa63vyaq1vx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08057",
  "fulltext_url": null,
  "id": "7hpa63vyaq1vx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08057",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7220981245",
  "pdf_url": "https://arxiv.org/pdf/2610.08057",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7hpa63vyaq1vx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.08057",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08057"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Webs and finite dimensional representation theory for quantum symmetric pairs",
  "title_text": "Webs and finite dimensional representation theory for quantum symmetric pairs",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/7hpa63vyaq1vx",
  "venue": "arXiv",
  "versions": []
}