{
  "id": "2308.01718",
  "version": "v1",
  "published": "2023-08-03T12:22:12.000Z",
  "updated": "2023-08-03T12:22:12.000Z",
  "title": "Symplectic tableaux and quantum symmetric pairs",
  "authors": [
    "Hideya Watanabe"
  ],
  "comment": "40 pages",
  "categories": [
    "math.RT",
    "math.CO",
    "math.QA"
  ],
  "abstract": "We provide a new branching rule from the general linear group $GL_{2n}(\\mathbb{C})$ to the symplectic group $Sp_{2n}(\\mathbb{C})$ by establishing a simple algorithm which gives rise to a bijection from the set of semistandard tableaux of a fixed shape to a disjoint union of several copies of sets of symplectic tableaux of various shapes. The algorithm arises from representation theory of a quantum symmetric pair of type $A\\mathrm{II}_{2n-1}$, which is a $q$-analogue of the classical symmetric pair $(\\mathfrak{gl}_{2n}(\\mathbb{C}), \\mathfrak{sp}_{2n}(\\mathbb{C}))$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-03T12:22:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "17B10",
      "17B37"
    ],
    "keywords": [
      "quantum symmetric pair",
      "symplectic tableaux",
      "general linear group",
      "classical symmetric pair",
      "representation theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}