{
  "id": "1704.01277",
  "version": "v1",
  "published": "2017-04-05T06:16:39.000Z",
  "updated": "2017-04-05T06:16:39.000Z",
  "title": "Crystal basis theory for a quantum symmetric pair $(\\mathbf{U},\\mathbf{U}^{\\jmath})$",
  "authors": [
    "Hideya Watanabe"
  ],
  "comment": "48 pages",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "We study the representation theory of a quantum symmetric pair $(\\mathbf{U},\\mathbf{U}^{\\jmath})$ with two parameters $p,q$ of type AIII, by using highest weight theory and a variant of Kashiwara's crystal basis theory. Namely, we classify the irreducible $\\mathbf{U}^{\\jmath}$-modules in a suitable category and associate with each of them a basis at $p=q=0$, the $\\jmath$-crystal basis. The $\\jmath$-crystal basis of a finite-dimensional $\\mathbf{U}$-module is thought of as a \"localization\" of the $\\jmath$-canonical basis, which was introduced by Huanchen Bao and Weiqiang Wang in 2013. Also, the $\\jmath$-crystal bases have nice combinatorial properties as the ordinary crystal bases do.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-05T06:16:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "05E10"
    ],
    "keywords": [
      "quantum symmetric pair",
      "kashiwaras crystal basis theory",
      "ordinary crystal bases",
      "nice combinatorial properties",
      "highest weight theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}