{
  "id": "1710.10375",
  "version": "v1",
  "published": "2017-10-28T02:48:45.000Z",
  "updated": "2017-10-28T02:48:45.000Z",
  "title": "The $q$-Schur algebras and $q$-Schur dualities of finite type",
  "authors": [
    "Li Luo",
    "Weiqiang Wang"
  ],
  "comment": "32 pages",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "We formulate a $q$-Schur algebra associated to an arbitrary $W$-invariant finite set $X_{\\texttt f}$ of integral weights for a complex simple Lie algebra with Weyl group $W$. We establish a $q$-Schur duality between the $q$-Schur algebra and Hecke algebra associated to $W$. We then realize geometrically the $q$-Schur algebra and duality, and construct a canonical basis for the $q$-Schur algebra with positivity. With suitable choices of $X_{\\texttt f}$ in classical types, we recover the $q$-Schur algebras in the literature. We study the $q$-Schur algebra of type $G_2$ in detail, and establish its connection to the category $\\mathcal O$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-28T02:48:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10"
    ],
    "keywords": [
      "schur algebra",
      "schur duality",
      "finite type",
      "complex simple lie algebra",
      "invariant finite set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}