{
  "id": "2005.08248",
  "version": "v1",
  "published": "2020-05-17T13:21:58.000Z",
  "updated": "2020-05-17T13:21:58.000Z",
  "title": "Categorification via blocks of modular representations II",
  "authors": [
    "Vinoth Nandakumar",
    "Gufang Zhao"
  ],
  "comment": "24 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Bernstein, Frenkel and Khovanov have constructed a categorification of tensor products of the standard representation of $\\mathfrak{sl}_2$ using singular blocks of category $\\mathcal{O}$ for $\\mathfrak{sl}_n$. In earlier work, we construct a positive characteristic analogue using blocks of representations of $\\mathfrak{sl}_n$ over a field $\\textbf{k}$ of characteristic $p > n$, with zero Frobenius character, and singular Harish-Chandra character. In the present paper, we extend these results and construct a categorical $\\mathfrak{sl}_k$-action, following Sussan's approach, by considering more singular blocks of modular representations of $\\mathfrak{sl}_n$. We consider both zero and non-zero Frobenius central character. In the former setting, we construct a graded lift of these categorifications which are equivalent to a geometric construction of Cautis, Kamnitzer and Licata. We establish a Koszul duality between two geometric categorificatons constructed in their work, and resolve a conjecture of theirs. For non-zero Frobenius central characters, we show that the geometric approach to categorical symmetric Howe duality by Cautis and Kamnitzer can be used to construct a graded lift of our categorification using singular blocks of modular representations of $\\mathfrak{sl}_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-17T13:21:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E47",
      "14M15",
      "14L35"
    ],
    "keywords": [
      "modular representations",
      "non-zero frobenius central character",
      "categorification",
      "singular blocks",
      "graded lift"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}