{
  "id": "1910.02294",
  "version": "v1",
  "published": "2019-10-05T16:47:26.000Z",
  "updated": "2019-10-05T16:47:26.000Z",
  "title": "Duflo-Serganova functor and superdimension formula for the periplectic Lie superalgebra",
  "authors": [
    "Inna Entova-Aizenbud",
    "Vera Serganova"
  ],
  "comment": "Comments welcome!",
  "categories": [
    "math.RT"
  ],
  "abstract": "In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo-Serganova functor. Given a simple $\\mathfrak{p}(n)$-module $L$ and a certain element $x\\in \\mathfrak{p}(n)$ of rank $1$, we give an explicit description of the composition factors of the $\\mathfrak{p}(n-1)$-module $DS_x(L)$, which is defined as the homology of the complex $$\\Pi M \\xrightarrow{x} M \\xrightarrow{x} \\Pi M.$$ In particular, we show that this $\\mathfrak{p}(n-1)$-module is multiplicity-free. We then use this result to give a simple explicit combinatorial formula for the superdimension of a simple integrable finite-dimensional $\\mathfrak{p}(n)$-module, based on its highest weight. In particular, this reproves the Kac-Wakimoto conjecture for $\\mathfrak{p}(n)$, which was proved earlier by the authors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-05T16:47:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "periplectic lie superalgebra",
      "duflo-serganova functor",
      "superdimension formula",
      "simple explicit combinatorial formula",
      "explicit description"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}