{
  "id": "0903.4239",
  "version": "v1",
  "published": "2009-03-25T05:23:54.000Z",
  "updated": "2009-03-25T05:23:54.000Z",
  "title": "Differential-Operator Representations of $S_n$ and Singular Vectors in Verma Modules",
  "authors": [
    "Xiaoping Xu"
  ],
  "comment": "22pages; This is a reformulation of our earlier manuscript \"Partial Differential Equations for Singular Vectors of sl(n)\" (arXiv:math/0305180)",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "Given a weight of $sl(n,\\mbb{C})$, we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module, and a differential-operator representation of the symmetric group $S_n$ on the related space of truncated power series. We prove that the solution space of the system of partial differential equations is exactly spanned by $\\{\\sgm(1)\\mid \\sgm\\in S_n\\}$. Moreover, the singular vectors of $sl(n,\\mbb{C})$ in the Verma module are given by those $\\sgm(1)$ that are polynomials. The well-known results of Verma, Bernstein-Gel'fand-Gel'fand and Jantzen for the case of $sl(n,\\mbb{C})$ are naturally included in our almost elementary approach of partial differential equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-25T05:23:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B20",
      "35C05"
    ],
    "keywords": [
      "singular vectors",
      "verma module",
      "differential-operator representation",
      "variable-coefficient second-order linear partial differential",
      "second-order linear partial differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.4239X"
    }
  }
}