{
  "id": "1912.02266",
  "version": "v1",
  "published": "2019-12-04T21:50:05.000Z",
  "updated": "2019-12-04T21:50:05.000Z",
  "title": "On the asymptotic behavior of the $q$-analog of Kostant's partition function",
  "authors": [
    "Pamela E. Harris",
    "Margaret Rahmoeller",
    "Lisa Schneider"
  ],
  "comment": "21 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "Kostant's partition function counts the number of distinct ways to express a weight of a classical Lie algebra $\\mathfrak{g}$ as a sum of positive roots of $\\mathfrak{g}$. We refer to each of these expressions as decompositions of a weight and our main result establishes that the (normalized) distribution of the number of positive roots in the decomposition of the highest root of a classical Lie algebra of rank $r$ converges to a Gaussian distribution as $r\\to\\infty$. We extend these results to an infinite family of weights, irrespective of Lie type, for which we establish a closed formula for the $q$-analog of Kostant's partition function and then prove that the analogous distribution also converges to a Gaussian distribution as the rank of the Lie algebra goes to infinity. We end our analysis with some directions for future research.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-04T21:50:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotic behavior",
      "classical lie algebra",
      "gaussian distribution",
      "kostants partition function counts",
      "positive roots"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}