{
  "id": "math/0307227",
  "version": "v1",
  "published": "2003-07-16T15:32:40.000Z",
  "updated": "2003-07-16T15:32:40.000Z",
  "title": "A vector partition function for the multiplicities of sl_k(C)",
  "authors": [
    "Sara Billey",
    "Victor Guillemin",
    "Etienne Rassart"
  ],
  "comment": "34 pages, 11 figures and diagrams; submitted to Journal of Algebra",
  "categories": [
    "math.CO",
    "math.RT",
    "math.SG"
  ],
  "abstract": "We use Gelfand-Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra sl_k(C) (type A_{k-1}) as a single partition function. This allows us to apply known results about partition functions to derive interesting properties of the weight diagrams. We relate this description to that of the Duistermaat-Heckman measure from symplectic geometry, which gives a large-scale limit way to look at multiplicity diagrams. We also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors. Using symplectic geometry, we prove that the partition of the permutahedron into domains of polynomiality of the Duistermaat-Heckman function is the same as that for the weight multiplicity function, and give an elementary proof of this for sl_4(C) (A_3).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-07-16T15:32:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E15",
      "52B20",
      "53D20",
      "68W30"
    ],
    "keywords": [
      "vector partition function",
      "weight multiplicity function",
      "weight diagrams",
      "symplectic geometry",
      "single partition function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......7227B"
    }
  }
}