{
  "id": "math/0309329",
  "version": "v2",
  "published": "2003-09-19T19:49:07.000Z",
  "updated": "2003-09-23T05:59:01.000Z",
  "title": "Vertices of Gelfand-Tsetlin Polytopes",
  "authors": [
    "Jesús A. De Loera",
    "Tyrrell B. McAllister"
  ],
  "comment": "14 pages, 3 figures, fixed attributions",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns arising in the representation theory $\\mathfrak{gl}_n \\C$ and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowest-dimensional face containing a given Gelfand-Tsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov about the integrality of all vertices of the Gelfand-Tsetlin polytopes. We can construct for each $n\\geq5$ a counterexample, with arbitrarily increasing denominators as $n$ grows, of a non-integral vertex. This is the first infinite family of non-integral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the non-integral vertices when $n$ is fixed.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2003-09-23T05:59:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gelfand-tsetlin polytopes",
      "non-integral polyhedra",
      "first infinite",
      "polyhedral geometry",
      "non-integral vertex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......9329D"
    }
  }
}