{
  "id": "0810.4314",
  "version": "v2",
  "published": "2008-10-23T19:40:55.000Z",
  "updated": "2010-05-16T03:38:23.000Z",
  "title": "Discrete Morse theory for totally non-negative flag varieties",
  "authors": [
    "Konstanze Rietsch",
    "Lauren Williams"
  ],
  "comment": "30 pages",
  "journal": "Adv. Math., 223, April 2010, 1855-1884",
  "categories": [
    "math.CO",
    "math.AG",
    "math.RT"
  ],
  "abstract": "In a seminal 1994 paper, Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_{\\geq 0} of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a \"remarkable polyhedral subspace\", and conjectured a decomposition into cells, which was subsequently proven by the first author. Subsequently the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's that (G/P)_{\\geq 0} -- the closure of the top-dimensional cell -- is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_{> 0} is homotopic to a sphere, is new for all G/P other than projective space.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-05-16T03:38:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "20G15"
    ],
    "keywords": [
      "discrete morse theory",
      "totally non-negative flag varieties",
      "top-dimensional cell",
      "flag variety g/p",
      "conjecture"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.4314R"
    }
  }
}