{
  "id": "1803.06901",
  "version": "v1",
  "published": "2018-03-19T13:14:43.000Z",
  "updated": "2018-03-19T13:14:43.000Z",
  "title": "Cyclic Sieving and Cluster Duality for Grassmannian",
  "authors": [
    "Linhui Shen",
    "Daping Weng"
  ],
  "comment": "40 pages",
  "categories": [
    "math.RT",
    "math-ph",
    "math.AG",
    "math.CO",
    "math.MP"
  ],
  "abstract": "We introduce a decorated configuration space $\\mathscr{C}onf_n^\\times(a)$ with a potential function $\\mathcal{W}$. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of $(\\mathscr{C}onf_n^\\times(a), \\mathcal{W})$ canonically parametrizes a linear basis of the homogenous coordinate ring of the Grassmannian ${\\rm Gr}_a(n)$. We prove that $(\\mathscr{C}onf_n^\\times(a), \\mathcal{W})$ is equivalent to the mirror Landau-Ginzburg model of Grassmannian considered by Marsh-Rietsch and Rietsch-Williams. As an application, we show a cyclic sieving phenomenon involving plane partitions under a sequence of piecewise-linear toggles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-19T13:14:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "grassmannian",
      "mirror landau-ginzburg model",
      "cluster duality conjecture",
      "plane partitions",
      "cyclic sieving phenomenon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}