{
  "id": "1804.01595",
  "version": "v2",
  "published": "2018-04-04T20:05:28.000Z",
  "updated": "2018-06-22T10:32:27.000Z",
  "title": "Matching fields and lattice points of simplices",
  "authors": [
    "Georg Loho",
    "Ben Smith"
  ],
  "comment": "33 pages, 18 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We show that the Chow covectors of a linkage matching field define a bijection of lattice points and we demonstrate how one can recover the linkage matching field from this bijection. This resolves two open questions from Sturmfels & Zelevinsky (1993) on linkage matching fields. For this, we give an explicit construction that associates a bipartite incidence graph of an ordered partition of a common set to all lattice points in a dilated simplex. Given a triangulation of a product of two simplices encoded by a set of bipartite trees, we similarly prove that the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex. Finally, we relate our findings to transversal matroids through the tropical Stiefel map.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2018-06-22T10:32:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "52B20",
      "52B40",
      "14T05"
    ],
    "keywords": [
      "lattice points",
      "prodsimplicial flag complex",
      "right degree vectors",
      "bipartite incidence graph",
      "linkage matching field define"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}