{
  "id": "2005.01787",
  "version": "v1",
  "published": "2020-05-04T18:52:44.000Z",
  "updated": "2020-05-04T18:52:44.000Z",
  "title": "Oriented Matroids from Triangulations of Products of Simplices",
  "authors": [
    "Marcel Celaya",
    "Georg Loho",
    "Chi Ho Yuen"
  ],
  "comment": "51 pages, 15 Figures",
  "categories": [
    "math.CO",
    "math.GT",
    "math.OC"
  ],
  "abstract": "We introduce a construction of oriented matroids from a triangulation of a product of two simplices. For this, we use the structure of such a triangulation in terms of polyhedral matching fields. The oriented matroid is composed of compatible chirotopes on the cells in a matroid subdivision of the hypersimplex, which might be of independent interest. In particular, we generalize this using the language of matroids over hyperfields, which gives a new approach to construct matroids over hyperfields. Using the polyhedral structure, we derive a topological representation of the oriented matroid. This relies on a variant of Viro's patchworking and insights on the structure of tropical oriented matroids. A recurring theme in our work is that various tropical constructions can be extended beyond tropicalization with new formulations and proof methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-04T18:52:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C40",
      "05E45",
      "12K99",
      "14T05",
      "52B40",
      "52C30",
      "57N60",
      "57Q99"
    ],
    "keywords": [
      "oriented matroid",
      "triangulation",
      "polyhedral matching fields",
      "matroid subdivision",
      "independent interest"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}