{
  "id": "1507.03221",
  "version": "v1",
  "published": "2015-07-12T12:04:40.000Z",
  "updated": "2015-07-12T12:04:40.000Z",
  "title": "Gorenstein Fano polytopes arising from order polytopes and chain polytopes",
  "authors": [
    "Takayuki Hibi",
    "Kazunori Matsuda",
    "Akiyoshi Tsuchiya"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "Richard Stanley introduced the order polytope $\\mathcal{O}(P)$ and the chain polytope $\\mathcal{C}(P)$ arising from a finite partially ordered set $P$, and showed that the Ehrhart polynomial of $\\mathcal{O}(P)$ is equal to that of $\\mathcal{C}(P)$. In addition, the unimodular equivalence problem of $\\mathcal{O}(P)$ and $\\mathcal{C}(P)$ was studied by the first author and Nan Li. In the present paper, three integral convex polytopes $\\Gamma(\\mathcal{O}(P), -\\mathcal{O}(Q))$, $\\Gamma(\\mathcal{O}(P), -\\mathcal{C}(Q))$ and $\\Gamma(\\mathcal{C}(P), -\\mathcal{C}(Q))$, where $P$ and $Q$ are partially ordered sets with $| P | = | Q |$, will be studied. First, it will be shown that the Ehrhart polynomial of $\\Gamma(\\mathcal{O}(P), -\\mathcal{C}(Q))$ coincides with that of $\\Gamma(\\mathcal{C}(P), -\\mathcal{C}(Q))$. Furthermore, when $P$ and $Q$ possess a common linear extension, it will be proved that these three convex polytopes have the same Ehrhart polynomial. Second, the problem of characterizing partially ordered sets $P$ and $Q$ for which $\\Gamma(\\mathcal{O}(P), -\\mathcal{O}(Q))$ or $\\Gamma(\\mathcal{O}(P), -\\mathcal{C}(Q))$ or $\\Gamma(\\mathcal{C}(P), -\\mathcal{C}(Q))$ is a smooth Fano polytope will be solved. Finally, when these three polytopes are smooth Fano polytopes, the unimodular equivalence problem of these three polytopes will be discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-12T12:04:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13P10",
      "52B20"
    ],
    "keywords": [
      "gorenstein fano polytopes arising",
      "order polytope",
      "chain polytope",
      "partially ordered set",
      "smooth fano polytope"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150703221H"
    }
  }
}