{
  "id": "2406.15803",
  "version": "v1",
  "published": "2024-06-22T10:12:11.000Z",
  "updated": "2024-06-22T10:12:11.000Z",
  "title": "Root polytopes, flow polytopes, and order polytopes",
  "authors": [
    "Konstanze Rietsch",
    "Lauren Williams"
  ],
  "comment": "32 pages, 12 figures, comments welcome!",
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "abstract": "In this paper we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points $\\{e_i-e_j \\ \\vert \\ i \\neq j\\} \\cup \\{\\pm e_i\\}$ in $\\mathbb{R}^n$, where $e_1,\\dots,e_n$ is the standard basis of $\\mathbb{R}^n$. Such a polytope can be encoded by a quiver $Q$ with vertices $V \\subseteq \\{v_1,\\dots,v_n\\} \\cup \\{\\star\\}$, where each edge $v_j\\to v_i$ or $\\star \\to v_i$ or $v_i\\to \\star$ gives rise to the point $e_i-e_j$ or $e_i$ or $-e_i$, respectively; we denote the corresponding polytope as $\\operatorname{Root}(Q)$. These polytopes have been studied extensively under names such as edge polytope and root polytope. We show that if the quiver $Q$ is star-connected (or strongly-connected if there is no $\\star$ vertex) then the root polytope $\\operatorname{Root}(Q)$ is reflexive and terminal; we moreover give a combinatorial description of the facets of $\\operatorname{Root}(Q)$. We also show that if $Q$ is planar, then $\\operatorname{Root}(Q)$ is (integrally equivalent to the) polar dual of the flow polytope of the dual quiver $Q^{\\vee}$. Finally we consider the case that $Q$ comes from the Hasse diagram of a ranked poset $P$, and show that $\\operatorname{Root}(Q)$ is polar dual to (a translation of) a marked poset polytope. Additionally, we show that the face fan $\\mathcal{F}_Q$ of $\\operatorname{Root}(Q)$ refines the normal fan $\\mathcal{N}(\\mathcal{O}(P))$ of the order polytope $\\mathcal{O}(P)$, and when $P$ is graded, these fans coincide. We moreover give a combinatorial description of the Picard group of the toric variety associated to $\\mathcal{F}_Q$ for any ranked poset $P$, in terms of a new ``canonical extension'' of $P$. These results have applications to mirror symmetry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-22T10:12:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B20",
      "14M25",
      "52B05",
      "05E14"
    ],
    "keywords": [
      "root polytope",
      "flow polytope",
      "order polytope",
      "combinatorial description",
      "polar dual"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}