{
  "id": "2404.03500",
  "version": "v1",
  "published": "2024-04-04T14:58:16.000Z",
  "updated": "2024-04-04T14:58:16.000Z",
  "title": "Total positivity and two inequalities by Athanasiadis and Tzanaki",
  "authors": [
    "Lili Mu",
    "Volkmar Welker"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\Delta$ be a (d-1)-dimensional simplicial complex and h^\\Delta = (h_0^ ,.. , h_d) its h-vector. For a face uniform subdivision operation \\F we write \\Delta_\\F for the subdivided complex and H_\\F for the matrix such that h^{\\Delta_\\F} = H_\\F h^\\Delta. In connection with the real rootedness of symmetric decompositions Athanasiadis and Tzanaki studied for strictly positive h-vectors the inequalities h_0 / h_1 \\leq h_1 / h_{d-1} \\leq .... \\leq h_d / h_0 and h_1 / h_{d-1} \\geq ... \\geq h_{d-2} / h_2 \\geq h_{d-1} / h_1. In this paper we show that if the inequalities holds for a simplicial complex $\\Delta$ and H_\\F is TP_2 (all entries and two minors are non-negative) then the inequalities hold for \\Delta_\\F. We prove that if \\F is the barycentric subdivision then H_\\F is TP_2. If \\F is the rth-edgewise subdivision then work of Diaconis and Fulman shows H_\\F is TP_2. Indeed in this case by work of Mao and Wang H_\\F is even TP.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-04T14:58:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "05A20"
    ],
    "keywords": [
      "total positivity",
      "face uniform subdivision operation",
      "simplicial complex",
      "inequalities hold",
      "symmetric decompositions athanasiadis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}