{
  "id": "2001.03922",
  "version": "v1",
  "published": "2020-01-12T12:41:02.000Z",
  "updated": "2020-01-12T12:41:02.000Z",
  "title": "Complements of Schubert polynomials",
  "authors": [
    "Neil J. Y. Fan",
    "Peter L. Guo",
    "Nicolas Y. Liu"
  ],
  "comment": "15 pages, 9 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\mathfrak{S}_w(x)=\\mathfrak{S}_w(x_1,\\ldots,x_n)$ be the Schubert polynomial for a permutation $w$ of $\\{1,2,\\ldots,n\\}$. For a composition $\\mu=(\\mu_1,\\ldots,\\mu_n)\\in \\mathbb{Z}_{\\geq 0}^n$, write $x^\\mu=x_1^{\\mu_1}\\cdots x_n^{\\mu_n}$. We say that $x^\\mu\\mathfrak{S}_w(x^{-1})=x^\\mu \\mathfrak{S}_w(x_1^{-1},\\ldots,x_n^{-1})$ is the complement of $\\mathfrak{S}_w(x)$ with respect to $\\mu$. Huh, Matherne, M\\'esz\\'aros and St.Dizier proved that $\\mathrm{N}(x_1^{n-1}\\cdots x_n^{n-1} \\mathfrak{S}_w(x^{-1}))$ is a Lorentzian polynomial, where $\\mathrm{N}$ is a linear operator sending a monomial $x_1^{\\mu_1}\\cdots x_n^{\\mu_n}$ to $\\frac{x_1^{\\mu_1}\\cdots x_n^{\\mu_n}}{\\mu_1!\\cdots \\mu_n!}$. They further conjectured that $\\mathrm{N}(\\mathfrak{S}_w(x))$ is Lorentzian. Motivated by this conjecture, we investigate the problem when $x^\\mu \\mathfrak{S}_w(x^{-1})$ is still a Schubert polynomial. If $x^\\mu \\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial, then $\\mathrm{N}(\\mathfrak{S}_w(x))$ will be Lorentzian. In this paper, we pay attention to the typical case that $\\mu=\\delta_n=(n-1,\\ldots, 1,0)$ is the staircase partition. Our result shows that $x^{\\delta_n} \\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial if and only if $w$ avoids the two patterns 132 and 312.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-12T12:41:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "schubert polynomial",
      "complement",
      "lorentzian polynomial",
      "linear operator",
      "pay attention"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}