{
  "id": "1910.05468",
  "version": "v1",
  "published": "2019-10-12T02:44:12.000Z",
  "updated": "2019-10-12T02:44:12.000Z",
  "title": "On $A_1^2$ restrictions of Weyl arrangements",
  "authors": [
    "Takuro Abe",
    "Hiroaki Terao",
    "Tan Nhat Tran"
  ],
  "comment": "28 pages",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "Let $\\mathcal{A}$ be a Weyl arrangement in an $\\ell$-dimensional Euclidean space. The freeness of restrictions of $\\mathcal{A}$ was first settled by a case-by-case method by Orlik-Terao (1993), and later by a uniform argument by Douglass (1999). Prior to this, Orlik-Solomon (1983) had completely determined the exponents of these arrangements by exhaustion. A classical result due to Orlik-Solomon-Terao (1986), asserts that the exponents of any $A_1$ restriction, i.e., the restriction of $\\mathcal{A}$ to a hyperplane, are given by $\\{m_1,\\ldots, m_{\\ell-1}\\}$, where $\\exp(\\mathcal{A})=\\{m_1,\\ldots, m_{\\ell}\\}$ with $m_1 \\le \\cdots\\le m_{\\ell}$. As a next step after Orlik-Solomon-Terao towards understanding the exponents of the restrictions, we will investigate the $A_1^2$ restrictions, i.e., the restrictions of $\\mathcal{A}$ to subspaces of the type $A_1^2$. In this paper, we give a combinatorial description of the exponents of the $A_1^2$ restrictions and describe bases for the modules of derivations in terms of the classical notion of related roots by Kostant (1955).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-12T02:44:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32S22",
      "17B22"
    ],
    "keywords": [
      "restriction",
      "weyl arrangement",
      "dimensional euclidean space",
      "combinatorial description",
      "orlik-solomon-terao"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}