{
  "id": "2207.05998",
  "version": "v1",
  "published": "2022-07-13T07:03:36.000Z",
  "updated": "2022-07-13T07:03:36.000Z",
  "title": "Combinatorial descriptions of biclosed sets in affine type",
  "authors": [
    "Grant T. Barkley",
    "David E Speyer"
  ],
  "comment": "24 pages, 3 figures",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "Let $W$ be a Coxeter group and let $\\Phi^+$ be its positive roots. A subset $B$ of $\\Phi^+$ is called biclosed if, whenever we have roots $\\alpha$, $\\beta$ and $\\gamma$ with $\\gamma \\in \\mathbb{R}_{>0} \\alpha + \\mathbb{R}_{>0} \\beta$, if $\\alpha$ and $\\beta \\in B$ then $\\gamma \\in B$ and, if $\\alpha$ and $\\beta \\not\\in B$, then $\\gamma \\not\\in B$. The finite biclosed sets are the inversion sets of the elements of $W$, and the containment between finite inversion sets is the weak order on $W$. Matthew Dyer suggested studying the poset of all biclosed subsets of $\\Phi^+$, ordered by containment, and conjectured that it is a complete lattice. As progress towards Dyer's conjecture, we classify all biclosed sets in the affine root systems. We provide both a type uniform description, and concrete models in the classical types $\\widetilde{A}$, $\\widetilde{B}$, $\\widetilde{C}$, $\\widetilde{D}$. We use our models to prove that biclosed sets form a complete lattice in types $\\widetilde{A}$ and $\\widetilde{C}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-13T07:03:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "17B22",
      "06B23"
    ],
    "keywords": [
      "affine type",
      "combinatorial descriptions",
      "complete lattice",
      "affine root systems",
      "type uniform description"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}