{
  "id": "1407.7507",
  "version": "v2",
  "published": "2014-07-28T19:27:11.000Z",
  "updated": "2015-06-10T12:20:25.000Z",
  "title": "SB-Labelings, Distributivity, and Bruhat Order on Sortable Elements",
  "authors": [
    "Henri Mühle"
  ],
  "comment": "13 pages, 2 figures",
  "journal": "The Electronic Journal of Combinatorics 22, 2 (2015) P2.40",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this article, we investigate the set of $\\gamma$-sortable elements, associated with a Coxeter group $W$ and a Coxeter element $\\gamma\\in W$, under Bruhat order, and we denote this poset by $\\mathcal{B}_{\\gamma}$. We show that this poset belongs to the class of SB-lattices recently introduced by Hersh and M\\'esz\\'aros, by proving a more general statement, namely that all join-distributive lattices are SB-lattices. The observation that $\\mathcal{B}_{\\gamma}$ is join-distributive is due to Armstrong. Subsequently, we investigate for which finite Coxeter groups $W$ and which Coxeter elements $\\gamma\\in W$ the lattice $\\mathcal{B}_{\\gamma}$ is in fact distributive. It turns out that this is the case for the \"coincidental\" Coxeter groups, namely the groups $A_{n},B_{n},H_{3}$ and $I_{2}(k)$. We conclude this article with a conjectural characteriziation of the Coxeter elements $\\gamma$ of said groups for which $\\mathcal{B}_{\\gamma}$ is distributive in terms of forbidden orientations of the Coxeter diagram.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-28T19:27:11.000Z",
      "abstract": "In this article, we investigate the set of $\\gamma$-sortable elements, associated with a Coxeter group $W$ and a Coxeter element $\\gamma\\in W$, under Bruhat order, and we denote this poset by $\\mathcal{B}_{\\gamma}$. We prove that this poset belongs to the class of SB-lattices recently introduced by Hersh and M\\'esz\\'aros. We use an observation of Armstrong, namely that $\\mathcal{B}_{\\gamma}$ is a join-distributive lattice, to generalize the previous result, and to show that all join-distributive lattices are SB-lattices. Subsequently, we investigate for which finite Coxeter groups $W$ and which Coxeter elements $\\gamma\\in W$ the lattice $\\mathcal{B}_{\\gamma}$ is distributive. It turns out that this is the case for the \"coincidental\" Coxeter groups, namely the groups $A_{n},B_{n},H_{3}$ and $I_{2}(k)$. We conclude this article with a conjectural characteriziation of the Coxeter elements $\\gamma$ of the said groups for which $\\mathcal{B}_{\\gamma}$ is distributive in terms of forbidden orientations of the Coxeter diagram.",
      "comment": "13 pages, 1 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-10T12:20:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "06D75",
      "06A07"
    ],
    "keywords": [
      "bruhat order",
      "sortable elements",
      "coxeter element",
      "sb-labelings",
      "distributivity"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.7507M"
    }
  }
}