{
  "id": "1502.03598",
  "version": "v1",
  "published": "2015-02-12T10:56:54.000Z",
  "updated": "2015-02-12T10:56:54.000Z",
  "title": "The Bruhat order on conjugation-invariant sets of involutions in the symmetric group",
  "authors": [
    "Mikael Hansson"
  ],
  "comment": "12 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $I_n$ be the set of involutions in the symmetric group $S_n$, and for $A \\subseteq \\{0,1,\\ldots,n\\}$, let \\[ F_n^A=\\{\\sigma \\in I_n \\mid \\text{$\\sigma$ has $a$ fixed points for some $a \\in A$}\\}. \\] We give a complete characterisation of the sets $A$ for which $F_n^A$, with the order induced by the Bruhat order on $S_n$, is a graded poset. In particular, we prove that $F_n^{\\{1\\}}$ (i.e., the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When $F_n^A$ is graded, we give its rank function. We also give a short new proof of the EL-shellability of $F_n^{\\{0\\}}$ (i.e., the set of fixed point-free involutions), which was recently proved by Can, Cherniavsky, and Twelbeck. Keywords: Bruhat order, symmetric group, involution, conjugacy class, graded poset, EL-shellability",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-12T10:56:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetric group",
      "bruhat order",
      "conjugation-invariant sets",
      "graded poset",
      "rank function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}