{
  "id": "1709.07996",
  "version": "v1",
  "published": "2017-09-23T03:45:10.000Z",
  "updated": "2017-09-23T03:45:10.000Z",
  "title": "On some actions of the 0-Hecke monoids of affine symmetric groups",
  "authors": [
    "Eric Marberg"
  ],
  "comment": "32 pages, 2 figures",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "There are left and right actions of the 0-Hecke monoid of the affine symmetric group $\\tilde{S}_n$ on involutions whose cycles are labeled periodically by nonnegative integers. Using these actions we construct two bijections, which are length-preserving in an appropriate sense, from the set of involutions in $\\tilde{S}_n$ to the set of $\\mathbb{N}$-weighted matchings in the $n$-element cycle graph. As an application, we show that the bivariate generating function counting the involutions in $\\tilde{S}_n$ by length and absolute length is a rescaled Lucas polynomial. The 0-Hecke monoid of $\\tilde{S}_n$ also acts on involutions (without any cycle labelling) by Demazure conjugation. The atoms of an involution $z \\in \\tilde{S}_n$ are the minimal length permutations $w$ which transform the identity to $z$ under this action. We prove that the set of atoms for an involution in $\\tilde{S}_n$ is naturally a bounded, graded poset, and give a formula for the set's minimum and maximum elements. Using these properties, we classify the covering relations in the Bruhat order restricted to involutions in $\\tilde{S}_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-23T03:45:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "affine symmetric group",
      "involution",
      "element cycle graph",
      "minimal length permutations",
      "bivariate generating function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}