{
  "id": "1507.00872",
  "version": "v1",
  "published": "2015-07-03T11:06:21.000Z",
  "updated": "2015-07-03T11:06:21.000Z",
  "title": "On involutions in symmetric groups and a conjecture of Lusztig",
  "authors": [
    "Jun Hu",
    "Jing Zhang"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $(W, S)$ be a Coxeter system equipped with a fixed automorphism $\\ast$ of order $\\leq 2$ which preserves $S$. Lusztig (and with Vogan in some special cases) have shown that the space spanned by set of \"twisted\" involutions was naturally endowed with a module structure of the Hecke algebra of $(W, S)$. Lusztig has conjectured that this module is isomorphic to the right ideal of the Hecke algebra (with Hecke parameter $u^2$) associated to $(W,S)$ generated by the element $X_{\\emptyset}:=\\sum_{w^\\ast=w}u^{-\\ell(w)}T_w$. In this paper we prove this conjecture in the case when $\\ast=\\text{id}$ and $W$ is the symmetric group on $n$ letters.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-03T11:06:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C08"
    ],
    "keywords": [
      "symmetric group",
      "involutions",
      "conjecture",
      "hecke algebra",
      "hecke parameter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150700872H"
    }
  }
}