{
  "id": "2303.07658",
  "version": "v2",
  "published": "2023-03-14T06:59:41.000Z",
  "updated": "2023-06-11T08:20:54.000Z",
  "title": "Sign-twisted generating functions of odd length over any parabolic quotients of the even hyperoctahedral groups",
  "authors": [
    "Haihang Gu",
    "Houyi Yu"
  ],
  "comment": "38 pages",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "The odd length on Weyl groups is a new statistic analogous to the classical Coxeter length function, and features combinatorial and parity conditions. We establish an explicit closed product formula of the sign-twisted generating functions of the odd length for any parabolic quotients of the even hyperoctahedral groups (Weyl groups of type $D$). As a consequence, we verify three conjectures posed by Brenti and Carnevale. We then give necessary and sufficient conditions for the sign-twisted generating functions to be not expressible as products of cyclotomic polynomials. It turns out that there is at most one non-cyclotomic factor, which always has the form $x^n + 2x^m + 1$ for some positive integers $m$ and $n$, settling a conjecture of Stembridge.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-06-11T08:20:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05A05",
      "05E16",
      "20F55"
    ],
    "keywords": [
      "sign-twisted generating functions",
      "odd length",
      "parabolic quotients",
      "hyperoctahedral groups",
      "weyl groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}