{
  "id": "1810.02703",
  "version": "v1",
  "published": "2018-10-04T09:20:47.000Z",
  "updated": "2018-10-04T09:20:47.000Z",
  "title": "On involutions in the Weyl group and $B$-orbit closures in the orthogonal case",
  "authors": [
    "Mikhail V. Ignatyev"
  ],
  "comment": "17 pages. arXiv admin note: text overlap with arXiv:1112.2624",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "We study coadjoint $B$-orbits on $\\mathfrak{n}^*$, where $B$ is a Borel subgroup of a complex orthogonal group $G$, and $\\mathfrak{n}$ is the Lie algebra of the unipotent radical of $B$. To each basis involution $w$ in the Weyl group $W$ of $G$ one can assign the associated $B$-orbit $\\Omega_w$. We prove that, given basis involutions $\\sigma$, $\\tau$ in $W$, if the orbit $\\Omega_{\\sigma}$ is contained in the closure of the orbit $\\Omega_{\\tau}$ then $\\sigma$ is less than or equal to $\\tau$ with respect to the Bruhat order on $W$. For a basis involution $w$, we also compute the dimension of $\\Omega_w$ and present a conjectural description of the closure of $\\Omega_w$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-04T09:20:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B22",
      "17B08",
      "17B30",
      "20F55"
    ],
    "keywords": [
      "weyl group",
      "orbit closures",
      "orthogonal case",
      "basis involution",
      "complex orthogonal group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}