{
  "id": "2107.10819",
  "version": "v1",
  "published": "2021-07-22T17:17:11.000Z",
  "updated": "2021-07-22T17:17:11.000Z",
  "title": "B_{n-1}-bundles on the flag variety, II",
  "authors": [
    "Mark Colarusso",
    "Sam Evens"
  ],
  "comment": "45 pages",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "This paper is the sequel to ``$B_{n-1}$-bundles on the flag variety, I\". We continue our study of the orbits of a Borel subgroup $B_{n-1}$ of $G_{n-1}=GL(n-1)$ (resp. $SO(n-1)$) acting on the flag variety $\\mathcal{B}_{n}$ of $G=GL(n)$ (resp. $SO(n)$). We begin by using the results of the first paper to obtain a complete combinatorial model of the $B_{n-1}$-orbits on $\\mathcal{B}_{n}$ in terms of partitions into lists. The model allows us to obtain explicit formulas for the number of orbits as well as the exponential generating functions for the sequences $\\{|B_{n-1}\\backslash \\mathcal{B}_{n}|\\}_{n\\geq 1}$ . We then use the combinatorial description of the orbits to construct a canonical set of representatives of the orbits in terms of flags. These representatives allow us to understand an extended monoid action on $B_{n-1}\\backslash \\mathcal{B}_{n}$ using simple roots of both $\\mathfrak{g}_{n-1}$ and $\\mathfrak{g}$ and show that the closure ordering on $B_{n-1}\\backslash \\mathcal{B}_{n}$ is the standard ordering of Richardson and Springer.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-22T17:17:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "14L30",
      "20G20",
      "05A15"
    ],
    "keywords": [
      "flag variety",
      "complete combinatorial model",
      "extended monoid action",
      "representatives",
      "explicit formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}