{
  "id": "2209.06201",
  "version": "v1",
  "published": "2022-09-13T17:54:50.000Z",
  "updated": "2022-09-13T17:54:50.000Z",
  "title": "Counting nearest faraway flats for Coxeter chambers",
  "authors": [
    "Theo Douvropoulos"
  ],
  "comment": "16 pages, comments very much welcome!",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "In a finite Coxeter group $W$ and with two given conjugacy classes of parabolic subgroups $[X]$ and $[Y]$, we count those parabolic subgroups of $W$ in $[Y]$ that are full support, while simultaneously being simple extensions (i.e., extensions by a single reflection) of some standard parabolic subgroup of $W$ in $[X]$. The enumeration is given by a product formula that depends only on the two parabolic types. Our derivation is case-free and combines a geometric interpretation of the \"full support\" property with a double counting argument involving Crapo's beta invariant. As a corollary, this approach gives the first case-free proof of Chapoton's formula for the number of reflections of full support in a real reflection group $W$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-13T17:54:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E99",
      "20F55"
    ],
    "keywords": [
      "counting nearest faraway flats",
      "coxeter chambers",
      "full support",
      "real reflection group",
      "standard parabolic subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}