{
  "id": "2109.04341",
  "version": "v1",
  "published": "2021-09-09T15:30:34.000Z",
  "updated": "2021-09-09T15:30:34.000Z",
  "title": "Counting chains in the noncrossing partition lattice via the W-Laplacian",
  "authors": [
    "Guillaume Chapuy",
    "Theo Douvropoulos"
  ],
  "comment": "17 pages, comments very much welcome!",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "We give an elementary, case-free, Coxeter-theoretic derivation of the formula $h^nn!/|W|$ for the number of maximal chains in the noncrossing partition lattice $NC(W)$ of a real reflection group $W$. Our proof proceeds by comparing the Deligne-Reading recursion with a parabolic recursion for the characteristic polynomial of the $W$-Laplacian matrix considered in our previous work. We further discuss the consequences of this formula for the geometric group theory of spherical and affine Artin groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-09T15:30:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "05E99",
      "20F36"
    ],
    "keywords": [
      "noncrossing partition lattice",
      "counting chains",
      "w-laplacian",
      "affine artin groups",
      "geometric group theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}