{
  "id": "1111.7172",
  "version": "v5",
  "published": "2011-11-30T14:13:07.000Z",
  "updated": "2014-08-12T13:55:34.000Z",
  "title": "EL-Shellability and Noncrossing Partitions Associated with Well-Generated Complex Reflection Groups",
  "authors": [
    "Henri Mühle"
  ],
  "comment": "37 pages, 4 figures. Moved the technical details of the proof of the EL-shellability of $NC_{G(d,d,n)}$ to the appendix. More references added",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this article we prove that the lattice of noncrossing partitions is EL-shellable when associated with the well-generated complex reflection group of type $G(d,d,n)$, for $d,n\\geq 3$, or with the exceptional well-generated complex reflection groups which are no real reflection groups. This result was previously established for the real reflection groups and it can be extended to the well-generated complex reflection group of type $G(d,1,n)$, for $d,n\\geq 3$, as well as to three exceptional groups, namely $G_{25},G_{26}$ and $G_{32}$, using a braid group argument. We thus conclude that the lattice of noncrossing partitions of any well-generated complex reflection group is EL-shellable. Using this result and a construction by Armstrong and Thomas, we conclude further that the poset of $m$-divisible noncrossing partitions is EL-shellable for every well-generated complex reflection group. Finally, we derive results on the M\\\"obius function of these posets previously conjectured by Armstrong, Krattenthaler and Tomie.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2014-08-12T13:55:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "06A07",
      "05E15"
    ],
    "keywords": [
      "noncrossing partitions",
      "real reflection groups",
      "el-shellability",
      "exceptional well-generated complex reflection groups",
      "braid group argument"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.7172M"
    }
  }
}