{
  "id": "1103.5179",
  "version": "v2",
  "published": "2011-03-27T02:29:58.000Z",
  "updated": "2011-10-17T23:55:13.000Z",
  "title": "Arrangements stable under the Coxeter groups",
  "authors": [
    "Hidehiko Kamiya",
    "Akimichi Takemura",
    "Hiroaki Terao"
  ],
  "journal": "Configuration Spaces: Geometry, Combinatorics and Topology, Scuola Normale Superiore Pisa, pp.327-354, 2012",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let B be a real hyperplane arrangement which is stable under the action of a Coxeter group W. Then B acts naturally on the set of chambers of B. We assume that B is disjoint from the Coxeter arrangement A=A(W) of W. In this paper, we show that the W-orbits of the set of chambers of B are in one-to-one correspondence with the chambers of C=A\\cup B which are contained in an arbitrarily fixed chamber of A. From this fact, we find that the number of W-orbits of the set of chambers of B is given by the number of chambers of C divided by the order of W. We will also study the set of chambers of C which are contained in a chamber b of B. We prove that the cardinality of this set is equal to the order of the isotropy subgroup W_b of b. We illustrate these results with some examples, and solve an open problem in Kamiya, Takemura and Terao [Ranking patterns of unfolding models of codimension one, Adv. in Appl. Math. (2010)] by using our results.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-10-17T23:55:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "32S22",
      "52C35"
    ],
    "keywords": [
      "coxeter group",
      "arrangements stable",
      "real hyperplane arrangement",
      "open problem",
      "one-to-one correspondence"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.5179K"
    }
  }
}