{
  "id": "math/0401404",
  "version": "v1",
  "published": "2004-01-28T19:15:03.000Z",
  "updated": "2004-01-28T19:15:03.000Z",
  "title": "Lattice congruences of the weak order",
  "authors": [
    "Nathan Reading"
  ],
  "comment": "26 pages, 4 figures",
  "journal": "Order, 21 (2004) no.4, 315-344.",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of join-irreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system (W,S) and a subset K of S, let \\eta_K:w \\mapsto w_K be the projection onto the parabolic subgroup W_K. We show that the fibers of \\eta_K constitute the smallest lattice congruence with 1\\equiv s for every s\\in(S-K). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type A or B we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of join-irreducibles of the congruence lattice of the weak order.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-01-28T19:15:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "06B10",
      "52C35"
    ],
    "keywords": [
      "weak order",
      "finite coxeter group",
      "congruence lattice",
      "directed graph",
      "smallest lattice congruence"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......1404R"
    }
  }
}