{
  "id": "math/0611106",
  "version": "v2",
  "published": "2006-11-04T22:33:12.000Z",
  "updated": "2007-10-18T22:02:21.000Z",
  "title": "Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups",
  "authors": [
    "Drew Armstrong"
  ],
  "comment": "Final version -- to appear in Memoirs of the American Mathematical Society. Many small improvements in exposition, especially in Sections 2.2, 4.1 and 5.2.1. Section 5.1.5 deleted. New references to recent work",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "This memoir constitutes the author's PhD thesis at Cornell University. It serves both as an expository work and as a description of new research. At the heart of the memoir, we introduce and study a poset $NC^{(k)}(W)$ for each finite Coxeter group $W$ and for each positive integer $k$. When $k=1$, our definition coincides with the generalized noncrossing partitions introduced by Brady-Watt and Bessis. When $W$ is the symmetric group, we obtain the poset of classical $k$-divisible noncrossing partitions, first studied by Edelman. Along the way, we include a comprehensive introduction to related background material. Before defining our generalization $NC^{(k)}(W)$, we develop from scratch the theory of algebraic noncrossing partitions $NC(W)$. This involves studying a finite Coxeter group $W$ with respect to its generating set $T$ of {\\em all} reflections, instead of the usual Coxeter generating set $S$. This is the first time that this material has appeared in one place. Finally, it turns out that our poset $NC^{(k)}(W)$ shares many enumerative features in common with the ``generalized nonnesting partitions'' of Athanasiadis and the ``generalized cluster complexes'' of Fomin and Reading. In particular, there is a generalized ``Fuss-Catalan number'', with a nice closed formula in terms of the invariant degrees of $W$, that plays an important role in each case. We give a basic introduction to these topics, and we describe several conjectures relating these three families of ``Fuss-Catalan objects''.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-10-18T22:02:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E15",
      "05E25",
      "05A18",
      "05-02"
    ],
    "keywords": [
      "generalized noncrossing partitions",
      "finite coxeter group",
      "combinatorics",
      "authors phd thesis",
      "usual coxeter generating set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....11106A"
    }
  }
}