{
  "id": "1108.2940",
  "version": "v2",
  "published": "2011-08-15T06:31:55.000Z",
  "updated": "2012-05-14T08:07:04.000Z",
  "title": "The dominance hierarchy in root systems of Coxeter groups",
  "authors": [
    "Xiang Fu"
  ],
  "journal": "J. Algebra, 366 (2012), 187-204",
  "doi": "10.1016/j.jalgebra.2012.05.013",
  "categories": [
    "math.RT"
  ],
  "abstract": "If $x$ and $y$ are roots in the root system with respect to the standard (Tits) geometric realization of a Coxeter group $W$, we say that $x$ \\emph{dominates} $y$ if for all $w\\in W$, $wy$ is a negative root whenever $wx$ is a negative root. We call a positive root \\emph{elementary} if it does not dominate any positive root other than itself. The set of all elementary roots is denoted by $\\E$. It has been proved by B. Brink and R. B. Howlett (Math. Ann. \\textbf{296} (1993), 179--190) that $\\E$ is finite if (and only if) $W$ is a finite-rank Coxeter group. Amongst other things, this finiteness property enabled Brink and Howlett to establish the automaticity of all finite-rank Coxeter groups. Later Brink has also given a complete description of the set $\\E$ for arbitrary finite-rank Coxeter groups (J. Algebra \\textbf{206} (1998)). However the set of non-elementary positive roots has received little attention in the literature. In this paper we answer a collection of questions concerning the dominance behaviour between such non-elementary positive roots. In particular, we show that for any finite-rank Coxeter group and for any non-negative integer $n$, the set of roots each dominating precisely $n$ other positive roots is finite. We give upper and lower bounds for the sizes of all such sets as well as an inductive algorithm for their computation.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-05-14T08:07:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "20F10",
      "20F65"
    ],
    "keywords": [
      "root system",
      "dominance hierarchy",
      "non-elementary positive roots",
      "arbitrary finite-rank coxeter groups",
      "finiteness property enabled brink"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.2940F"
    }
  }
}