{
  "id": "2012.11015",
  "version": "v1",
  "published": "2020-12-20T20:22:57.000Z",
  "updated": "2020-12-20T20:22:57.000Z",
  "title": "Structure of conjugacy classes in Coxeter groups",
  "authors": [
    "Timothée Marquis"
  ],
  "comment": "99 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "This paper gives a definitive solution to the problem of describing conjugacy classes in arbitrary Coxeter groups in terms of cyclic shifts. Let $(W,S)$ be a Coxeter system. A cyclic shift of an element $w\\in W$ is a conjugate of $w$ of the form $sws$ for some simple reflection $s\\in S$ such that $\\ell_S(sws)\\leq\\ell_S(w)$. The cyclic shift class of $w$ is then the set of elements of $W$ that can be obtained from $w$ by a sequence of cyclic shifts. Given a subset $K\\subseteq S$ such that $W_K:=\\langle K\\rangle\\subseteq W$ is finite, we also call two elements $w,w'\\in W$ $K$-conjugate if $w,w'$ normalise $W_K$ and $w'=w_0(K)ww_0(K)$, where $w_0(K)$ is the longest element of $W_K$. Let $\\mathcal O$ be a conjugacy class in $W$, and let $\\mathcal O^{\\min}$ be the set of elements of minimal length in $\\mathcal O$. Then $\\mathcal O^{\\min}$ is the disjoint union of finitely many cyclic shift classes $C_1,\\dots,C_k$. We define the structural conjugation graph associated to $\\mathcal O$ to be the graph with vertices $C_1,\\dots,C_k$, and with an edge between distinct vertices $C_i,C_j$ if they contain representatives $u\\in C_i$ and $v\\in C_j$ such that $u,v$ are $K$-conjugate for some $K\\subseteq S$. In this paper, we compute explicitely the structural conjugation graph associated to any conjugacy class in $W$, and show in particular that it is connected (that is, any two conjugate elements of $W$ differ only by a sequence of cyclic shifts and $K$-conjugations). Along the way, we obtain several results of independent interest, such as a description of the centraliser of an infinite order element $w\\in W$, as well as the existence of natural decompositions of $w$ as a product of a \"torsion part\" and of a \"straight part\", with useful properties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-20T20:22:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "20E45"
    ],
    "keywords": [
      "conjugacy class",
      "cyclic shift class",
      "structural conjugation graph",
      "arbitrary coxeter groups",
      "infinite order element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 99,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}