{
  "id": "1102.4353",
  "version": "v1",
  "published": "2011-02-21T21:38:56.000Z",
  "updated": "2011-02-21T21:38:56.000Z",
  "title": "Word-Induced Measures on Compact Groups",
  "authors": [
    "Gene S. Kopp",
    "John D. Wiltshire-Gordon"
  ],
  "comment": "15 pages, 3 figures",
  "categories": [
    "math.GR",
    "math.GN",
    "math.PR",
    "math.RT"
  ],
  "abstract": "Consider a group word w in n letters. For a compact group G, w induces a map G^n \\rightarrow G$ and thus a pushforward measure {\\mu}_w on G from the Haar measure on G^n. We associate to each word w a 2-dimensional cell complex X(w) and prove in Theorem 2.5 that {\\mu}_w is determined by the topology of X(w). The proof makes use of non-abelian cohomology and Nielsen's classification of automorphisms of free groups [Nie24]. Focusing on the case when X(w) is a surface, we rediscover representation-theoretic formulas for {\\mu}_w that were derived by Witten in the context of quantum gauge theory [Wit91]. These formulas generalize a result of Erd\\H{o}s and Tur\\'an on the probability that two random elements of a finite group commute [ET68]. As another corollary, we give an elementary proof that the dimension of an irreducible complex representation of a finite group divides the order of the group; the only ingredients are Schur's lemma, basic counting, and a divisibility argument.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-02-21T21:38:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54H11",
      "05E15"
    ],
    "keywords": [
      "compact group",
      "word-induced measures",
      "rediscover representation-theoretic formulas",
      "quantum gauge theory",
      "finite group divides"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.4353K"
    }
  }
}