{
  "id": "0809.2427",
  "version": "v2",
  "published": "2008-09-15T00:02:28.000Z",
  "updated": "2010-12-04T01:05:24.000Z",
  "title": "On Coxeter Diagrams of complex reflection groups",
  "authors": [
    "Tathagata Basak"
  ],
  "comment": "27 pages, 4 figures. Major addition to the previous version. Section 4 is new. Organization of the paper modified. Stylistic changes. Small errors and typos corrected",
  "categories": [
    "math.GR",
    "math.RT"
  ],
  "abstract": "We study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $\\cE = \\ZZ[e^{2 \\pi i/3}]$: there are only four such lattices, namely, the $\\cE$-lattices whose real forms are $A_2$, $D_4$, $E_6$ and $E_8$. Next, we address the issue of characterizing the diagrams for unitary reflection groups, a question that was raised by Brou\\'{e}, Malle and Rouquier. To this end, we describe an algorithm which, given a unitary reflection group $G$, picks out a set of complex reflections. The algorithm is based on an analogy with Weyl groups. If $G$ is a Weyl group, the algorithm immediately yields a set of simple roots. Experimentally we observe that if $G$ is primitive and $G$ has a set of roots whose $\\ZZ$--span is a discrete subset of the ambient vector space, then the algorithm selects a minimal generating set for $G$. The group $G$ has a presentation on these generators such that if we forget that the generators have finite order then we get a (Coxeter-like) presentation of the corresponding braid group. For some groups, such as $G_{33}$ and $G_{34}$, new diagrams are obtained. For $G_{34}$, our new diagram extends to an \"affine diagram\" with $\\ZZ/7\\ZZ$ symmetry.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-12-04T01:05:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "20F05",
      "20F65",
      "51F25"
    ],
    "keywords": [
      "complex reflection groups",
      "unitary reflection group",
      "weyl group",
      "ambient vector space",
      "study coxeter diagrams"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.2427B"
    }
  }
}