{
  "id": "1003.0719",
  "version": "v3",
  "published": "2010-03-03T19:05:57.000Z",
  "updated": "2010-08-31T20:24:06.000Z",
  "title": "Abelianization of Subgroups of Reflection Group and their Braid Group; an Application to Cohomology",
  "authors": [
    "Vincent Beck"
  ],
  "comment": "16 pages, new results on the stabilizer of a hyperplane added in section 1 and 2, new organisation of the paper, tables and GAP instructions added",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "The final result of this article gives the order of the extension $$\\xymatrix{1\\ar[r] & P/[P,P] \\ar^{j}[r] & B/[P,P] \\ar^-{p}[r] & W \\ar[r] & 1}$$ as an element of the cohomology group $H^2(W,P/[P,P])$ (where $B$ and $P$ stands for the braid group and the pure braid group associated to the complex reflection group $W$). To obtain this result, we describe the abelianization of the stabilizer $N_H$ of a hyperplane $H$. Contrary to the case of Coxeter groups, $N_H$ is not in general a reflection subgroup of the complex reflection group $W$. So the first step is to refine Stanley-Springer's theorem on the abelianization of a reflection group. The second step is to describe the abelianization of various types of big subgroups of the braid group $B$ of $W$. More precisely, we just need a group homomorphism from the inverse image of $N_H$ by $p$ with values in $\\QQ$ (where $p : B \\ra W$ is the canonical morphism) but a slight enhancement gives a complete description of the abelianization of $p^{-1}(W')$ where $W'$ is a reflection subgroup of $W$ or the stabilizer of a hyperplane. We also suggest a lifting construction for every element of the centralizer of a reflection in $W$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-08-31T20:24:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36",
      "20F55",
      "20E22"
    ],
    "keywords": [
      "abelianization",
      "complex reflection group",
      "cohomology",
      "reflection subgroup",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.0719B"
    }
  }
}