{
  "id": "math/9912013",
  "version": "v1",
  "published": "1999-12-02T02:22:14.000Z",
  "updated": "1999-12-02T02:22:14.000Z",
  "title": "Representations of the braid group B_3 and of SL(2,Z)",
  "authors": [
    "Imre Tuba",
    "Hans Wenzl"
  ],
  "comment": "To appear in the Pacific Journal of Mathematics",
  "categories": [
    "math.RT",
    "math.GR",
    "math.QA",
    "math.RA"
  ],
  "abstract": "We give a complete classification of simple representations of the braid group B_3 with dimension $\\leq 5$ over any algebraically closed f ield. In particular, we prove that a simple d-dimensional representation $\\rho: B_3 \\to GL(V)$ is determined up to isomorphism by the eigenvalues $\\lambda_1, \\lambda_2, ..., \\lambda_d$ of the image of the generators for d=2,3 and a choice of a $\\delta=\\sqrt{\\det \\rho(\\sigma_1)}$ for d=4 or a choice of $\\delta=\\sqrt[5]{\\det \\rho(\\sigma_1)}$ for d=5. We also s howed that such representations exist whenever the eigenvalues and $\\delta$ are not roots of certain polynomials $Q_{ij}^{(d)}$, which are explicitly given. In this case, we construct the matrices via which the generators act on V. As an application of our techniques, we also obtain nontrivial q-versions of some of Deligne's formulas for dimensions of representations of exceptional Lie groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1999-12-02T02:22:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36",
      "20C07",
      "81R10",
      "16S34",
      "15A69"
    ],
    "keywords": [
      "braid group",
      "exceptional lie groups",
      "simple d-dimensional representation",
      "simple representations",
      "delignes formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math.....12013T"
    }
  }
}