{
  "id": "math/9908110",
  "version": "v3",
  "published": "1999-08-20T05:23:19.000Z",
  "updated": "1999-12-01T22:29:57.000Z",
  "title": "Low-Dimensional Unitary Representations of B_3",
  "authors": [
    "Imre Tuba"
  ],
  "comment": "Added sections + some minor updates",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "We characterize all simple unitarizable representations of the braid group $B_3$ on complex vector spaces of dimension $d \\leq 5$. In particular, we prove that if $\\sigma_1$ and $\\sigma_2$ denote the two generating twists of $B_3$, then a simple representation $\\rho:B_3 \\to \\gl(V)$ (for $\\dim V \\leq 5$) is unitarizable if and only if the eigenvalues $\\lambda_1, \\lambda_2, ..., \\lambda_d$ of $\\rho(\\sigma_1)$ are distinct, satisfy $|\\lambda_i|=1$ and $\\mu^{(d)}_{1i} > 0$ for $2 \\leq i \\leq d$, where the $\\mu^{(d)}_{1i}$ are functions of the eigenvalues, explicitly described in this paper.",
  "revisions": [
    {
      "version": "v3",
      "updated": "1999-12-01T22:29:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36",
      "20C07",
      "81R10",
      "20H20",
      "16S34"
    ],
    "keywords": [
      "low-dimensional unitary representations",
      "complex vector spaces",
      "eigenvalues",
      "simple unitarizable representations",
      "simple representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math......8110T"
    }
  }
}