{
  "id": "math/0301042",
  "version": "v1",
  "published": "2003-01-06T16:28:59.000Z",
  "updated": "2003-01-06T16:28:59.000Z",
  "title": "On the symmetry classes of the first covariant derivatives of tensor fields",
  "authors": [
    "B. Fiedler"
  ],
  "comment": "21 pages. Sent in to Seminaire Lotharingien de Combinatoire: http://www.mat.univie.ac.at/~slc/",
  "journal": "Seminaire Lotharingien de Combinatoire, 49 (2003) Article B49f",
  "categories": [
    "math.CO",
    "cs.SC",
    "math.DG"
  ],
  "abstract": "We show that the symmetry classes of torsion-free covariant derivatives $\\nabla T$ of r-times covariant tensor fields T can be characterized by Littlewood-Richardson products $\\sigma [1]$ where $\\sigma$ is a representation of the symmetric group $S_r$ which is connected with the symmetry class of T. If $\\sigma = [\\lambda]$ is irreducible then $\\sigma [1]$ has a multiplicity free reduction $[\\lambda][1] = \\sum [\\mu]$ and all primitive idempotents belonging to that sum can be calculated from a generating idempotent e of the symmetry class of T by means of the irreducible characters or of a discrete Fourier transform of $S_{r+1}$. We apply these facts to derivatives $\\nabla S$, $\\nabla A$ of symmetric or alternating tensor fields. The symmetry classes of the differences $\\nabla S - sym(\\nabla S)$ and $\\nabla A - alt(\\nabla A)$ are characterized by Young frames (r, 1) and (2, 1^{r-1}), respectively. However, while the symmetry class of $\\nabla A - alt(\\nabla A)$ can be generated by Young symmetrizers of (2, 1^{r-1}), no Young symmetrizer of (r, 1) generates the symmetry class of $\\nabla S - sym(\\nabla S)$. Furthermore we show in the case r = 2 that $\\nabla S - sym(\\nabla S)$ and $\\nabla A - alt(\\nabla A)$ can be applied in generator formulas of algebraic covariant derivative curvature tensors. For certain symbolic calculations we used the Mathematica packages Ricci and PERMS.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-01-06T16:28:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53B20",
      "15A72",
      "05E10",
      "16D60",
      "05-04"
    ],
    "keywords": [
      "symmetry classes",
      "first covariant derivatives",
      "r-times covariant tensor fields",
      "algebraic covariant derivative curvature tensors",
      "young symmetrizer"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......1042F"
    }
  }
}