{
  "id": "math/0312171",
  "version": "v1",
  "published": "2003-12-08T19:31:51.000Z",
  "updated": "2003-12-08T19:31:51.000Z",
  "title": "Short formulas for algebraic covariant derivative curvature tensors via Algebraic Combinatorics",
  "authors": [
    "Bernd Fiedler"
  ],
  "comment": "38 pages",
  "categories": [
    "math.CO",
    "cs.SC",
    "math.DG"
  ],
  "abstract": "We consider generators of algebraic covariant derivative curvature tensors R' which can be constructed by a Young symmetrization of product tensors W*U or U*W, where W and U are covariant tensors of order 2 and 3. W is a symmetric or alternating tensor whereas U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2,1). Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 16 or 20 summands if W is symmetric or skew-symmetric, respectively. In special cases these numbers can be reduced to the minima 12 or 10. If these minima occur then U admits an index commutation symmetry. Furthermore minimal lengths are possible if U is formed from torsion-free covariant derivatives of symmetric or alternating 2-tensor fields. Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the symmetry class of algebraic covariant derivative curvature tensors. Furthermore we apply ideals and idempotents in group rings C[S_r] and discrete Fourier transforms for symmetric groups S_r. For symbolic calculations we used the Mathematica packages Ricci and PERMS.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-12-08T19:31:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53B20",
      "15A72",
      "05E10",
      "16D60",
      "05-04"
    ],
    "keywords": [
      "algebraic covariant derivative curvature tensors",
      "algebraic combinatorics",
      "short formulas",
      "symmetry class",
      "discrete fourier transforms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....12171F"
    }
  }
}