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On the symmetry classes of the first covariant derivatives of tensor fields

Bernd Fiedler
Organization: Mathematical Institute, University of Leipzig
Journal / Anthology

Seminaire Lotharingien de Combinatoire (submitted)
Year: 2003

We show that the symmetry classes of torsion-free covariant derivatives ∇T of r-times covariant tensor fields T can be characterized by Littlewood-Richardson products σ[1] where σ is a representation of the symmetric group Sr which is connected with the symmetry class of T. If σ ∼ [λ] is irreducible then σ[1] has a multiplicity free reduction [λ][1] ∼ ∑λ⊂μ [μ] 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 ∇S, ∇A of symmetric or alternating tensor fields. The symmetry classes of the differences ∇S - sym(∇S) and ∇A - alt(∇A) are characterized by Young frames (r, 1) and (2, 1^{r-1}), respectively. However, while the symmetry class of ∇A - alt(∇A) can be generated by Young symmetrizers of (2, 1^{r-1}), no Young symmetrizer of (r, 1) generates the symmetry class of ∇S - sym(∇S). Furthermore we show in the case r = 2 that ∇S - sym(∇S) and ∇A - alt(∇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.

*Mathematics > Algebra > Group Theory