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

We show that the symmetry classes of torsion-free covariant derivatives ∇T of r-times covariant tensor fields T can be characterized by LittlewoodRichardson 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 Sr+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) = ∇A−dA are characterized by Young frames (r, 1) ⊢ 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), 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.