  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.
