On the symmetry classes of the first covariant derivatives of tensor fields
<|reference_start|>On the symmetry classes of the first covariant derivatives of tensor fields: 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.<|reference_end|>
@article{fiedler2003on,
    title={On the symmetry classes of the first covariant derivatives of tensor
  fields},
    author={B. Fiedler},
    journal={Seminaire Lotharingien de Combinatoire, 49 (2003) Article B49f},
    year={2003},
    archivePrefix={arXiv},
    eprint={math/0301042},
    primaryClass={math.CO cs.SC math.DG}
}