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

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

2003-01-06

arxiv.org/abs/math/0301042v1

Seminaire Lotharingien de Combinatoire, 49 (2003) Article B49f

Mathematics

Combinatorics

21 pages. Sent in to Seminaire Lotharingien de Combinatoire: http://www.mat.univie.ac.at/~slc/

Scientific paper

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.

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