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

TL;DR

This paper characterizes the symmetry classes of covariant derivatives of tensor fields using Littlewood-Richardson products and Young symmetrizers, providing explicit methods for symmetric and alternating tensors and applications to curvature tensors.

Contribution

It introduces a novel characterization of symmetry classes of covariant derivatives via Littlewood-Richardson products and computes primitive idempotents using Fourier transforms, with applications to curvature tensor generators.

Findings

Symmetry classes of derivatives are described by Littlewood-Richardson products.

Primitive idempotents can be calculated from generating idempotents using characters.

Applications include generator formulas for algebraic covariant derivative curvature tensors.

Abstract

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.