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Séminaire Lotharingien de Combinatoire, B49f (2003), 22 pp.

# Bernd Fiedler

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On the Symmetry Classes of the First Covariant Derivatives of Tensor Fields

**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 \sim* [*\lambda*]
is irreducible then
*\sigma*[1] has a multiplicity free reduction [*\lambda*][1]
*\sim \sum_{\lambda \subset \mu}* [*\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*)
= *\nabla A* - *dA* are characterized by Young frames
(*r*,1) *\vdash r*+1 and (2,1^{r-1}) *\vdash 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.

Received: January 23, 2003.
Revised: December 17, 2003.
Accepted: December 30, 2003.

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