Séminaire Lotharingien de Combinatoire, B49f (2003), 22 pp.
Bernd Fiedler
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 Sr
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 Sr+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 (2,1r-1) \vdash r+1,
respectively. However, while the symmetry class of \nabla A -
alt(\nabla A) can be generated by Young symmetrizers
of (2,1r-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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