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

Bernd Fiedler

Séminaire lotharingien de combinatoire, Tome 49 (2002-2004)

Résumé

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.