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On the symmetry classes of the first covariant derivatives of tensor fields. (English) Zbl 1043.53017
Summary: 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.
MSC:
53B20 Local Riemannian geometry
15A72 Vector and tensor algebra, theory of invariants
05E10 Combinatorial aspects of representation theory
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
05-04 Software, source code, etc. for problems pertaining to combinatorics
Software:
PERMS; Mathematica; Ricci
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