








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






Organization:  Mathematical Institute, University of Leipzig 






Seminaire Lotharingien de Combinatoire (submitted) 






We show that the symmetry classes of torsionfree covariant derivatives ∇T of rtimes covariant tensor fields T can be characterized by LittlewoodRichardson products σ[1] where σ is a representation of the symmetric group S_{r} which is connected with the symmetry class of T. If σ ∼ [λ] is irreducible then σ[1] has a multiplicity free reduction [λ][1] ∼ ∑_{λ⊂μ [μ] 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 ∇S, ∇A of symmetric or alternating tensor fields. The symmetry classes of the differences ∇S  sym(∇S) and ∇A  alt(∇A) are characterized by Young frames (r, 1) and (2, 1^{r1}), respectively. However, while the symmetry class of ∇A  alt(∇A) can be generated by Young symmetrizers of (2, 1^{r1}), no Young symmetrizer of (r, 1) generates the symmetry class of ∇S  sym(∇S). Furthermore we show in the case r = 2 that ∇S  sym(∇S) and ∇A  alt(∇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.} 











http://arxiv.org/abs/math.CO/0301042







   
 
