Generators of algebraic covariant derivative curvature tensors and Young symmetrizers

We show that the space of algebraic covariant derivative curvature tensors R is generated by Young symmetrized product tensors T ⊗ T̂ or T̂ ⊗ T , where T and T̂ are covariant tensors of order 2 and 3 whose symmetry classes are irreducible and characterized by the following pairs of partitions: {(2), (3)}, {(2), (2 1)} or {(1), (2 1)}. Each of the partitions (2), (3) and (1) describes exactly one symmetry class, whereas the partition (2 1) characterizes an infinite set S of irreducible symmetry classes. This set S contains exactly one symmetry class S0 ∈ S whose elements T̂ ∈ S0 can not play the role of generators of tensors R . The tensors T̂ of all other symmetry classes from S \ {S0} can be used as generators for tensors R. Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the symmetry class of algebraic covariant derivative curvature tensors. Furthermore we apply ideals and idempotents in group rings C[Sr], the Littlewood-Richardson rule and discrete Fourier transforms for symmetric groups Sr. For certain symbolic calculations we used the Mathematica packages Ricci and PERMS.