Generators of algebraic covariant derivative curvature tensors and Young symmetrizers
<|reference_start|>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 tensor products W*U or U*W, where W and U 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 1),(2 1)}. Each of the partitions (2), (3) and (1 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 S_0 whose elements U can not play the role of generators of tensors R'. The tensors U of all other symmetry classes from S\{S_0} 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.<|reference_end|>
@article{fiedler2003generators,
    title={Generators of algebraic covariant derivative curvature tensors and Young
  symmetrizers},
    author={B. Fiedler},
    journal={In: Leading-Edge Computer Science, S. Shannon (ed.), Nova Science
  Publishers, Inc. New York, 2006. pp. 219-239. ISBN: 1-59454-526-X},
    year={2003},
    archivePrefix={arXiv},
    eprint={math/0310020},
    primaryClass={math.CO cs.SC math.DG}
}