Generators of algebraic covariant derivative curvature tensors and Young symmetrizers

Details Generators of algebraic covariant derivative curvature tensors and Young symmetrizers Generators of algebraic covariant derivative curvature tensors and Young symmetrizers

2003-10-02

arxiv.org/abs/math/0310020v1

In: Leading-Edge Computer Science, S. Shannon (ed.), Nova Science Publishers, Inc. New York, 2006. pp. 219-239. ISBN: 1-59454-

Mathematics

Combinatorics

18 pages. Chapter for a book "Progress in Computer Science Research", in preparation by Nova Science Publishers, Inc.: http:

Scientific paper

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.

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