Generators of algebraic curvature tensors based on a (2,1)-symmetry

Abstract

We consider generators of algebraic curvature tensors R which can be constructed by a Young symmetrization of product tensors U*w or w*U, where U and w are covariant tensors of order 3 and 1. We assume that U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2,1). We show that the 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. Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 8 summands. In special cases these numbers can be reduced to the minimum 4. If this minimum occurs then U admits an index commutation symmetry. Furthermore minimal lengths are possible if U is formed from torsion-free covariant derivatives of alternating 2-tensor fields. We apply ideals and idempotents of group rings C[S_r] of symmetric groups S_r, Young symmetrizers, discrete Fourier transforms and Littlewood-Richardson products. For symbolic calculations we used the Mathematica packages Ricci and PERMS.