  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.
