  We consider generators of algebraic covariant derivative curvature tensors R'
which can be constructed by a Young symmetrization of product tensors W*U or
U*W, where W and U are covariant tensors of order 2 and 3. W is a symmetric or
alternating tensor whereas U belongs to a class of the infinite set S of
irreducible symmetry classes characterized by the partition (2,1). 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 16 or 20 summands if W is symmetric or
skew-symmetric, respectively. In special cases these numbers can be reduced to
the minima 12 or 10. If these minima occur then U admits an index commutation
symmetry. Furthermore minimal lengths are possible if U is formed from
torsion-free covariant derivatives of symmetric or alternating 2-tensor fields.
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[S_r] and discrete Fourier
transforms for symmetric groups S_r. For symbolic calculations we used the
Mathematica packages Ricci and PERMS.
