The identities of the algebraic invariants of the four-dimensional Riemann tensor. II

Abstract

This paper makes use of several known results from invariant theory to further investigate the invariants of the Riemann tensor and the identities between them. The work also relies on the idea of expressing the Riemann tensor in terms of two complex matrices in the space of self-dual bivectors. It is shown that all invariants that are of even degree in the trace-free Ricci tensor can be written as polynomial functions of a set of 28 invariants. It is believed that this set is a complete set for this type of invariant. Several identities for matrix polynomials of 3×3 matrices are also obtained. These are used to find a large number of the identities between the 28 invariants. While some of these identities are quite complicated, it is shown how, in the general case, they might be used to obtain all invariants from a knowledge of a smaller set.