The application of graph theory towards determining syzygies of the dependent polynomial invariants of the Riemann tensor
by
Allan Lim
Deakin University
Coauthors: John Carminati

The invariants of the Riemann tensor are of interest owing to their role in the classification of spacetimes in General Relativity. While there has been some progress towards the determination of a complete set of invariants, the Syzygy Problem: namely the explicit determination of the polynomial relationships (syzygies) relating the invariants outside the complete set to invariants within this set, remains unresolved. In this paper, we present a graph-theoretic notation for the invariants of the Riemann tensor and use this notation to derive constructive completeness proofs for the pure invariants, which lead to an algorithm for the explicit determination of the syzygies relating any pure invariant to the complete set. Progress towards proving similar theorems for the mixed invariants will be discussed.

Date received: September 20, 2003