On Reconstructing Configurations of Points in ${\mathbb{P}}^2$ from a Joint Distribution of Invariants

Consider the diagonal action of the projective group ${\text{PGL}}_3$ on $n$ copies of ${\mathbb{P}}^2$. In addition, consider the action of the symmetric group ${\mathrm{\Sigma }}_n$ by permuting the copies. In this paper we find a set of generators for the invariant field of the combined group ${\mathrm{\Sigma }}_n\times {\text{PGL}}_3$. As the main application, we obtain a reconstruction principle for point configurations in ${\mathbb{P}}^2$ from their sub-configurations of five points. Finally, we address the question of how such reconstruction principles pass down to subgroups.