Separation of the orbits in representations of SO2 and O2 over R and C

Abstract

I provide a minimal set of invariant polynomials separating the orbits for representations of ${\mathrm{SO}}_2$ and $O_2$ over $C$ and $R$. The idea is to select only polynomials of support of size 2 for ${\mathrm{SO}}_2$ and 4 for $O_2$. I thus obtain cardinalities in respectively $O(\dim {\left(V\right)}^2)$ and $O(\dim {\left(V\right)}^4)$. These cardinalities are much smaller than for generating sets, which require polynomials of arbitrary large supports. Yet a separating set is sufficient for most of the applications. It appears also that real separating sets are smaller than the complex ones, which helps significantly for applications over $R$. I finally use the obtained separating set to stratify the real representations by isotropy classes.