Stable Separation of Orbits for Finite Abelian Group Actions

Abstract

In this paper we construct two new families of invariant maps that separate the orbits of the action of a finite Abelian group on a finite dimensional complex vector space. One of these families is Lipschitz continuous with respect to the quotient metric on the space of orbits, but involves computing large powers of the components of the vectors which can lead to instabilities. The other family avoids this issue by putting the powers only on the phase of the components, but in turn is not continuous. However, we show that they are Lipschitz continuous on the set of vectors with fixed support, so in particular they are Lipschitz on the set of vectors with no zero entries. Furthermore, the target dimension of these maps is small, i.e., linear in the original dimension.