The incidence class and the hierarchy of orbits

TL;DR

This paper explores the incidence class of orbits in representation spaces, establishing conditions under which the incidence class indicates orbit hierarchy, with applications to quiver representations and singularities.

Contribution

It generalizes the notion of incidence class to orbits in representations, providing positivity conditions that determine orbit hierarchy, especially in Dynkin quiver cases.

Findings

Positivity ensures incidence class non-vanishing iff orbit closure inclusion.

For Dynkin quivers, positivity holds for all orbits.

In singularity cases, positivity may not hold universally.

Abstract

R. Rim\'anyi defined the incidence class of two singularities X and Y as $[X]|_Y$, the restriction of the Thom polynomial of X to Y. He conjectured that (under mild conditions) the incidence is not zero if and only if Y is in the closure of X. Generalizing this notion we define the incidence class of two orbits X and Y of a representation. We give a sufficient condition (positivity) for Y to have the property that the incidence class $[X]|_Y$ is not zero if and only if Y is in the closure of X for any other orbit X. We show that for many interesting cases, e.g. the quiver representations of Dynkin type positivity holds for all orbits. In other words in these cases the incidence classes completely determine the hierarchy of the orbits. We also study the case of singularities where positivity doesn't hold for all orbits.