The incidence class and the hierarchy of orbits

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.

💡 Research Summary

The paper revisits the notion of incidence class originally introduced by R. Rimányi for singularities, where for two singularities (X) and (Y) the incidence class is defined as the restriction of the Thom polynomial of (X) to (Y), denoted (