Adjoint action of automorphism groups on radical endomorphisms, generic equivalence and Dynkin quivers

TL;DR

This paper characterizes Dynkin quivers by the existence of dense open orbits under automorphism group actions on radical endomorphisms of projective representations, providing a new perspective on their structure.

Contribution

It introduces a novel characterization of Dynkin quivers through the action of automorphism groups on radical endomorphisms, linking orbit structure to quiver classification.

Findings

Dense open orbits exist for all projective representations iff the quiver is Dynkin.

Provides a new criterion to identify Dynkin quivers based on automorphism group actions.

Connects geometric orbit properties with algebraic quiver classification.

Abstract

Let $Q$ be a connected quiver with no oriented cycles, $k$ the field of complex numbers and $P$ a projective representation of $Q$. We study the adjoint action of the automorphism group $\Aut_{kQ} P$ on the space of radical endomorphisms $\radE_{kQ}P$. Using generic equivalence, we show that the quiver $Q$ has the property that there exists a dense open $\Aut_{kQ} P$-orbit in $\radE_{kQ} P$, for all projective representations $P$, if and only if $Q$ is a Dynkin quiver. This gives a new characterisation of Dynkin quivers.