On total stability conditions for Dynkin quivers

TL;DR

This paper demonstrates that for a specific Dynkin quiver of type E7, no total stability condition of a certain form exists, providing a counterexample to an existing conjecture.

Contribution

It establishes the non-existence of a particular class of stability conditions for a specific Dynkin quiver, countering Reineke's conjecture.

Findings

No slope function of the form μ=θ/dim defines a total stability condition for the E7 quiver.

Provides a counterexample to Reineke's conjecture.

Highlights limitations in the existence of certain stability conditions.

Abstract

We show that for a Dynkin quiver $Q$ of type $E_7$ with a specific orientation, the path algebra $KQ$ has no slope function of the form $\mu=\frac{\theta}{\dim}$ that defines a total stability condition. This gives a counterexample to a conjecture of Reineke.