Total stability and Auslander-Reiten theory for Dynkin quivers

This paper concerns stability functions for Dynkin quivers, in the generality introduced by Rudakov. We show that relatively few inequalities need to be satisfied for a stability function to be totally stable (i.e. to make every indecomposable stable). Namely, a stability function $\mu$ is totally stable if and only if $\mu (\tau V)<\mu (V)$ for every almost split sequence $0\to \tau V\to E\to V\to 0$ where $E$ is indecomposable. These can be visualized as those sequences around the “border” of the Auslander-Reiten quiver.