Total stability and Auslander-Reiten theory for Dynkin quivers

TL;DR

This paper characterizes total stability functions for Dynkin quivers, showing they are determined by inequalities involving almost split sequences and the Auslander-Reiten quiver, simplifying the stability analysis.

Contribution

It provides a precise criterion for total stability in Dynkin quivers based on inequalities from almost split sequences, connecting stability to Auslander-Reiten theory.

Findings

Total stability is characterized by inequalities involving almost split sequences.

The stability condition relates to the position of sequences around the Auslander-Reiten quiver.

Few inequalities are needed to verify total stability for Dynkin quivers.

Abstract

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.