Cluster fans, stability conditions, and domains of semi-invariants

TL;DR

This paper explores the geometric structure of stability conditions and semi-invariants in quiver theory, revealing connections to cluster fans and providing new proofs of key results, especially for Dynkin quivers.

Contribution

It establishes that the cone of finite stability conditions is covered by the dual of the cluster fan and offers new proofs of Schofield's results, linking stability, semi-invariants, and cluster complexes.

Findings

The cone of stability conditions is covered by the dual cluster fan.

New proofs of Schofield's results on perpendicular categories.

Reconfirmation of the Igusa-Orr-Todorov-Weyman theorem for Dynkin quivers.

Abstract

We show that the cone of finite stability conditions of a quiver Q without oriented cycles has a fan covering given by (the dual of) the cluster fan of Q. Along the way, we give new proofs of Schofield's results on perpendicular categories. We also study domains of semi-invariants of quivers via quiver exceptional sequences. In particular, we recover Igusa-Orr-Todorov-Weyman's theorem on cluster complexes and domains of semi-invariants for Dynkin quivers.