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

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.