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

Chindris, Calin

doi:10.1090/S0002-9947-2010-05184-0

Cluster fans, stability conditions, and domains of semi-invariants
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Trans. Amer. Math. Soc. 363 (2011), 2171-2190 Request permission

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 (1991) on perpendicular categories. From our results, we recover Igusa-Orr-Todorov-Weyman’s theorem on cluster complexes and domains of semi-invariants for Dynkin quivers. For arbitrary quivers, we also give a description of the domains of semi-invariants labeled by real Schur roots in terms of quiver exceptional sets.

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