Stability conditions, g-vectors, and associahedra

We will use King stability conditions (which are simply elements in a vector space) to construct torsion classes. For representations of Dynkin quivers, we can get every torsion class by varying the stability condition. This gives an intuitive way of building torsion classes. It is also interesting to consider the set of stability conditions giving rise to a particular torsion class; it turns out that this set is a convex cone. These cones partition the vector space, giving a polyhedral fan. In type A, this fan is the normal fan to the associahedron. The rays of this fan correspond naturally to certain modules (the tau-rigid indecomposables). The generator for a ray is called the g-vector of the corresponding module, and has an algebraic interpretation. We will work all of this out explicitly in the case of the path quiver, where everything has natural interpretations in terms of the Tamari lattice.

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Date published: Wednesday, March 18, 2026