Tame algebras have dense \(\mathbf{g}\)-vector fans

Abstract

The \(\mathbf{g}\)-vector fan of a finite-dimensional algebra is a fan whose rays are the \(\mathbf{g}\)-vectors of its \(2\)-term presilting objects. We prove that the \(\mathbf{g}\)-vector fan of a tame algebra is dense. We then apply this result to obtain a near classification of quivers for which the closure of the cluster \(\mathbf{g}\)-vector fan is dense or is a half-space, using the additive categorification of cluster algebras by means of Jacobian algebras. As another application, we prove that for quivers with potentials arising from once-punctured closed surfaces, the stability and cluster scattering diagrams only differ by wall-crossing functions on the walls contained in a separating hyperplane. The appendix is devoted to the construction of truncated twist functors and their adjoints.