C-vectors and dimension vectors for cluster-finite quivers

Abstract

Let $(Q,W)$ be a quiver with a non degenerate potential. We give a new description of the \textbf{c}-vectors of $Q$. We use it to show that, if $Q$ is mutation equivalent to a Dynkin quiver, then the set of positive $\mathbf{c}$-vectors of the cluster algebra associated to $Q^{\text{op}}$ coincides with the set of dimension vectors of the indecomposable modules over the Jacobian algebra of $(Q,W)$.