C-vectors and dimension vectors for cluster-finite quivers

TL;DR

This paper provides a new description of c-vectors for quivers with potentials and establishes a correspondence between positive c-vectors and indecomposable module dimension vectors for Dynkin-type quivers.

Contribution

It introduces a novel characterization of c-vectors and proves their equivalence to indecomposable module dimension vectors in certain cluster algebras.

Findings

Positive c-vectors match indecomposable module dimension vectors for Dynkin quivers.

New description of c-vectors in the context of quivers with potentials.

Connection established between cluster algebra combinatorics and representation theory.

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 $(Q,W)$.