Modules Over Cluster-Tilted Algebras Determined by Their Dimension Vectors

We prove that indecomposable transjective modules over cluster-tilted algebras are uniquely determined by their dimension vectors. Similarly, we prove that for cluster-concealed algebras, rigid modules lifting to rigid objects in the corresponding cluster category are uniquely determined by their dimension vectors. Finally, we apply our results to a conjecture of Fomin and Zelevinsky on denominators of cluster variables.     

Caldero–Keller Approach to the Denominators of Cluster Variables

Buan, Marsh, and Reiten proved that if a cluster-tilting object T in a cluster category 𝒞 associated to an acyclic quiver Q satisfies certain conditions with respect to the exchange pairs in 𝒞, then the denominator in its reduced form of every cluster variable in the cluster algebra associated to Q has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of T. In this article, we give an alternative proof of this result using the Caldero–Keller approach to acyclic cluster algebras and the work of Palu on cluster characters.