Modules over cluster-tilted algebras determined by their dimension vectors

TL;DR

This paper proves that indecomposable modules over certain algebras are uniquely identified by their dimension vectors and applies these results to a conjecture in cluster algebra theory.

Contribution

It establishes the uniqueness of modules determined by dimension vectors over cluster-tilted and cluster-concealed algebras, advancing understanding in cluster algebra representation theory.

Findings

Indecomposable transjective modules are uniquely determined by their dimension vectors.

Rigid modules in cluster-concealed algebras are uniquely determined by their dimension vectors.

Results contribute to a conjecture on denominators of cluster variables.

Abstract

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.