Denominators of cluster variables

TL;DR

This paper explores the relationship between cluster variables and modules in cluster categories, providing conditions under which denominators of cluster variables relate to module dimension vectors.

Contribution

It establishes conditions on cluster-tilting objects that determine when denominators of cluster variables match module dimension vectors.

Findings

Denominator exponents correspond to module dimension vectors under certain conditions.

Provides a criterion linking cluster algebra denominators to module theory.

Enhances understanding of the algebraic structure of cluster variables.

Abstract

Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in the cluster algebra and exceptional indecomposable objects in the cluster category inducing a correspondence between clusters and cluster-tilting objects. Fix a cluster-tilting object T and a corresponding initial cluster. By the Laurent phenomenon, every cluster variable can be written as a Laurent polynomial in the initial cluster. We give conditions on T equivalent to the fact that the denominator in the reduced form for every cluster variable in the cluster algebra has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of T.