Caldero-Keller approach to the denominators of cluster variables

TL;DR

This paper provides an alternative proof for the formula of denominators of cluster variables in acyclic cluster algebras, utilizing the Caldero-Keller approach and cluster characters, building on prior work by Buan, Marsh, and Reiten.

Contribution

It offers a new proof of the denominator formula for cluster variables using Caldero-Keller methods and Palu's cluster characters, expanding the theoretical understanding.

Findings

Alternative proof of denominator formula established

Utilizes Caldero-Keller approach and cluster characters

Connects module dimension vectors to cluster variable denominators

Abstract

Buan, Marsh and Reiten proved that if a cluster-tilting object $T$ in a cluster category $\mathcal C$ associated to an acyclic quiver $Q$ satisfies certain conditions with respect to the exchange pairs in $\mathcal C$, 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 paper, 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.