Positivity for Regular Cluster Characters in Acyclic Cluster Algebras

TL;DR

This paper proves the positivity of cluster characters associated with regular modules in acyclic cluster algebras, covering tame and wild cases with specific module conditions, advancing understanding of positivity in cluster theory.

Contribution

It establishes positivity results for cluster characters of regular modules in acyclic cluster algebras, including new cases for tame and wild hereditary algebras.

Findings

Positivity holds for regular modules in tame cases.

Positivity holds for certain regular Schur modules in wild cases.

Results extend positivity understanding in cluster algebra theory.

Abstract

Let $Q$ be an acyclic quiver and let $\mathcal A(Q)$ be the corresponding cluster algebra. Let $H$ be the path algebra of $Q$ over an algebraically closed field and let $M$ be an indecomposable regular $H$-module. We prove the positivity of the cluster characters associated to $M$ expressed in the initial seed of $\mathcal A(Q)$ when either $H$ is tame and $M$ is any regular $H$-module, or $H$ is wild and $M$ is a regular Schur module which is not quasi-simple.