Locally free Caldero-Chapoton functions via reflections

TL;DR

This paper explores the behavior of locally free Caldero-Chapoton functions in cluster algebras, establishing new formulas and proofs for their expressions in various algebraic settings.

Contribution

It introduces a reflection-based approach to express non-initial cluster variables as locally free Caldero-Chapoton functions, providing new proofs and extending formulas.

Findings

Non-initial cluster variables in rank 2 are expressed via locally free Caldero-Chapoton functions.

A new proof of Caldero-Chapoton formulas in Dynkin cases is provided.

The formula is extended to general acyclic skew-symmetrizable cluster algebras.

Abstract

We study the reflections of locally free Caldero-Chapoton functions associated to representations of Geiss-Leclerc-Schr\"oer's quivers with relations for symmetrizable Cartan matrices. We prove that for rank 2 cluster algebras, non-initial cluster variables are expressed as locally free Caldero-Chapoton functions of locally free indecomposable rigid representations. Our method gives rise to a new proof of the locally free Caldero-Chapoton formulas obtained by Geiss-Leclerc-Schr\"oer in Dynkin cases. For general acyclic skew-symmetrizable cluster algebras, we prove the formula for any non-initial cluster variable obtained by almost sink and source mutations.