Locally free Caldero-Chapoton functions via reflections
Representation Theory
2023-02-27 v2 Combinatorics
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.
Cite
@article{arxiv.2206.02289,
title = {Locally free Caldero-Chapoton functions via reflections},
author = {Lang Mou},
journal= {arXiv preprint arXiv:2206.02289},
year = {2023}
}
Comments
22 pages; v2 improvements in proofs of Prop. 4.7 and Prop. 4.9