Component cluster for acyclic quiver

TL;DR

This paper explores the structure of component clusters in Caldero-Chapoton algebras related to acyclic quivers, comparing them to classical clusters and analyzing mutation relations especially for affine and wild quivers.

Contribution

It introduces a definition of mutation for component clusters and determines their relations for affine quivers, extending the understanding of cluster structures.

Findings

Component clusters are compared to classical clusters in acyclic quivers.

Mutation relations for component clusters are established for affine quivers.

Bounds on the size of component clusters are provided for wild quivers.

Abstract

The theory of Caldero-Chapoton algebras of Cerulli-Irelli, Labardini-Fragoso and Schroer leads to a refinement of the notions of cluster variables and clusters, via so called component clusters. In this paper we compare component clusters to classical clusters for the cluster algebra of an acyclic quiver. We propose a definition of mutation between component clusters and determine the mutation relations of component clusters for affine quivers. In the case of a wild quiver, we provide bounds for the size of component clusters.