The cluster character for cyclic quivers

TL;DR

This paper introduces a new cluster character for cyclic quivers' cluster categories, proving its properties and constructing a basis for related algebras, advancing the understanding of cluster algebra structures.

Contribution

It defines an analogue of the Caldero-Chapoton map for cyclic quivers and proves it functions as a cluster character with a constructed algebra basis.

Findings

The map is a cluster character satisfying inductive multiplication formulas.

A $ extbf{Z}$-basis for the algebra generated by generalized cluster variables is constructed.

The work extends cluster algebra theory to cyclic quivers.

Abstract

We define an analogue of the Caldero-Chapoton map (\cite{CC}) for the cluster category of finite dimensional nilpotent representations over a cyclic quiver. We prove that it is a cluster character (in the sense of \cite{Palu}) and satisfies some inductive formulas for the multiplication between the generalized cluster variables (the images of objects of the cluster category under the map). Moreover, we construct a $\mathbb{Z}$-basis for the algebras generated by all generalized cluster variables.