Cluster characters for cyclic quivers

Abstract

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

Keywords

Cyclic quiver / cluster algebra / \mathrm{\mathbb{Z}}-basis

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg