Generic cluster characters

TL;DR

This paper introduces a set of generic values of the cluster character in 2-Calabi-Yau categories, linking them to cluster monomials and providing bases for acyclic cluster algebras.

Contribution

It establishes the existence of a generic set of cluster character values under certain conditions, connecting them to cluster monomials and bases in acyclic cluster algebras.

Findings

The set of generic values contains all cluster monomials.

In finite cases, the generic set coincides with all cluster variables.

Provides a method to construct Z-linear bases in acyclic cluster algebras.

Abstract

Let $\CC$ be a Hom-finite triangulated 2-Calabi-Yau category with a cluster-tilting object $T$. Under a constructibility condition we prove the existence of a set $\mathcal G^T(\CC)$ of generic values of the cluster character associated to $T$. If $\CC$ has a cluster structure in the sense of Buan-Iyama-Reiten-Scott, $\mathcal G^T(\CC)$ contains the set of cluster monomials of the corresponding cluster algebra. Moreover, these sets coincide if $\mathcal C$ has finitely many indecomposable objects. When $\CC$ is the cluster category of an acyclic quiver and $T$ is the canonical cluster-tilting object, this set coincides with the set of generic variables previously introduced by the author in the context of acyclic cluster algebras. In particular, it allows to construct $\Z$-linear bases in acyclic cluster algebras.