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Generic cluster characters

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.