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.Comment: 24 pages. Final Version. In particular, a new section studying an explicit example was adde

Cluster algebras as Hall algebras of quiver representations

Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding quiver representation category. This also provides some explicit formulas for cluster variables.Comment: 17 pages ; 2 figures ; the title has changed ! some other minor modification

From triangulated categories to cluster algebras

The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of the cluster category. This realization provides a natural basis for A. We prove new results and formulate conjectures on `good basis' properties, positivity, denominator theorems and toric degenerations.Comment: 31 pages, typos correcte