Positivity for Regular Cluster Characters in Acyclic Cluster Algebras

Let $Q$ be an acyclic quiver and let $\mathcal A(Q)$ be the corresponding cluster algebra. Let $H$ be the path algebra of $Q$ over an algebraically closed field and let $M$ be an indecomposable regular $H$-module. We prove the positivity of the cluster characters associated to $M$ expressed in the initial seed of $\mathcal A(Q)$ when either $H$ is tame and $M$ is any regular $H$-module, or $H$ is wild and $M$ is a regular Schur module which is not quasi-simple.
Comment: v2 : 15 pages. Title changed. The paper was entirely rewritten and shortened. For the sake of simplicity, the context was reduced to acyclic cluster algebras and the section on the A-double-infinite quiver was removed. Nevertheless, the methods and the results remain essentially unchanged. To appear in the Journal of Algebra and its Applications