Bases for cluster algebras from surfaces

We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, for example, principal coefficients.Comment: 53 pages; v2 references update

On cluster algebras arising from unpunctured surfaces II

We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. Furthermore, we obtain direct formulas for F-polynomials and g-vectors and show that F-polynomials have constant term equal to 1. As an application, we compute the Euler-Poincar\'e characteristic of quiver Grassmannians in Dynkin type $A$ and affine Dynkin type $\tilde A$.Comment: 36 pages, 9 figure