Mutation classes of finite type cluster algebras with principal coefficients

Cluster algebras of finite type is a fundamental class of algebras whose classification is identical to the famous Cartan–Killing classification. More recently, Fomin and Zelevinsky introduced another central notion of cluster algebras with principal coefficients. These algebras are determined combinatorially by mutation classes of certain rectangular matrices. It was conjectured, by Fomin and Zelevinsky, that finite type cluster algebras with principal coefficients are characterized by the mutation classes which are finite. In this paper, we prove this conjecture.