Denominator vectors and compatibility degrees in cluster algebras of finite type

Abstract

We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra. They provide two simple proofs of the known fact that the denominator vector of any non-initial cluster variable with respect to any initial cluster seed has non-negative entries and is different from zero.