Positivity in skew-symmetric cluster algebras of finite type

Abstract

We prove that the basis of cluster monomials of a skew-symmetric cluster algebra A of finite type is the atomic basis of A. This means that an element of A is positive if and only if it has a non-negative expansion in the basis of cluster monomials. In particular cluster monomials are positive indecomposable, i.e. they cannot be written as a sum of positive elements.