Cluster algebras of type A (1)2

In this paper we study cluster algebras A of type affine-A2. We solve the recurrence relations among the cluster variables (which form a T-system of type affine-A2). We solve the recurrence relations among the coefficients of A (which form a Y-system of type affine-A2). In A, there is a natural notion of positivity. We find linear bases B of A such that positive linear combinations of elements of B coincide with the cone of positive elements. We call these bases atomic bases of A. These are the analogue of the “canonical bases” found by Sherman and Zelevinsky in type affine-A1. Every atomic basis consists of cluster monomials together with extra elements. We provide explicit expressions for the elements of such bases in every cluster. We prove that the elements of B are parameterized by Z3 via their g-vectors in every cluster. We prove that the denominator vector map in every acyclic seed of A restricts to a bijection between B and Z3. We find explicit recurrence relations to express every element of A as linear combinations of elements of B.