Semicanonical Basis Generators of the Cluster Algebra of Type $A_1^{(1)}$

Abstract

We study the cluster variables and "imaginary" elements of the semicanonical basis for the coefficient-free cluster algebra of affine type $A_1^{(1)}$. A closed formula for the Laurent expansions of these elements was given by P.Caldero and the author. As a by-product, there was given a combinatorial interpretation of the Laurent polynomials in question, equivalent to the one obtained by G.Musiker and J.Propp. The original argument by P.Caldero and the author used a geometric interpretation of the Laurent polynomials due to P.Caldero and F.Chapoton. This note provides a quick, self-contained and completely elementary alternative proof of the same results.