Semicanonical basis generators of the cluster algebra of type \(A_1^{(1)}\). (Q870090)

Summary: 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 \textit{P. Caldero} and the author [Mosc. Math. J. 6, No. 3, 411-429 (2006; Zbl 1133.16012)]. As a by-product, there was given a combinatorial interpretation of the Laurent polynomials in question, equivalent to the one obtained by \textit{G. Musiker} and \textit{J. Propp} [Electron. J. Comb. 14, No. 1, Research paper R15 (2007; Zbl 1140.05053)]. The original argument by P. Caldero and the author used a geometric interpretation of the Laurent polynomials due to \textit{P. Caldero} and \textit{F. Chapoton} [Comment. Math. Helv. 81, No. 3, 595-616 (2006; Zbl 1119.16013)]. This note provides a quick, self-contained and completely elementary alternative proof of the same results.