Generic bases of skew-symmetrizable affine type cluster algebras

Abstract

Geiss, Leclerc and Schr\"oer introduced a class of 1-Iwanaga-Gorenstein algebras $H$ associated to symmetrizable Cartan matrices with acyclic orientations, generalizing the path algebras of acyclic quivers. They also proved that indecomposable rigid $H$-modules of finite projective dimension are in bijection with non-initial cluster variables of the corresponding Fomin-Zelevinsky cluster algebra. In this article, we prove in all affine types that their conjectural Caldero-Chapoton type formula on these modules coincide with the Laurent expression of cluster variables. By taking generic Caldero-Chapoton functions on varieties of modules of finite projective dimension, we obtain bases for affine type cluster algebras with full-rank coefficients containing all cluster monomials.