Generic bases of skew-symmetrizable affine type cluster algebras

TL;DR

This paper proves a conjectural formula for modules over certain algebras in affine types, leading to the construction of bases for affine cluster algebras that include all cluster monomials.

Contribution

It confirms a Caldero-Chapoton type formula in affine types and constructs bases containing all cluster monomials for these cluster algebras.

Findings

Caldero-Chapoton formula matches Laurent expressions of cluster variables in affine types.

Constructs bases for affine cluster algebras with full-rank coefficients.

Includes all cluster monomials in the constructed bases.

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.