An affine almost positive roots model

Abstract

We generalize the almost positive roots model for cluster algebras from finite type to a uniform finite/affine type model. We define a subset ${\Phi }_c$ of the root system and a compatibility degree on ${\Phi }_c$, given by a formula that is new even in finite type. The clusters (maximal pairwise compatible sets of roots) define a complete fan Fan${}_c(\Phi )$. Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of Fan${}_c(\Phi )$ induced by real roots to the $\mathbf{g}$-vector fan of the associated cluster algebra. We show that ${\Phi }_c$ is the set of denominator vectors of the associated acyclic cluster algebra and conjecture that the compatibility degree also describes denominator vectors for non-acyclic initial seeds. We extend results on exchangeability of roots to the affine case.