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 the almost positive Schur roots $\Phi_c$ and a compatibility degree, given by a formula that is new even in finite type. The clusters define a complete fan $\operatorname{Fan}_c(\Phi)$. Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of $\operatorname{Fan}_c(\Phi)$ 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.