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 ${\mathrm{Fan}}_c(\Phi )$. Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of ${\mathrm{Fan}}_c(\Phi )$ 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.

45 pages. *Version 4 addresses concerns from a referee * Version 3 corrects typesetting errors caused by the order of packages in the preamble * Version 2 is a major revision and contains only the results concerning the affine almost positive roots model; the discussion on orbits of coxeter elements is now arXiv:1808.05090