We generalize the almost positive roots model for cluster algebras from finite type to a uniform finite/affine type model. We define a subset of the root system and a compatibility degree on , given by a formula that is new even in finite type. The clusters (maximal pairwise compatible sets of roots) define a complete fan Fan. Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of Fan induced by real roots to the -vector fan of the associated cluster algebra. We show that 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.
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Nathan Reading, Salvatore Stella, An affine almost positive roots model. J. Comb. Algebra 4 (2020), no. 1, pp. 1–59DOI 10.4171/JCA/37