An affine almost positive roots model

2017/07/02 by Reading, Nathan, Stella, Salvatore · 1 citation
#13F60 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1707.00340

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 Φ_c and a compatibility degree, given by a formula that is new even in finite type. The clusters define a complete fan Fan_c(Φ). Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of Fan_c(Φ) induced by real roots to the \mathbf g-vector fan of the associated cluster algebra. We show that Φ_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.

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