An affine almost positive roots model

  • Nathan Reading

    North Carolina State University, Raleigh, USA
  • Salvatore Stella

    University of Leicester, UK and Università di Roma La Sapienza, Italy
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We generalize the almost positive roots model for cluster algebras from finite type to a uniform finite/affine type model. We define a subset Φc\Phi_c of the root system and a compatibility degree on Φc\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 Fanc(Φ)_c(\Phi). Equivalently, every vector has a unique cluster expansion. We give a piecewise linear isomorphism from the subfan of Fanc(Φ)_c(\Phi) induced by real roots to the g\mathbf g-vector fan of the associated cluster algebra. We show that Φc\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.

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Nathan Reading, Salvatore Stella, An affine almost positive roots model. J. Comb. Algebra 4 (2020), no. 1, pp. 1–59

DOI 10.4171/JCA/37