# 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

## Abstract

We generalize the almost positive roots model for cluster algebras from finite type to a uniform finite/affine type model. We define a subset $Φ_{c}$ of the root system and a compatibility degree on $Φ_{c}$, given by a formula that is new even in finite type. The clusters (maximal pairwise compatible sets of roots) 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 $g$-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.

## Cite this article

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