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Cluster Structures and Subfans in Scattering Diagrams
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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We give more precise statements of Fock–Goncharov duality conjecture for cluster varieties parametrizing ${\rm SL}_{2}/{\rm PGL}_{2}$-local systems on the once punctured torus. Then we prove these statements. Along the way, using distinct subfans in the scattering diagram, we produce an example of a cluster variety with two non-equivalent cluster structures. To overcome the technical difficulty of infinite non-cluster wall-crossing in the scattering diagram, we introduce quiver folding into the machinery of scattering diagrams and give a quotient construction of scattering diagrams.
Keywords: cluster varieties, Donaldson–Thomas transformations, Markov quiver, non-equivalent cluster structures, scattering diagrams, quiver folding. 
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     author = {Yan Zhou},
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}
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Yan Zhou. Cluster Structures and Subfans in Scattering Diagrams. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a12/

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