Unistructurality of cluster algebras from unpunctured surfaces

Bazier-Matte, Véronique; Plamondon, Pierre-Guy

doi:10.1090/proc/14932

Unistructurality of cluster algebras from unpunctured surfaces
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by Véronique Bazier-Matte and Pierre-Guy Plamondon PDF

Proc. Amer. Math. Soc. 148 (2020), 2397-2409 Request permission

Abstract:

A cluster algebra is unistructural if the set of its cluster variables determines its clusters and seeds. It is conjectured that all cluster algebras are unistructural. In this paper, we show that any cluster algebra arising from a triangulation of a marked surface without punctures is unistructural. Our proof relies on the existence of a positive basis known as the bracelet basis and on the skein relations. We also prove that a cluster algebra defined from a disjoint union of quivers is unistructural if and only if the cluster algebras defined from the connected components of the quiver are unistructural.

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