On Cluster Algebras Arising from Unpunctured Surfaces

Abstract

Abstract. We study cluster algebras that are associated to unpunctured surfaces, with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. In the special case where the cluster algebra is acyclic, we also give a formula for the expansion of cluster variables as a polynomial whose indeterminates are the cluster variables contained in the union of an arbitrary acyclic cluster and all its neighbouring clusters in the mutation graph.

Atomic bases of cluster algebras of types A and Ã

“…The positivity conjecture asserts that every cluster monomial in A Q is a positive element of A Q ; see [18]. This conjecture was, in particular, established for cluster algebras with a bipartite seed in [29], and for cluster algebras arising from surfaces [28,36].…”

Positive basis for surface skein algebras

“…The bracelets basis was considered by Musiker, Schiffler, and Williams (12), who proved a weaker form of positivity. This weaker positivity and explicit combinatorial formulas have been well studied (13)(14)(15)(16).…”

Positivity for cluster algebras

“…• Cluster algebras from surfaces. In this case, positivity has been shown in [24] building on [28], [30], [29], using the fact that each cluster variable in such a cluster algebra corresponds to a curve in an oriented Riemann surface, and the Laurent expansion of the cluster variable is determined by the crossing pattern of the curve with a fixed triangulation of the surface [11], [12]. The construction and the proof of the positivity conjecture have been generalized to non skew-symmetric cluster algebras from orbifolds in [10].…”