Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials

TL;DR

This paper extends the theory of quivers with potentials associated to triangulated surfaces by including tagged triangulations, establishing mutation invariance, and applying these results to prove properties of cluster monomials in surface cluster algebras.

Contribution

It introduces a construction of quivers with potentials for tagged triangulations, proves their mutation invariance, and applies these to analyze cluster monomials.

Findings

Quivers with potentials are associated to tagged triangulations.

Mutations of QPs correspond to flips of tagged triangulations.

Cluster monomials are proven to be linearly independent.

Abstract

To each tagged triangulation of a surface with marked points and non-empty boundary we associate a quiver with potential, in such a way that whenever we apply a flip to a tagged triangulation, the Jacobian algebra of the QP associated to the resulting tagged triangulation is isomorphic to the Jacobian algebra of the QP obtained by mutating the QP of the original one. Furthermore, we show that any two tagged triangulations are related by a sequence of flips compatible with QP-mutation. We also prove that for each of the QPs constructed, the ideal of the non-completed path algebra generated by the cyclic derivatives is admissible and the corresponding quotient is isomorphic to the Jacobian algebra. These results, which generalize some of the second author's previous work for ideal triangulations, are then applied to prove properties of cluster monomials, like linear independence, in the cluster algebra associated to the given surface by Fomin-Shapiro-Thurston (with an arbitrary system of coefficients).