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

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).

v4: Added: Comparison of our results on linear independence of cluster monomials with results obtained by other authors; a couple of examples and figures; references. Some typos corrected. Final version, to appear in Compositio Mathematica. 27 pages, 9 figures

References in corpus (5)

• Cluster algebras and triangulated surfaces. Part II: Lambda lengths
• Positivity in skew-symmetric cluster algebras of finite type
• Atomic bases in cluster algebras of types A and $\tilde{A}$
• Mutations for quivers with potentials: Oberwolfach talk, April 2007
• Quiver Grassmannians and their Euler characteristics: Oberwolfach talk, May 2010

Cited by in corpus (19)

• Cluster algebras and triangulated surfaces. Part II: Lambda lengths
• BPS Quivers and Spectra of Complete N=2 Quantum Field Theories
• Bases for cluster algebras from surfaces
• The representation type of Jacobian algebras
• Cluster categories for marked surfaces: punctured case
• Quivers with potentials associated to triangulated surfaces, part IV: Removing boundary assumptions
• Ice quivers with potentials associated with triangulations and Cohen-Macaulay modules over orders
• Bases for cluster algebras from orbifolds
• The enough $g$-pairs property and denominator vectors of cluster algebras
• Density of $g$-vector cones from triangulated surfaces
• T-Path Formula and Atomic Bases for Cluster Algebras of Type D
• Ice quivers with potential arising from once-punctured polygons and Cohen-Macaulay modules
• $q$-Stability conditions via $q$-quadratic differentials for Calabi-Yau-$\mathbb{X}$ categories
• Unfolding of acyclic sign-skew-symmetric cluster algebras and applications to positivity and $F$-polynomials
• Derived invariants for surface cut algebras II: the punctured case
• Arc representations
• A geometric model for the module category of a skew-gentle algebra
• Cluster Expansions: T-walks, Labeled Posets and Matrix Calculations
• Laminations of punctured surfaces as $\tau$-regular irreducible components