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

By Giovanni Cerulli Irelli, Daniel Labardini-Fragoso. Compositio Mathematica, 2011

58 citations recorded by the source corpus.

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 quiver with potential (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 and Thurston (with an arbitrary system of coefficients).

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b1253ed9bfe391b53a75549ac701b9dafabd32ba

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Citations in the corpus

Quivers with potentials associated to triangulated surfaces, part IV: removing boundary assumptions (2012, background): “ness or non-emptiness of the boundary (cf. [15]; and in the unpunctured non-empty boundary case, [3]), and for all tagged triangulations of arbitrarily-punctured surfaces with non-empty boundary (cf. [9]), but not for non-ideal tagged triangulations of (necessarily punctured)surfaces with empty boun”
Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths (2012, background): “A construction of quivers with potentials associated to triangulations was proposed in [41], and further studied in [34].”