Bases for cluster algebras from surfaces

By Gregg Musiker, Ralf Schiffler, Lauren K. Williams. Compositio Mathematica, 2011

176 citations recorded by the source corpus.

Abstract We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, such as principal coefficients.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c1b624d5484875860f26799f1a824245f857283b

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Introduction to Cluster Algebras (2018, background): “Here is a partial list of related topics: combinatorics [39], hyperbolic geometry [11, 12, 40], Lie theory [19], Poisson geometry [24], integrable systems [9, 26], representations of associative algebras [6, 7, 5, 46, 45, 47], mathematical physics [10, 1], and quantum groups [33, 20, 34, 3].”
Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials (2011, background): “e with boundary, Theorem 6.4 and Corollary 6.5 indicate that, if an atomic basis exists, it should contain the cluster monomials. Remark 6.8. After this paper was submitted have the preprints [8] and [23] appeared. • In [8] it is shown that cluster monomials are always linearly independent in skew-s”
Cluster algebras and derived categories (2012, background): “He is very grateful to B. Leclerc for correcting an error in a previous version of example 4.3 and to H. Nakajima for a helpful conversation on the results of [30].”
Triangular bases in quantum cluster algebras and monoidal categorification conjectures (2015, background): “is, which is a variant of the dual canonical basis, cf. [Lus00]. (ii) For commutative cluster algebras arising from triangulated surfaces, Musiker, Schiﬄer and Williams constructed various bases, cf. [MSW13]. Their positivity were discussed in [Thu14]. Fock and Goncharov also constructed “canonical ”
Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths (2012, background): “ (for surfaces without punctures) and [44] (in full generality). Even more general formulas of this kind, for elements of certain explicit combinatorial bases in these cluster algebras, were given in [45] (making use of [46]). Another family of bases, closely related to dual canonical bases for quan”
Cluster categories for marked surfaces: punctured case (2013, background): “gebras in the surface case: • Felikson-Shapiro-Tumarkin [16] classiﬁed cluster algebras of ﬁnite mutation type, that they are all from marked surfaces expect for few cases. • Musiker-Schiﬄer-Williams [37] constructed two canonical bases by two types of collections of curves. • Musiker-Schiﬄer-Willia”
Linear independence of cluster monomials for skew-symmetric cluster algebras (2012, background): “The conjecture has been checked for several classes, notably for finite type cluster algebras [7] [12], for skew-symmetric acyclic cluster algebras [20] and for cluster algebras associated with surfaces [9] (see also [15] [26]).”