Matrix formulae and skein relations for cluster algebras from surfaces

By Gregg Musiker, Lauren K. Williams. 2011

63 citations recorded by the source corpus.

This paper concerns cluster algebras with principal coefficients A(S,M) associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [MSW2]. Given any (generalized) arc or loop in the surface -- with or without self-intersections -- we associate an element of (the fraction field of) A(S,M), using products of elements of PSL_2(R). We give a direct proof that our matrix formulas for arcs and loops agree with the combinatorial formulas for arcs and loops in terms of matchings, which were given in [MSW, MSW2]. Finally, we use our matrix formulas to prove skein relations for the cluster algebra elements associated to arcs and loops. Our matrix formulas and skein relations generalize prior work of Fock and Goncharov [FG1, FG2, FG3], who worked in the coefficient-free case. The results of this paper will be used in [MSW2] in order to show that certain collections of arcs and loops comprise a vector-space basis for A(S,M).

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

View Paper: https://www.semanticscholar.org/paper/26d1889005e4d7ae0329be8adfde7c895d115d75

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths (2012, background): “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]).”
Skein algebras and cluster algebras of marked surfaces (2012, methodology): “e equality A 1 () = Sk o() for triangulable surfaces with at least two marked points (the commutative specialization of Theorem 9.8) has been independently proven by Musiker, Schier and Williams (in [MW13],[MSW13]) using more explicit methods than this paper. Remark 1.3. The commutative cluster alge”
Cluster categories for marked surfaces: punctured case (2013, background): “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-Williams [36], Musiker-Williams [38] and Canakci-Schiﬄer [9, 10, 11] gave combinatorial formulas for cluster variables and relations”
Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings (2012, unknown): “It is known [20, 19] that if the two arcs cross, then one can rewrite the product of the cluster variables as a linear combination of elements corresponding to arcs without crossings.”