Matrix formulae and skein relations for cluster algebras from surfaces

TL;DR

This paper develops matrix formulas and skein relations for cluster algebras from surfaces, generalizing previous work and providing tools for basis construction in these algebraic structures.

Contribution

It introduces explicit matrix formulas for arcs and loops in cluster algebras from surfaces and proves skein relations, extending prior coefficient-free results to principal coefficients.

Findings

Matrix formulas for arcs and loops match combinatorial matchings formulas.

Skein relations for cluster algebra elements are established.

Results generalize Fock-Goncharov's work to include principal coefficients.

Abstract

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).