[Paper Review] Infinite rank surface cluster algebras

̛İlke Çanakçı, Anna Felikson|Durham Research Online (Durham University)|Apr 6, 2017

Algebraic structures and combinatorial models26 references3 citations

TL;DR

This paper introduces infinite rank surface cluster algebras for infinite bordered surfaces with finitely many accumulation points of boundary marked points, generalizing finite surface cluster algebras via infinite mutation sequences and lambda lengths in hyperbolic geometry. The key contribution is an expansion formula for cluster variables using infinite snake graphs and Laurent series, along with skein relations and proof of the Laurent phenomenon and positivity in the infinite setting.

ABSTRACT

We generalise surface cluster algebras to the case of infinite surfaces where the surface contains finitely many accumulation points of boundary marked points. To connect different triangulations of an infinite surface, we consider infinite mutation sequences. We show transitivity of infinite mutation sequences on triangulations of an infinite surface and examine different types of mutation sequences. Moreover, we use a hyperbolic structure on an infinite surface to extend the notion of surface cluster algebras to infinite rank by giving cluster variables as lambda lengths of arcs. Furthermore, we study the structural properties of infinite rank surface cluster algebras in combinatorial terms, namely we extend "snake graph combinatorics" to give an expansion formula for cluster variables. We also show skein relations for infinite rank surface cluster algebras.

Motivation & Objective

• To extend surface cluster algebras to infinite surfaces with finitely many accumulation points of boundary marked points.
• To develop a framework for infinite mutation sequences that allow transitivity between triangulations of infinite surfaces.
• To define cluster variables using lambda lengths in a hyperbolic structure, enabling infinite rank cluster algebra construction.
• To generalize snake graph combinatorics to infinite settings and derive Laurent series expansions for cluster variables.
• To establish skein relations and structural properties such as the Laurent phenomenon and positivity in the infinite rank case.

Proposed method

• Use of infinite triangulations and infinite mutation sequences to connect different triangulations of an infinite surface.
• Introduction of a hyperbolic structure on the infinite surface to define lambda lengths as cluster variables.
• Construction of infinite snake graphs from lifts of generalised arcs in the universal cover.
• Definition of Laurent series associated with infinite snake graphs, parametrized by perfect matchings.
• Application of Ptolemy relations and skein relations to derive algebraic identities in the cluster algebra.
• Adaptation of the snake graph expansion formula to infinite settings, proving Laurent phenomenon and positivity via combinatorial coefficients.

Experimental results

Research questions

• RQ1 Can infinite mutation sequences be used to achieve transitivity between any two triangulations of an infinite surface?
• RQ2 How can cluster variables be defined in infinite rank surface cluster algebras when finite mutation sequences are insufficient?
• RQ3 What is the infinite analog of the snake graph combinatorics for Laurent expansion of cluster variables?
• RQ4 Do skein relations hold in the context of infinite rank surface cluster algebras?
• RQ5 Is the Laurent phenomenon and positivity preserved in infinite rank surface cluster algebras?

Key findings

• Infinite mutation sequences are transitive on triangulations of infinite surfaces with finitely many accumulation points of boundary marked points.
• Cluster variables in the infinite rank surface cluster algebra are given by lambda lengths of arcs, which are expressed as Laurent series via infinite snake graphs.
• The Laurent phenomenon holds for infinite rank surface cluster algebras, with cluster variables expressed as Laurent series in any initial cluster.
• Positivity of coefficients in the Laurent series expansion is established, as coefficients are positive integers parametrized by perfect matchings of infinite snake graphs.
• Skein relations hold in the form $x_1x_2 = x_3x_4 + x_5x_6$ for crossing generalised arcs, generalizing the finite case.
• For unpunctured surfaces, the denominator exponents in the Laurent expansion of a cluster variable equal the intersection numbers with arcs in the initial triangulation.