Cluster Expansions: T-walks, Labeled Posets and Matrix Calculations

TL;DR

This paper introduces two combinatorial methods, T-walks and labeled posets, for computing cluster expansion formulas in cluster algebras from punctured surfaces, enhancing computational techniques in the field.

Contribution

It presents novel combinatorial approaches using T-walks and labeled posets for calculating cluster expansions, extending existing models to punctured surfaces and matrix calculations.

Findings

T-walks extend T-path models to punctured surfaces

Labeled posets provide an alternative computation method

Matrix techniques enable quick calculations

Abstract

We give two new combinatorial methods for computing cluster expansion formulas for arcs coming from possibly punctured surfaces. The first is by using $T$-walks, an extension of the $T$-path model for unpunctured surfaces to general surfaces. We also introduce a new way of generating $T$-paths. The second method is by using order ideals of labeled posets associated to arcs. We also use the theory of oriented posets to give a quick way to calculate the expressions using $2$ by $2$ matrices. The techniques introduced are applicable to different settings in cluster algebras and beyond.