Combinatorics of $\mathcal{X}$-variables in finite type cluster algebras

TL;DR

This paper calculates the number of -variables in finite type cluster algebras with the universal semifield, linking these counts to geometric structures in classical types and conjecturing broader applicability.

Contribution

It provides explicit counts of -variables for finite type cluster algebras and establishes a geometric bijection for classical types, proposing a conjecture for general surfaces.

Findings

Number of -variables computed for finite type cluster algebras.

Bijection between coefficients and quadrilaterals in triangulations for classical types.

Conjecture extending results to arbitrary marked surfaces.

Abstract

We compute the number of $\mathcal{X}$-variables (also called coefficients) of a cluster algebra of finite type when the underlying semifield is the universal semifield. For classical types, these numbers arise from a bijection between coefficients and quadrilaterals (with a choice of diagonal) appearing in triangulations of certain marked surfaces. We conjecture that similar results hold for cluster algebras from arbitrary marked surfaces, and obtain corollaries regarding the structure of finite type cluster algebras of geometric type.