Skew-symmetrizable cluster algebras from surfaces and symmetric quivers

TL;DR

This paper provides a geometric and algebraic framework for skew-symmetrizable cluster algebras from surfaces, linking cluster variables to arcs and modules, and establishing a cluster expansion formula using perfect matchings.

Contribution

It introduces a geometric realization of skew-symmetrizable cluster algebras via surface arcs and triangulations, and connects these algebras to symmetric quivers and representation theory.

Findings

Cluster variables correspond to $\sigma$-orbits of arcs.

A ring homomorphism to skew-symmetric cluster algebras is constructed.

A cluster expansion formula using perfect matchings is established.

Abstract

We study skew-symmetrizable cluster algebras $\mathcal{A}$ associated with unpunctured surfaces $\tilde{\mathbf{S}}$ endowed with an orientation-preserving involution $\sigma$. We give a geometric realization of such cluster algebras by showing that cluster variables of $\mathcal{A}$ correspond to $\sigma$-orbits of arcs of $\tilde{\mathbf{S}}$, while clusters are given by admissible $\sigma$-invariant triangulations. We establish a ring homomorphism from $\mathcal{A}$ to a skew-symmetric cluster algebra of the same rank, which is combinatorially derived from $\mathcal{A}$. We use this result to provide a cluster expansion formula for any $\sigma$-orbit $[\gamma]$ in terms of perfect matchings of some labeled modified snake graphs constructed from the arcs of $[\gamma]$. Then, we associate a symmetric finite-dimensional algebra $A$ to any seed of $\mathcal{A}$, such that non-initial cluster variables bijectively correspond to orthogonal indecomposable $A$-modules. Finally, we exhibit a purely representation-theoretic map from the category of orthogonal $A$-modules to $\mathcal{A}$, providing a Caldero-Chapoton map in this setting.