Cluster automorphisms and the marked exchange graphs of skew-symmetrizable cluster algebras

TL;DR

This paper extends the understanding of cluster automorphisms from skew-symmetric to skew-symmetrizable cluster algebras by introducing a marked exchange graph and exploring their behavior under unfoldings.

Contribution

It generalizes known results by defining a marking on exchange graphs for skew-symmetrizable algebras and analyzing automorphism behavior through unfoldings and orbifold coverings.

Findings

Marked exchange graphs facilitate the study of automorphisms in skew-symmetrizable cases.

Unfoldings relate skew-symmetrizable matrices to skew-symmetric ones, aiding analysis.

Applications include coverings of orbifolds by surfaces.

Abstract

Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras. In this paper we aim to generalise these results to the skew-symmetrizable case by introducing a marking on the exchange graph. Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how cluster automorphisms behave under this unfolding with applications to coverings of orbifolds by surfaces.