Cluster automorphism groups and automorphism groups of exchange graphs

TL;DR

This paper proves that for most finite type and skew-symmetric finite mutation type cluster algebras, the automorphism group of the algebra is isomorphic to that of its exchange graph, revealing a deep structural connection.

Contribution

It establishes the isomorphism between cluster automorphism groups and exchange graph automorphism groups for broad classes of cluster algebras, except some specific types.

Findings

Automorphism groups are isomorphic in finite type (excluding rank two and F4)

Automorphism groups are isomorphic in skew-symmetric finite mutation type

Provides structural insight into cluster algebra symmetries

Abstract

For a coefficient free cluster algebra $\mathcal{A}$, we study the cluster automorphism group $Aut(\mathcal{A})$ and the automorphism group $Aut(E_{\mathcal{A}})$ of its exchange graph $E_{\mathcal{A}}$. We show that these two groups are isomorphic with each other, if $\mathcal{A}$ is of finite type excepting types of rank two and type $F_4$, or if $\mathcal{A}$ is of skew-symmetric finite mutation type.