Cluster automorphism groups of cluster algebras of finite type

TL;DR

This paper characterizes the automorphism groups of finite type cluster algebras, linking them to root system symmetries and providing explicit group computations, including exceptional cases.

Contribution

It establishes a comprehensive description of cluster automorphism groups for finite type cluster algebras, including exceptional cases, and relates them to root system automorphisms and folding techniques.

Findings

Automorphisms correspond to root system transformations for most types.

Exceptional automorphisms exist in type D_{2n}, not generated by standard transformations.

Automorphism groups are isomorphic to those of associated universal cluster algebras.

Abstract

We study the cluster automorphism group $Aut(\mathcal{A})$ of a coefficient free cluster algebra $\mathcal{A}$ of finite type. A cluster automorphism of $\mathcal{A}$ is a permutation of the cluster variable set $\mathscr{X}$ that is compatible with cluster mutations. We show that, on the one hand, by the well-known correspondence between $\mathscr{X}$ and the almost positive root system $\Phi_{\geq -1}$ of the corresponding Dynkin type, the piecewise-linear transformations $\tau_+$ and $\tau_-$ on $\Phi_{\geq -1}$ induce cluster automorphisms $f_+$ and $f_-$ of $\mathcal{A}$ respectively; on the other hand, excepting type $D_{2n},(n\geqslant 2)$, all the cluster automorphisms of $\mathcal{A}$ are compositions of $f_+$ and $f_-$. For a cluster algebra of type $D_{2n},(n\geqslant 2)$, there exists exceptional cluster automorphism induced by a permutation of negative simple roots in $\Phi_{\geq -1}$, which is not a composition of $\tau_+$ and $\tau_-$. By using these results and folding a simply laced cluster algebra, we compute the cluster automorphism group for a non-simply laced finite type cluster algebra. As an application, we show that $Aut(\mathcal{A})$ is isomorphic to the cluster automorphism group of the $FZ$-universal cluster algebra of $\mathcal{A}$.