Cluster automorphism groups and automorphism groups of exchange graphs

Wen Chang and Bin Zhu

Vol. 307 (2020), No. 2, 283–302

DOI: 10.2140/pjm.2020.307.283

Abstract

For a coefficient-free cluster algebra $\mathcal{𝒜}$, we study the cluster automorphism group $Aut\left(\mathcal{𝒜}\right)$ and the automorphism group $Aut\left(E_{\mathcal{𝒜}}\right)$ 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 2 and type $F_4$, or if $\mathcal{𝒜}$ is of skew-symmetric finite mutation type.

Keywords

cluster algebra, cluster automorphism group, exchange graph

Mathematical Subject Classification 2010

Primary: 13F60

Secondary: 05E40

Milestones

Received: 20 June 2019

Revised: 27 January 2020

Accepted: 4 May 2020

Published: 4 September 2020