Vol. 307, No. 2, 2020

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Cluster automorphism groups and automorphism groups of exchange graphs

Wen Chang and Bin Zhu

Vol. 307 (2020), No. 2, 283–302
Abstract

For a coefficient-free cluster algebra $\mathsc{𝒜}$, we study the cluster automorphism group $Aut\left(\mathsc{𝒜}\right)$ and the automorphism group $Aut\left({E}_{\mathsc{𝒜}}\right)$ of its exchange graph  ${E}_{\mathsc{𝒜}}$. We show that these two groups are isomorphic with each other, if $\mathsc{𝒜}$ is of finite type excepting types of rank 2 and type ${F}_{4}$, or if $\mathsc{𝒜}$ is of skew-symmetric finite mutation type.

Keywords
cluster algebra, cluster automorphism group, exchange graph
Primary: 13F60
Secondary: 05E40
Milestones
Received: 20 June 2019
Revised: 27 January 2020
Accepted: 4 May 2020
Published: 4 September 2020
Authors
 Wen Chang School of Mathematics and Information Science Shaanxi Normal University Xian China Department of Mathematical Sciences Tsinghua University Beijing China Bin Zhu Department of Mathematical Sciences Tsinghua University Beijing China