The paper proves that any Z-algebra homomorphism of a cluster algebra mapping one cluster to another is a cluster automorphism, confirming Chang-Schiffler's conjecture.
Cluster automorphism groups and automorphism groups of exchange graphs
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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.
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A conjecture on cluster automorphisms of cluster algebras
The paper proves that any Z-algebra homomorphism of a cluster algebra mapping one cluster to another is a cluster automorphism, confirming Chang-Schiffler's conjecture.