Cluster automorphism groups of cluster algebras of finite type

Abstract

We study the cluster automorphism group Aut(A) of a coefficient free cluster algebra A of finite type. A cluster automorphism of A is a permutation of the cluster variable set X that is compatible with cluster mutations. We show that, on the one hand, by the well-known correspondence between X and the almost positive root system Φ≥−1 of the corresponding Dynkin type, the piecewise-linear transformations τ+ and τ− on Φ≥−1 induce cluster automorphisms f+ and f− of A respectively; on the other hand, excepting type D2n (n⩾2), all the cluster automorphisms of A are compositions of f+ and f−. For a cluster algebra of type D2n (n⩾2), there exists an exceptional cluster automorphism induced by a permutation of negative simple roots in Φ≥−1, which is not a composition of τ+ and τ−. 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(A) is isomorphic to the cluster automorphism group of the FZ-universal cluster algebra of A.