Cluster automorphism groups of cluster algebras of finite type

We study the cluster automorphism group Aut(A)Aut(A) of a coefficient free cluster algebra AA of finite type. A cluster automorphism of AA is a permutation of the cluster variable set XX that is compatible with cluster mutations. We show that, on the one hand, by the well-known correspondence between XX and the almost positive root system Φ≥−1Φ≥−1 of the corresponding Dynkin type, the piecewise-linear transformations τ+τ+ and τ−τ− on Φ≥−1Φ≥−1 induce cluster automorphisms f+f+ and f−f− of AA respectively; on the other hand, excepting type D2nD2n (n⩾2n⩾2), all the cluster automorphisms of AA are compositions of f+f+ and f−f−. For a cluster algebra of type D2nD2n (n⩾2n⩾2), there exists an exceptional cluster automorphism induced by a permutation of negative simple roots in Φ≥−1Φ≥−1, which is not a composition of τ+τ+ and τ−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)Aut(A) is isomorphic to the cluster automorphism group of the FZ  -universal cluster algebra of AA.