Exchange Maps of Cluster Algebras

Every two seeds in a field of fractions $\mathcal{F}$ together with a symmetric group element gives rise to an automorphism of $\mathcal{F}$ called an exchange automorphism. For positive cluster algebras, we provide equivalent conditions for exchange automorphisms to be cluster isomorphisms of the corresponding cluster algebras. The equivalent conditions are given in terms of a symmetric group action on the set of seeds, which is also studied. Presentations of groups of cluster automorphisms of some seeds of ranks 1 and 2 are given.