Labelled seeds and the mutation group

Abstract: We study the set S of labelled seeds of a cluster algebra of rank n inside a field F as a homogeneous space for the group M_n of (globally defined) mutations and relabellings. Regular equivalence relations on S are associated to subgroups W of Aut_{M_n}(S), and we thus obtain groupoids W \ S. We show that for two natural choices of equivalence relation, the corresponding groups W^c and W^+ act on F, and the groupoids W^c \ S and W^+ \ S on the model field K=Q(x_1,...,x_n). The groupoid W^+ \ S is equivalent to Fock-Goncharov's cluster modular groupoid. Moreover, W^c is isomorphic to the group of cluster automorphisms, and W^+ to the subgroup of direct cluster automorphisms, in the sense of Assem-Schiffler-Shramchenko. We also prove that, for mutation classes whose seeds have mutation finite quivers, the stabilizer of a labelled seed under M_n determines the quiver of the seed up to 'similarity', meaning up to taking opposites of some of the connected components. Consequently, the subgroup W^c is the entire automorphism group of S in these cases.