## Abstract

We study the set S of labelled seeds of a cluster algebra of rank n inside a field 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_{Mn} (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 , and the groupoids W^{c}\S and W^{+}\S act on the model field K = ℚ(x_{1}, ⋯, x_{n}). The groupoid W^{+}\S is equivalent to Fock-Goncharov's cluster modular groupoid. Moreover, Wc 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 stabiliser 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.

Original language | English |
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Pages (from-to) | 193-217 |

Number of pages | 25 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 163 |

Issue number | 2 |

Early online date | 11 Oct 2016 |

DOIs | |

Publication status | Published - 30 Sep 2017 |

## ASJC Scopus subject areas

- Mathematics(all)