Labelled seeds and the mutation group

King, A. and Pressland, M. (2017) Labelled seeds and the mutation group. Mathematical Proceedings of the Cambridge Philosophical Society, 163(2), pp. 193-217. (doi: 10.1017/s0305004116000918)

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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 Mn 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 Wc and W+ act on F, and the groupoids Wc \S and W+\S act on the model field K = Q(x1,..., xn). 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 Mn determines the quiver of the seed up to ‘similarity’, meaning up to taking opposites of some of the connected components. Consequently, the subgroup Wc is the entire automorphism group of S in these cases.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Pressland, Dr Matthew
Authors: King, A., and Pressland, M.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Mathematical Proceedings of the Cambridge Philosophical Society
Publisher:Cambridge University Press
ISSN (Online):1469-8064
Published Online:11 October 2016

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