King, A and Pressland, M orcid.org/0000-0002-9631-3583 (2017) Labelled seeds and the mutation group. Mathematical Proceedings of the Cambridge Philosophical Society, 163 (2). pp. 193-217. ISSN 0305-0041
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 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 W\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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | "This article has been published in a revised form in Mathematical Proceedings of the Cambridge Philosophical Society (https://doi.org/10.1017/S0305004116000918). This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © Cambridge Philosophical Society 2016 |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 03 Apr 2020 11:19 |
Last Modified: | 03 Apr 2020 11:19 |
Status: | Published |
Publisher: | Cambridge University Press |
Identification Number: | 10.1017/s0305004116000918 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:159077 |