A lattice isomorphism theorem for cluster groups of mutation-Dynkin type A
A lattice isomorphism theorem for cluster groups of mutation-Dynkin type A
Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of finite type, the associated cluster groups are isomorphic to finite reflection groups. As for finite Coxeter groups, we can consider parabolic subgroups of cluster groups. We prove that, in the type An case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups. Moreover, each parabolic subgroup has a presentation given by restricting the presentation of the whole group.
5409-5427
Webster, Isobel
8b54431c-6550-43ac-9806-7474d4b5350e
23 April 2019
Webster, Isobel
8b54431c-6550-43ac-9806-7474d4b5350e
Webster, Isobel
(2019)
A lattice isomorphism theorem for cluster groups of mutation-Dynkin type A.
Journal of Pure and Applied Algebra, 223 (12), .
(doi:10.1016/j.jpaa.2019.04.005).
Abstract
Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of finite type, the associated cluster groups are isomorphic to finite reflection groups. As for finite Coxeter groups, we can consider parabolic subgroups of cluster groups. We prove that, in the type An case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups. Moreover, each parabolic subgroup has a presentation given by restricting the presentation of the whole group.
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Accepted/In Press date: 1 April 2016
e-pub ahead of print date: 16 April 2019
Published date: 23 April 2019
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Local EPrints ID: 437747
URI: http://eprints.soton.ac.uk/id/eprint/437747
ISSN: 0022-4049
PURE UUID: 62ec0767-ae9d-4e15-a8c8-6fa0715a9f50
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Date deposited: 13 Feb 2020 17:32
Last modified: 16 Mar 2024 06:19
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Author:
Isobel Webster
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