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Cluster groups: groups with presentations arising from cluster algebras

Cluster groups: groups with presentations arising from cluster algebras
Cluster groups: groups with presentations arising from cluster algebras
Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, 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, thus are finite Coxeter groups. There are many well-established results for Coxeter presentations and we are interested in whether cluster group presentations possess comparable properties. As for finite Coxeter groups, we can consider parabolic subgroups of cluster groups. We prove that, in the type A case, an isomorphism exists between the lattice of subsets of the set of defining generators of the cluster group and the lattice of its parabolic subgroups. Moreover, we show each parabolic subgroup has a presentation given by restricting the presentation of the whole group. In addition, we provide a method for obtaining a positive companion basis of the quiver. For more general quivers, we prove an alternative exchange lemma for the associated cluster group by showing that each element has a factorisation with respect to a given parabolic subgroup. In the type A case, we also consider elements whose reduced expressions all begin with a certain fixed generator and examine the form of these reduced expressions. Finally, we provide an alternative proof to the known fact that the length function on a parabolic subgroup of a Coxeter group of type A agrees with the length function on the whole group and discuss an analogous conjecture for cluster groups.
University of Leeds
Webster, Isobel
8b54431c-6550-43ac-9806-7474d4b5350e
Webster, Isobel
8b54431c-6550-43ac-9806-7474d4b5350e
Marsh, Robert
aa7cc3c7-6437-4469-844f-e471095b335f

Webster, Isobel (2019) Cluster groups: groups with presentations arising from cluster algebras. University of Leeds, Doctoral Thesis, 135pp.

Record type: Thesis (Doctoral)

Abstract

Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, 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, thus are finite Coxeter groups. There are many well-established results for Coxeter presentations and we are interested in whether cluster group presentations possess comparable properties. As for finite Coxeter groups, we can consider parabolic subgroups of cluster groups. We prove that, in the type A case, an isomorphism exists between the lattice of subsets of the set of defining generators of the cluster group and the lattice of its parabolic subgroups. Moreover, we show each parabolic subgroup has a presentation given by restricting the presentation of the whole group. In addition, we provide a method for obtaining a positive companion basis of the quiver. For more general quivers, we prove an alternative exchange lemma for the associated cluster group by showing that each element has a factorisation with respect to a given parabolic subgroup. In the type A case, we also consider elements whose reduced expressions all begin with a certain fixed generator and examine the form of these reduced expressions. Finally, we provide an alternative proof to the known fact that the length function on a parabolic subgroup of a Coxeter group of type A agrees with the length function on the whole group and discuss an analogous conjecture for cluster groups.

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Webster Thesis Final - Version of Record
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Published date: October 2019

Identifiers

Local EPrints ID: 437729
URI: http://eprints.soton.ac.uk/id/eprint/437729
PURE UUID: e236b483-6207-466e-a611-2f3d0ee827ed
ORCID for Isobel Webster: ORCID iD orcid.org/0000-0001-8997-8280

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Date deposited: 13 Feb 2020 17:30
Last modified: 16 Mar 2024 06:19

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Contributors

Author: Isobel Webster ORCID iD
Thesis advisor: Robert Marsh

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