White Rose University Consortium logo
University of Leeds logo University of Sheffield logo York University logo

Cluster groups: groups with Presentations Arising from Cluster Algebras.

Webster, Isobel (2019) Cluster groups: groups with Presentations Arising from Cluster Algebras. PhD thesis, University of Leeds.

[img]
Preview
Text
ThesisFinal.pdf - Final eThesis - complete (pdf)
Available under License Creative Commons Attribution-Noncommercial-Share Alike 2.0 UK: England & Wales.

Download (1524Kb) | Preview

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.

Item Type: Thesis (PhD)
Keywords: Cluster groups and Coxeter groups
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds)
The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds)
Identification Number/EthosID: uk.bl.ethos.789469
Depositing User: Miss Isobel Webster
Date Deposited: 08 Nov 2019 16:21
Last Modified: 18 Feb 2020 12:51
URI: http://etheses.whiterose.ac.uk/id/eprint/25149

You do not need to contact us to get a copy of this thesis. Please use the 'Download' link(s) above to get a copy.
You can contact us about this thesis. If you need to make a general enquiry, please see the Contact us page.

Actions (repository staff only: login required)