# Algebra seminar in Lincoln: talk by Isobel Webster

On the 25th of September 2019 **Isobel Webster (University of Leeds)** visited Charlotte Scott Centre for Algebra and gave a talk “A lattice isomorphism theorem for cluster groups of type A”.

*Abstract*: Each quiver appearing in a seed of a skew-symmetric 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 skew-symmetric cluster algebras of finite type, the associated cluster groups are isomorphic to finite reflection groups and thus are finite Coxeter groups. There are many well-established results for Coxeter presentations and it is natural to ask whether the cluster group presentations possess comparable properties.

I will define a cluster group associated to a cluster quiver and explain how the theory of cluster algebras forms the basis of research into cluster groups. As for Coxeter groups, we can consider parabolic subgroups of cluster groups. I will outline a proof which shows that, in the type A 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.

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