# Algebra Seminar by Isobel Webster

On Wednesday the 25th of September 2019, Isobel Webster (University of Leeds) will be visiting the Charlotte Scott Centre for Algebra and giving a seminar. Her talk will be at 3pm in INB3305, and the details are as follows:

Title: 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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