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.
Charlotte Scott Centre for Algebra 
Reblogged this on Maths & Physics News.
LikeLike