# Pure Maths Colloquium: Isobel Webster

This talk is part of the Pure Maths Colloquium at the University of St Andrews. Check out our upcoming talks at https://theran.lt/pure-colloquium/.

Where: | Zoom 885 2324 8736 |

When: | Nov 05 2020 @ 16.00 |

Video: | Link (internal) |

Speaker: | Isobel Webster University of St Andrews |

Title: | A Lattice Isomorphism Theorem for Cluster Groups of Type A |

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.