LMS Society Meeting: Mary Cartwright Lecture 2024

Opening of the meeting and Society Business

Sara Lombardo (LMS Committee for Women and Diversity in Mathematics Chair) will give some background on Mary Cartwright and introduce the first speaker.

Bethany Marsh (University of Leeds)

Presentations of reflection groups (with Francesca Fedele)
 
A reflection group is a group generated by reflections of Euclidean space. In 1934, Coxeter classified the finite reflection groups and showed that they have beautiful presentations, known as Coxeter presentations. More recently, cluster algebras, introduced by Fomin-Zelevinsky in 2001, have been used by several authors to construct new families of presentations of reflection groups and braid groups.
 
There is a natural generalisation of the notion of reflection to a finite-dimensional complex vector space, used to define the complex reflection groups. The irreducible finite complex reflection groups were classified by Shephard-Todd in 1954: there is an infinite family G(m,p,n), where m, p and n are positive integers and p divides m, together with 34 exceptional cases.
 
In these two talks, we will present new families of presentations of the complex reflection group G(d,d,n) and its corresponding braid group. We show that the braid group associated to the complex reflection group G(d,d,n) is an index d subgroup of the braid group of the orbifold quotient O(d) of a disk by a cyclic group of order d. We also associate a presentation of G(d,d,n) and its braid group to each (tagged) triangulation of O(d) in such a way that the presentation given by Broué-Malle-Rouquier corresponds to a special tagged triangulation.