Colloquium talk at IMSc by Ben Elias on Thursday 24 Jan, 2014
1400--1500 hrs, room 123
Title: Categorical actions of Coxeter groups and braid groups

Abstract: We all know what it means for a group G to act on a vector
space V: one has an endomorphism of V for each element of G, with the
obvious compatibility relation. However, one rarely defines a group
action in this way! Instead, one simplifies the data required by
choosing a presentation of G by generators and relations, and only
defining an endomorphism for each generator, checking the relations.

There is also a notion of what it means for a group G to act on a
category C. However, given a presentation of a group, it is not at
all clear what compatibilities are required to define a categorical
action using this presentation. We discuss this problem, which is
too difficult to solve in general. Coxeter groups are groups with
a particular kind of presentation, generalizing the Weyl groups
which appear in representation theory; potential actions of Coxeter
groups and braid groups on categories abound in the literature, but
are rarely shown to satisfy compatibility. In joint work with Geordie
Williamson, we have found the correct compatibility relations for
Coxeter groups (and conjecturally for their braid groups). The
answer is actually a topological one, dealing with the topology of
the Coxeter complex.