Hall 123

Categorical actions of Coxeter groups and braid groups

Ben Elias

Massachussets Institute of Technology

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.