Constructing linear representations for families of groups
Arthur Soulié (University of Cambridge)
Monday 18th November 16:00-17:00 Maths 311B
Many families of groups, such as braid groups, have a representation theory of wild type, in the sense that there is no known classification schema. Hence it is useful to shape constructions of linear representations for such family of groups to understand its representation theory.
In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of Bn with a representation of Bn+1. This construction complexifies in a sense the initial representation: for instance, starting from a dimension one representation, one obtains the unreduced Burau representation. I will present this construction and its generalizations from a functorial point of view and explain that each construction defines an endofunctor, called a Long-Moody functor. I will give generalizations of this construction to other families of groups such as automorphism groups of free groups, mapping class groups of orientable and non-orientable surfaces or loop braid groups. After defining notions of polynomial functors in this context, I will prove that the Long-Moody functors increase by one the degree of polynomiality. This tool could be used to classify families of linear representations of these families of groups.
Then, I will also present a unified functorial construction of homological representations of families of groups, which is a joint work in progress with Martin Palmer. For instance, this construction provides the family of Lawrence-Bigelow representations for braid groups. Under some additional assumptions, general notions of polynomiality on functors are a useful tool to classify these representations.