Representations of braid groups
Abstract
In this paper we survey some work on representations of given by the induced action on a homology module of some space. One of these, called the Lawrence-Krammer representation, recently came to prominence when it was shown to be faithful for all . We will outline the methods used, applying them to a closely related representation for which the proof is slightly easier. The main tool is the Blanchfield pairing, a sesquilinear pairing between elements of relative homology. We discuss two other applications of the Blanchfield pairing, namely a proof that the Burau representation is not faithful for large , and a homological definition of the Jones polynomial. Finally, we discuss possible applications to the representation theory of the Hecke algebra, and ultimately of the symmetric group over fields of non-zero characteristic.
Cite
@article{arxiv.math/0304212,
title = {Representations of braid groups},
author = {Stephen Bigelow},
journal= {arXiv preprint arXiv:math/0304212},
year = {2007}
}