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Representations of braid groups

Stephen Bigelow
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Abstract

In this paper we survey some work on representations of $B_n$ 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 $n$. 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 $n$, 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.

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Stephen Bigelow. 2003-04-15. Representations of braid groups. https://arxiv.org/abs/math/0304212

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