On Krammer's Representation of the Braid Group

TL;DR

This paper proves that Krammer's representation of the braid group is equivalent to a specific algebraic representation, establishing its irreducibility and faithfulness through algebraic and dimension arguments.

Contribution

It demonstrates the equivalence between Krammer's representation and an algebraic representation derived from the Birman-Murakami-Wenzl algebra, confirming key properties.

Findings

Krammer's representation is identical to an algebraic representation after rescaling.

The representation is proven to be irreducible.

The representation is shown to be faithful.

Abstract

A connection is made between the Krammer representation and the Birman-Murakami-Wenzl algebra. Inspired by a dimension argument, a basis is found for a certain irrep of the algebra, and relations which generate the matrices are found. Following a rescaling and change of parameters, the matrices are found to be identical to those of the Krammer representation. The two representations are thus the same, proving the irreducibility of one and the faithfulness of the other.