Monodromy Representations of Mixed Braid Groups

TL;DR

This paper explicitly describes unitary representations of mixed braid groups acting on the cohomology of Abelian branched covers of the complex projective line, revealing their structure and relation to known representations.

Contribution

It provides a new explicit description of unitary representations of mixed braid groups and connects them to complex reflections and the multivariate Burau representation.

Findings

The image of the representation is generated by complex reflections.

The representations are related to the multivariate Burau representation.

Explicit descriptions of these representations are provided.

Abstract

We explicitly describe unitary representations of mixed braid groups on the cohomology of Abelian branched covers of $\mathbf{CP}^1$ . We show that the image of the representation is generated by complex reflections and relate it to the multivariate Burau representation.