Some remarks about the faithfulness of the Burau representation of Artin--Tits groups

TL;DR

This paper investigates the faithfulness of the Burau representation for Artin--Tits groups, demonstrating non-faithfulness in specific affine and finite types using categorical and algorithmic methods.

Contribution

It extends the faithfulness analysis of the Burau representation from braid groups to broader Artin--Tits groups, providing new non-faithfulness results.

Findings

Burau representation is not faithful in affine type A_3.

Non-faithfulness over several finite rings in type D_4.

Uses categorical methods to generalize curve strategies outside type A.

Abstract

We discuss the extension of the faithfulness question for the Burau representation of braid groups to the case of Artin--Tits groups. We prove that the Burau representation is not faithful in affine type $\tilde{A_3}$, and not faithful over several finite rings in type $D_4$, using an algorithmic approach based on categorical methods that generalize Bigelow's curve strategy outside of type $A$.