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

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 $\widetilde{A_3}$, 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 $A$.