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 A₃, and not faithful over several finite rings in type D₄, using an algorithmic approach based on categorical methods that generalize Bigelow's curve strategy outside of type A.