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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.
Bapat et al. (Fri,) studied this question.