The loop Hecke algebra has dimension 1/2 * binom(2n,n) for z ≠ ±1, and is isomorphic to the endomorphism algebra of a tensor power for the negative half of quantum gl(1|1).
Some remarks about the faithfulness of the Burau representation of Artin--Tits groups
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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 $\tilde{A_3}$, 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 $A$.
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A basis and Schur-Weyl duality for the loop Hecke algebra
The loop Hecke algebra has dimension 1/2 * binom(2n,n) for z ≠ ±1, and is isomorphic to the endomorphism algebra of a tensor power for the negative half of quantum gl(1|1).