On the injectivity of the braid group in the Hecke algebra (Q2777724)

Let \((W,S)\) be a Coxeter system with braid group \(B_W\) and Hecke algebra \(H_W=H_W(q)\). An interesting question in the theory of braid groups concerns the existence of faithful finite dimensional representations of \(B_W\). As the authors explain, the question of whether \(B_W\) has a faithful finite dimensional linear representation has been recently settled in the case where \(W\) is finite; if \(W\) is in addition crystallographic, the degree of such a representation may be taken to be equal to the number of positive roots. Another well-known representation of braid groups is the Burau representation, which is now known not to be faithful in type \(A_n\) for \(n\geq 4\).NEWLINENEWLINENEWLINEThe main representation of interest in this paper is the reduced Burau representation, which is the composition \(\psi\) of the natural surjection \(\varphi\colon B_W\twoheadrightarrow H_W\) with the reflection representation of \(H_W\). The main result is that \(\psi\), and therefore, \(\varphi\), is injective whenever \((W,S)\) is of rank 2. The authors speculate that this result also holds in type \(A_3\).