The Burau Representation is Unitary

Abstract

Abstract. A slight modification of the Burau representation of the braid group is shown to be unitary relative to an explicitly defined Hermitian form. This gives a partial answer to the problem of identifying the image of the Burau representation and provides a tool for attacking the question of whether or not the Burau representation is faithful.In [2, problem 14, p. An important open question is whether or not the Burau representation is faithful. This is known for n < 3 [4] (also see [2]). It seems likely that the results presented here will help answer this question for n > 4.In §1 we establish some conventions about matrices and in §2 we describe the Burau representation. In §3 we prove the theorem and justify the remarks following its statement above. Finally, in §4 we offer two conjectures which, taken together, would imply that the Burau representation is faithful.1. Preliminaries. Let Z denote the ring of (rational) integers, let t be an indeterminate and let Z[t, /"'] denote the ring of Laurent polynomials with coefficients in Z. We consider infinite-dimensional matrices A = (a¡ ) with entries in Z [f, /"']

On the Linear Representation of Braid Groups

“…We also show the existence of invariant symmetric 2-tensors, first observed in [15]. This comes from writing down a height function which is clearly invariant for braid group diffeomorphisms.…”

“…The latter was first observed in the case of the braid group in [15] by Squier. We also show that one of the conjectures of [15] is true.…”

“…In particular, in the first case one obtains a one-parameter family of symmetric matrices, which essentially describes the form of [15]; in the second case, one obtains an «-parameter family of symmetric matrices. The word "nondegenerate" here is interpreted as "generically nondegenerate".…”

Matrix Representations of Artin Groups

“…We note that in the special case of the braid groups our representation is a version of the Burau representation ( [5] or see [2]). The results below are first, a generalization to arbitrary Artin groups of the author's observation [10] that the Burau representation of Bn is unitary and second, a generalization to arbitrary rank 2 Artin groups of the well-known fact (see [9 or 2] ) that the Burau representation of B3 is faithful. Let M be an n x n Coxeter matrix.…”

Low-dimensional unitary representations of 𝐵₃

“…[1], [2], and [4]), and specializations of the reduced Burau and Gassner representations in [5]. Such representations easily lead to representations of PSL(2, Z) = B 3 /Z, where Z is the center of B 3 , and PSL(2, Z) = SL(2, Z)/{±1}, where {±1} is the center of SL(2, Z).…”