Dimension n Representations of the Braid Group on n Strings

In 1996, E. Formanek classified all the irreducible complex representations of Bn of dimension at most n − 1, where Bn is the Artin braid group on n strings. In this paper we extend this classification to the representations of dimension n, for n ≥ 9. We prove that all such representations are equivalent to the tensor product of a one-dimensional representation and a specialization of a certain one-parameter family of n-dimensional representations which was first discovered in 1996 by Tong, Yang, and Ma. In order to do this, we classify all the irreducible complex representations ρ of Bn for which rank(ρ(σi) − 1) = 2, where the σi are the standard generators.