Dimension $n$ Representations of the Braid Group on $n$ Strings

In 1996 E. Formanek classified all the irreducible complex representations of $B_n$ of dimension at most $n-1,$ where $B_n$ is the Artin braid group on $n$ strings. In this paper we extend this classification to the representations of dimension $n,$ for $n\geq 9$. We prove that all such representations are equivalent to a tensor product of a one-dimensional representation and a specialization of a certain one-parameter family of $n-$dimensional representations, that was first discovered in 1996 by Tong, Yang, and Ma. We use our classification of irreducible complex representations of corank two, in the preprint math.GR/0003047.