On the irreducibility of local representations of the braid group \(B_n\) (Q6611201)

In this paper, the authors study the irreducibility of some local homogeneous multi-parameter representations of the braid group. In particular, they show that for \(n\) greater than \(6\), any homogeneous local representation of type \(1\) or \(2\) is irreducible. Then, the authors prove that any homogeneous local representation of type \(3\) is equivalent to a complex specialization of the standard representation. Also, they prove that any three dimensional local representation of type \(1\) is reducible to a representation of Burau type and that any three-dimensional local representation of type \(2\) is equivalent to a complex specialization of the standard representation. The paper finishes by concluding that the obtained nine-dimensional multi-parameter representation is a direct sum of a complex specialization of the standard representation and a six-dimensional representation.