On the irreducibility of local representations of the Braid group \(B_n\)

We prove that any homogeneous local representation \(\varphi :B_n \rightarrow GL_n(\mathbb {C})\) of type 1 or 2 of dimension \(n\ge 6\) is reducible. Then, we prove that any representation \(\varphi :B_n \rightarrow GL_n(\mathbb {C})\) of type 3 is equivalent to a complex specialization of the standard representation \(\tau _n\). Also, we study the irreducibility of all local linear representations of the braid group \(B_3\) of degree 3. We prove that any local representation of type 1 of \(B_3\) is reducible to a Burau type representation and that any local representation of type 2 of \(B_3\) is equivalent to a complex specialization of the standard representation. Moreover, we construct a representation of \(B_3\) of degree 6 using the tensor product of local representations of type 2. Let \(u_i\), \(i=1,2\), be non-zero complex numbers on the unit circle. We determine a necessary and sufficient condition that guarantees the irreducibility of the obtained representation.