On the irreducibility of local representations of the Braid group Bₙ

We prove that any homogeneous local representation φ :Bₙ → GLₙ(C) φ : B n → G L n ( C ) of type 1 or 2 of dimension n≥ 6 n ≥ 6 is reducible. Then, we prove that any representation φ :Bₙ → GLₙ(C) φ : B n → G L n ( C ) of type 3 is equivalent to a complex specialization of the standard representation τ ₙ τ n . Also, we study the irreducibility of all local linear representations of the braid group B₃ B 3 of degree 3. We prove that any local representation of type 1 of B₃ B 3 is reducible to a Burau type representation and that any local representation of type 2 of B₃ B 3 is equivalent to a complex specialization of the standard representation. Moreover, we construct a representation of B₃ B 3 of degree 6 using the tensor product of local representations of type 2. Let uᵢ u i , $$i=1,2$$ 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.