On the irreducibility of local representations of the Braid group Bₙ

Abstract We prove that any homogeneous local representation: 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: Bₙ GLₙ (C) φ: B n → G L n (C) of type 3 is equivalent to a complex specialization of the standard representation 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.