On the irreducibility of local representations of the Braid group $$B_n$$

Abstract

AbstractWe prove that any homogeneous local representation $$\varphi :B_n \rightarrow GL_n(\mathbb {C})$$ φ : B n → G L n ( C ) of type 1 or 2 of dimension $$n\ge 6$$ n ≥ 6 is reducible. Then, we prove that any representation $$\varphi :B_n \rightarrow GL_n(\mathbb {C})$$ φ : B n → G L n ( C ) of type 3 is equivalent to a complex specialization of the standard representation $$\tau _n$$ τ n . Also, we study the irreducibility of all local linear representations of the braid group $$B_3$$ B 3 of degree 3. We prove that any local representation of type 1 of $$B_3$$ B 3 is reducible to a Burau type representation and that any local representation of type 2 of $$B_3$$ B 3 is equivalent to a complex specialization of the standard representation. Moreover, we construct a representation of $$B_3$$ B 3 of degree 6 using the tensor product of local representations of type 2. Let $$u_i$$ 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.