Necessary and sufficient conditions for the irreducibility of a linear representation of the braid group Bₙ

Valerij G. Bardakov and P. Bellingeri introduced a new linear representation ρ ̄F ρ ¯ F of degree $$n+1$$ n + 1 of the braid group Bₙ B n . We study the irreducibility of this representation. We prove that ρ ̄F ρ ¯ F is reducible to the degree $$n-1$$ n - 1 . Moreover, we give necessary and sufficient conditions for the irreducibility of the complex specialization of its $$n-1$$ n - 1 degree composition factor φ ̄F ϕ ¯ F .