Valerij G. Bardakov and P. Bellingeri introduced a new linear representation ρ_F of degree n+1 of the braid group B_n. We study the irreducibility of this representation. We prove that ρ_F is reducible to the degree n−1. Moreover, we give necessary and sufficient conditions for the irreducibility of the complex specialization of its n−1 degree composition factor φ_F .