In this paper, we determine a sufficient condition for the irreducibility of the family of representations of the braid group constructed by C.M. Egea and E. Galina without requiring that the representations are self-adjoint. Then we construct a new subfamily of multi-parameter representations $(\psi_m,V_m), $ $1\le m< n$, of dimension $ V_m =\binom {n} {m} $. Finally, we study the irreducibility of $(\psi_m,V_m) $