An irreducibility criterion for group representations with arithmetic applications

We prove a criterion for the irreducibility of an integral group representation \rho over the fraction field of a Noetherian domain R in terms of suitably defined reductions of \rho at prime ideals of R. As applications, we give irreducibility results for universal deformations of residual representations, with special attention to universal deformations of residual Galois representations associated with modular forms of weight at least 2