Lepton mixing and discrete symmetries

The pattern of lepton mixing can emerge from breaking a flavor symmetry in different ways in the neutrino and charged lepton Yukawa sectors. In this framework, we derive the model-independent conditions imposed on the mixing matrix by the structure of discrete groups of the von Dyck type which include A₄, S₄, and A₅. We show that, in general, these conditions lead to at least two equations for the mixing parameters (angles and $CP$ phase δ). These constraints, which correspond to unbroken residual symmetries, are consistent with nonzero 13 mixing and deviations from maximal 2--3 mixing. For the simplest case, which leads to an S₄ model and reproduces the allowed values of the mixing angles, we predict δ=(90^∘--120^∘).