Discrete symmetries and mixing of Dirac neutrinos

We study mixing of the Dirac neutrinos in the residual symmetries approach. The key difference from the Majorana case is that the Dirac mass matrix may have larger symmetries: $G_\nu=\mathbf{Z}_{n}$ with $n\geq3$. The symmetry group relations have been generalized to the case of Dirac neutrinos. Using them we have found all new relations between mixing parameters and corresponding symmetry assignments which are in agreement with the present data. The viable relations exist only for the charged lepton residual symmetry $G_{\ell} = \mathbf{Z}_{2}$. The relations involve elements of the rows of the PMNS matrix and lead to precise predictions of the 2-3 mixing angle and certain ranges of the CP violation phase. For larger symmetries $G_{\ell}$, an agreement with data can be achieved if $\sim10\%$ corrections related to breaking of $G_{\ell}$ and $G_\nu$ are included.