Neutrino mass, mixing and discrete symmetries

Abstract: Status of the discrete symmetry approach to explanation of the lepton masses and mixing is summarized in view of recent experimental results, in particular, establishing relatively large 1-3 mixing. The lepton mixing can originate from breaking of discrete flavor symmetry $G_f$ to different residual symmetries $G_{\ell}$ and $G_\nu$ in the charged lepton and neutrino sectors. In this framework the {\it symmetry group condition} has been derived which allows to get relations between the lepton mixing elements immediately without explicit model building. The condition has been applied to different residual neutrino symmetries $G_\nu$. For generic (mass independent) $G_\nu = {\bf Z}2$ the condition leads to two relations between the mixing parameters and fixes one column of the mixing matrix. In the case of $G\nu = {\bf Z}2 \times {\bf Z}_2$ the condition fixes the mixing matrix completely. The non-generic (mass spectrum dependent) $G\nu$ lead to relations which include mixing angles, neutrino masses and Majorana phases. The symmetries $G_{\ell}$, $G_\nu$, $G_f$ are identified which lead to the experimentally observed values of the mixing angles and allow to predict the CP phase.