The Flavor Symmetry

Assuming that the lepton, quark and Higgs fields belong to the three-dimensional reducible representation of the permutation group $S_3$, we suggest a minimal $S_3$ invariant extension of the standard model. We find that in the leptonic sector the exact $S_3\times Z_2$ symmetry, which allows 6 real independent parameters, is consistent with experimental data and predicts the bi-maximal mixing of the left-handed neutrinos and that the third neutrino is the lightest neutrino. $Z_2$ is anomaly-free, but forbids CP-violations in the leptonic as well as in the hadronic sector. Therefore, we may identify the origin of the CP-violations with the breaking of the $Z_2$ symmetry, which may be understood in a more fundamental theory. With the exact $S_3$ only, there are 10 real independent parameters and one independent phase, which the Cabibbo-Kobayashi-Maskawa mixing matrix $V_{\rm CKM}$ depends on. A set of values of these parameters that are consistent with the experimental observations is given.