Non-Abelian Discrete Symmetries in Particle Physics

We review pedagogically non-Abelian discrete groups, which play an important role in particle physics. We show group-theoretical aspects for many concrete groups, such as representations and their tensor products. We explain how to derive, conjugacy classes, characters, representations, and tensor products for these groups (with a finite number). We discuss them explicitly for S-N, A(N), T', D-N, Q(N), E(2N(2)), Delta(3N(2)), T-7, Sigma(3N(3)), and Delta(6N(2)), which have been applied for model building in particle physics. We also present typical flavor models by using A(4), S-4, and Delta(54) groups. Breaking patterns of discrete groups and decompositions of multiplets are important for applications of the non-Abelian discrete symmetry. We discuss these breaking patterns of the non-Abelian discrete group, which are a powerful tool for model buildings. We also review briefly anomalies of non-Abelian discrete symmetries by using the path integral approach.