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, Σ(2N^2), Δ(3N^2), T_7, Σ(3N^3), and Δ(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 Δ (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 integr

al approach.