Non-Abelian Discrete Symmetries in Particle Physics

We review pedagogically non-Abelian discrete groups, which play an important role in the particle physics. We show group-theoretical aspects for many concrete groups, such as representations, their tensor products. We explain how to derive, conjugacy classes, characters, representations, and tensor products for these groups (with a finite number). We discussed 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 the 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 about anomalies of non-Abelian discrete symmetries by using the path integral approach.

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