Non-Abelian Discrete Symmetries in Particle Physics

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Abstract
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 {itS}{in{itN}}, {itA}{in{itN}}, {itT}', {itD}{nt{itN}}, {itQ}{in{itN}}, {itΣ}(2{itN}{su2}), Δ(3{itN}{su2}), {itT}{in7}, Σ(3{itN}{su3}), and Δ(6{itN}{su2}), which have been applied for model building in particle physics. We also present typical flavor models by using {itA}{in4}, {itS}{in4}, 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 integral approach.

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