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Non-Abelian Discrete Symmetries in Particle Physics

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arxiv 1003.3552 v2 pith:LKU3DIME submitted 2010-03-18 hep-th hep-ph

Non-Abelian Discrete Symmetries in Particle Physics

classification hep-th hep-ph
keywords discretegroupsnon-abeliandeltaparticlephysicsbreakingimportant
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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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 $S_N$, $A_N$, $T'$, $D_N$, $Q_N$, $\Sigma(2N^2)$, $\Delta(3N^2)$, $T_7$, $\Sigma(3N^3)$ and $\Delta(6N^2)$, which have been applied for model building in the particle physics. We also present typical flavor models by using $A_4$, $S_4$, and $\Delta (54)$ 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 about anomalies of non-Abelian discrete symmetries by using the path integral approach.

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