an INTRODUCTION TO NON-ABELIAN DISCRETE SYMMETRIES FOR PARTICLE PHYSICISTS

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Preface to the Second Edition Preface to the First Edition Contents 1 Introduction 2 Basics of Finite Groups 3 SN 3.1 S3 3.2 S4 4 AN 4.1 A4 4.2 A5 5 T' 6 DN 6.1 DN with N= Even 6.2 DN with N= Odd 6.3 D4 6.4 D5 7 QN 7.1 QN with N=4n 7.2 QN with N=4n+2 7.3 Q4 7.4 Q6 8 QD2N 8.1 Generic Aspects 8.2 QD16 9 Σ(2N2) 9.1 Generic Aspects 9.2 Σ(18) 9.3 Σ(32) 9.4 Σ(50) 10 Δ(3N2) 10.1 Δ(3N2) with N/3 neq Integer 10.2 Δ(3N2) with N/3 = Integer 10.3 Δ(27) 10.4 Δ(48) 10.4.1 Conjugacy Classes and Tensor Products 10.4.2 The Different Bases of Δ(48) 11 TN 11.1 Generic Aspects 11.2 T7 11.3 T13 Group Theory 11.4 T19 Group Theory 12 Σ(3N3) 12.1 Generic Aspects 12.2 Σ(81) 13 Δ(6N2) 13.1 Δ(6N2) with N/3= Integer 13.2 Δ(6N2) with N/3= Integer 13.3 Δ(54) 13.4 Δ(96) 13.4.1 Conjugacy Classes 13.4.2 Characters and Representations 13.4.3 Tensor Products 14 Subgroups and Decompositions of Multiplets 14.1 S3 14.2 S4 14.3 A4 14.4 A5 14.5 T' 14.6 General DN 14.7 D4 14.8 General QN 14.9 Q4 14.10 QD2N 14.11 General Σ(2N2) 14.12 Σ(32) 14.13 General Δ(3N2) 14.14 Δ(27) 14.15 General TN 14.16 T7 14.17 General Σ(3N3) 14.18 Σ(81) 14.19 General Δ(6N2) 14.20 Δ(54) 15 Finite Subgroups of Continuous Groups 15.1 Finite Subgroups of SO(3) 15.1.1 SO(3)toDN 15.1.2 SO(3)toA4 15.1.3 SO(3)toS4 15.2 Finite Subgroups of SU(2) 15.2.1 SU(2)toQN 15.2.2 SU(2)toT' 15.3 Finite Subgroups of SU(3) 15.3.1 SU(3)toA4 15.3.2 SU(3)toA5 15.3.3 SU(3)toS4 15.3.4 SU(3)toDN 15.3.5 SU(3)toT' 15.3.6 SU(3)toΔ(3N2) 15.3.7 SU(3)toΔ(6N2) 16 Modular Symmetry 17 Automorphism 17.1 Z3 17.2 Z2timesZ'2 17.3 A4 17.4 Σ(18) 17.5 Δ(27) 18 Anomalies 18.1 Generic Aspects 18.2 Explicit Calculations 18.3 Comments on Anomalies 19 Non-Abelian Discrete Symmetry in Quark/Lepton Flavor Models 19.1 Neutrino Flavor Mixing and Neutrino Mass Matrix 19.2 A4 Flavor Symmetry 19.2.1 Realizing the Tri-Bimaximal Mixing of Flavors 19.2.2 Breaking Tri-Bimaximal Mixing 19.3 S4 Flavor Model 19.4 Alternative Flavor Mixing 19.5 Modular A4 Invariance and Neutrino Mixing 19.6 Comments on Other Applications 19.7 Comment on Origins of Flavor Symmetries 20 Generalized CP Symmetry 20.1 CP Transformation in Flavor Space 20.2 CP Violating Phase and Its Group Theoretical Origin 20.3 Modular Symmetry with Generalized CP Symmetry 20.3.1 CP Transformation of the Modulus τ 20.3.2 CP Transformation of Modular Multiplets 20.4 CP Violation by Modulus τ in A4 Model of Leptons A Useful Theorems B Representations of S4 in Several Bases B.1 The Basis I B.2 The Basis II B.3 The Basis III B.4 The Basis IV C Representations of A4 in Different Basis D Representations of A5 in Different Basis D.1 The Basis I D.2 The Basis II E Representations of T' in Different Basis E.1 The Basis I E.2 The Basis II F Representations of Δ(96) in Different Basis G Other Smaller Groups G.1 Z 4 Z4 G.2 Z 8 Z2 G.3 (Z2timesZ4)Z2 (I) G.4 (Z2timesZ4)Z2 (II) G.5 Z3 Z8 G.6 (Z6timesZ2) Z2 G.7 Z9 Z3 H Generators of the Modular Group I Modular Forms