《粒子物理学家用非阿贝尔离散对称导论=aN lntroduction to Non Abelian Discrete Symmetries for Particle Physicists》求取 ⇩

1Introduction1

References3

2Basics of Finite Groups13

References20

3SN21

3.1S321

3.1.1 Conjugacy Classes21

3.1.2 Characters and Representations22

3.1.3 Tensor Products22

3.2S425

3.2.1 Conjugacy Classes27

3.2.2 Characters and Representations27

3.2.3 Tensor Products29

References30

4AN31

4.1 A431

4.2A534

4.2.1 Conjugacy Classes35

4.2.2 Characters and Representations35

4.2.3 Tensor Products37

References41

5T′43

5.1 Conjugacy Classes43

5.2 Characters and Representations44

5.3 Tensor Products47

6DN51

6.1DN with N Even51

6.1.1 Conjugacy Classes52

6.1.2 Characters and Representations52

6.1.3 Tensor Products54

6.2DN with N Odd56

6.2.1 Conjugacy Classes56

6.2.2 Characters and Representations56

6.2.3 Tensor Products57

6.3 D458

6.4 D559

7QN61

7.1QN with N=4n61

7.1.1 Conjugacy Classes62

7.1.2 Characters and Representations62

7.1.3 Tensor Products62

7.2QN with N=4n+264

7.2.1 Conjugacy Classes64

7.2.2 Characters and Representations64

7.2.3 Tensor Products65

7.3 Q466

7.4 Q667

8QD2N69

8.1Generic Aspects69

8.1.1 Conjugacy Classes70

8.1.2 Characters and Representations70

8.1.3 Tensor Products71

8.2 QD1672

9∑(2N2)75

9.1Generic Aspects75

9.1.1 Conjugacy Classes75

9.1.2 Characters and Representations76

9.1.3 Tensor Products77

9.2 ∑(18)78

9.3 ∑(32)80

9.4 ∑(50)84

10△(3N2)87

10.1△(3N2) with N/3≠Integer87

10.1.1 Conjugacy Classes88

10.1.2 Characters and Representations89

10.1.3 Tensor Products89

10.2△(3N2) with N/3 Integer91

10.2.1 Conjugacy Classes91

10.2.2 Characters and Representations92

10.2.3 Tensor Products93

10.3 △(27)94

References95

11TN97

11.1Generic Aspects97

11.1.1 Conjugacy Classes98

11.1.2 Characters and Representations99

11.1.3 Tensor Products99

11.2 T7100

11.3 T13102

11.4 T19104

References108

12∑(3N3)109

12.1Generic Aspects109

12.1.1 Conjugacy Classes110

12.1.2 Characters and Representations111

12.1.3 Tensor Products112

12.2 ∑(81)113

References121

13△(6N2)123

13.1△(6N2)with N/3≠Integer123

13.1.1 Conjugacy Classes123

13.1.2 Characters and Representations126

13.1.3 Tensor Products128

13.2△(6N2) with N/3 Integer131

13.2.1 Conjugacy Classes131

13.2.2 Characters and Representations133

13.2.3 Tensor Products134

13.3△(54)138

13.3.1 Conjugacy Classes138

13.3.2 Characters and Representations139

13.3.3 Tensor Products141

References145

14Subgroups and Decompositions of Multiplets147

14.1S3147

14.1.1 S3→Z3148

14.1.2 S3→Z2148

14.2S4149

14.2.1 S4→S3150

14.2.2 S4→A4151

14.2.3 S4→∑(8)151

14.3A4152

14.3.1 A4→Z3152

14.3.2 A4→Z2×Z2153

14.4A5153

14.4.1 A5→A4153

14.4.2 A5→D5153

14.4.3 A5→S3？D3154

14.5T′154

14.5.1 T′→Z6154

14.5.2 T′→Z4155

14.5.3 T′→Q4155

14.6General DN155

14.6.1 DN→Z2156

14.6.2 DN→ZN157

14.6.3 DN→DM157

14.7D4158

14.7.1 D4→Z4158

14.7.2 D4→Z2×Z2159

14.7.3 D4→Z2159

14.8General QN159

14.8.1 QN→Z4160

14.8.2 QN→ZN161

14.8.3 QN→QM161

14.9Q4162

14.9.1 Q4→Z4162

14.10QD2N162

14.10.1 QD2N→Z2163

14.10.2 QD2N→ZN163

14.10.3 QD2N→DN/2163

14.11General∑(2N2)164

14.11.1 ∑(2N2)→Z2N164

14.11.2 ∑(2N2)→ZN×ZN164

14.11.3 ∑(2N2)→DN165

14.11.4 ∑(2N2)→QN166

14.11.5 ∑(2N2)→∑(2M2)166

14.12 ∑(32)167

14.13General △(3N2)168

14.13.1 △(3N2)→Z3169

14.13.2 △(3N2)→ZN×ZN169

14.13.3 △(3N2)→TN170

14.13.4 △(3N2)→△(3M2)170

14.14△(27)172

14.14.1 △(27)→Z3172

14.14.2 △(27)→Z3×Z3172

14.15General TN173

14.15.1 TN→Z3173

14.15.2 TN→ZN173

14.16T7174

14.16.1 T7→Z3174

14.16.2 T7→Z7175

14.17General ∑(3N3)175

14.17.1 ∑(3N2)→ZN×ZN×ZN175

14.17.2 ∑(3N3)→△(3N2)175

14.17.3 ∑(3N3)→∑(3M3)176

14.18∑(81)176

14.18.1 ∑(81)→Z3×Z3×Z3177

14.18.2 ∑(81)→△(27)177

14.19General△(6N2)178

14.19.1 △(6N2)→∑(2N2)179

14.19.2 △(6N2)→△(3N2)180

14.19.3 △(6N2)→△(6M2)180

14.20△(54)181

14.20.1 △(54)→S3×Z3182

14.20.2 △(54)→∑(18)182

14.20.3 △(54)→△(27)183

15Anomalies185

15.1 Generic Aspects185

15.2Explicit Calculations189

15.2.1 53189

15.2.2 S4190

15.2.3 A4190

15.2.4 A5191

15.2.5 T′192

15.2.6 DN (N Even)193

15.2.7 DN (N Odd)194

15.2.8QN(N=4n)194

15.2.9 QN(N=4n+2)195

15.2.10 QD2N196

15.2.11 ∑(2N2)197

15.2.12 △(3N2)(N/3≠Integer)198

15.2.13 △(3N2)(N/3 Integer)199

15.2.14 TN200

15.2.15 ∑(3N3)201

15.2.16 △(6N2)(N/3≠Integer)202

15.2.17 △(6N2)(N/3 Integer)203

15.3 Comments on Anomalies203

References204

16Non-Abellan Discrete Symmetry in Quark/Lepton Flavor Models205

16.1 Neutrino Flavor Mixing and Neutrino Mass Matrix205

16.2A4 Flavor Symmetry207

16.2.1 Realizing Tri-Bimaximal Mixing of Flavors207

16.2.2 Breaking Tri-Bimaximal Mixing209

16.3 S4 Flavor Model211

16.4 Alternative Flavor Mixing219

16.5 Comments on Other Applications222

16.6 Comment on Origins of Flavor Symmetries223

References224

Appendix AUseful Theorems229

References235

Appendix BRepresentations of S4 in Different Bases237

B.1 Basis Ⅰ237

B.2 Basis Ⅱ238

B.3 Basis Ⅲ240

B.4 Basis Ⅳ242

References244

Appendix CRepresentations of A4 in Different Bases245

C.1 Basis Ⅰ245

C.2 Basis Ⅱ245

References246

Appendix DRepresentations of A5 in Different Bases247

D.1 Basis Ⅰ247

D.2 Basis Ⅱ253

References259

Appendix ERepresentations of T'in Different Bases261

E.1 Basis Ⅰ262

E.2 Basis Ⅱ263

References264

Appendix FOther Smaller Groups265

F.1 Z4？Z4265

F.2 Z8？Z2268

F.3 (Z2×Z4)？Z2(Ⅰ)270

F.4 (Z2×Z4)？Z2(Ⅱ)272

F.5 Z3？Z8275

F.6 (Z6×Z2)？Z2277

F.7 Z9？Z3281

References283

Index285