9789812380654

Foreword to the first edition	p. v
Preface to the first edition	p. vii
Preface to the second edition	p. ix
Addendum	p. xi
Contents	p. xiii
Glossary	p. xxiii
Introduction	p. 1
Elements of Group Theory
The Definition of a Group	p. 4
The Permutation Group S[subscript n]	p. 7
Subgroups	p. 9
Isomorphism and Homomorphism	p. 10
Conjugate Classes	p. 12
Cosets and Lagrange's Theorem	p. 14
Invariant Subgroups	p. 16
Factor Groups	p. 17
Direct Product and Semi-Direct Product Groups	p. 17
Group Representation Theory
Linear Vector Spaces	p. 19
Linear Operators and their Representations	p. 22
Complete Sets of Commuting Operators	p. 24
Group Representations	p. 26
Unitary Representation	p. 29
Regular Representation and the Group Algebra	p. 31
The Space of Functions on the Group	p. 33
Equivalent Representations and Characters	p. 34
Reducible and Irreducible Representations	p. 35
Subduced and Induced Representations	p. 38
Schur's Lemma	p. 39
Appendix: Non-Orthogonal Bases	p. 40
Representation Theory for Finite Groups
The Class Space and Class Operators	p. 46
The First Kind of CSCO of G (CSCO-I)	p. 51
The Projection Operator P[superscript (v)]	p. 57
The Reduction of the Representations of C[subscript 3v], S[subscript 2] and S[subscript 3]	p. 62
The State Permutation Group	p. 67
Reduction of the Regular Rep of S[subscript 3]	p. 69
The Intrinsic Group	p. 71
CSCO-II and CSCO-III of G	p. 77
Full Reduction of the Regular Representation	p. 79
The Projection Operator P[superscript (v) subscript mk] and the Generalized Projection Operator P[superscript (v) subscript mk]	p. 89
The Eigenfunction Method for Characters	p. 94
Applications of Simple Characters	p. 95
Reduction of Non-Regular Reps (EFM for Irreducible Bases)	p. 96
Irreducible Basis Vectors in a Non-Orthogonal Reducible Basis	p. 101
Kronecker Product of Representations	p. 102
The Clebsch-Gordan (CG) Coefficients	p. 103
Isoscalar Factors	p. 106
Irreducible Tensors for a Group G	p. 107
Symmetries of the CG Coefficients and Isoscalar Factors	p. 110
Applications of Group Theory in Quantum Mechanics	p. 112
Summary	p. 116
Representation Theory of the Permutation Group
Partitions, Young Diagrams and Eigenvalues of CSCO-I	p. 119
Characters of Permutation Group	p. 120
Branching Laws, the Young-Yamanouchi Basis and Young Tableaux	p. 121
Yamanouchi Matrix Elements	p. 122
The CSCO-II of Permutation Groups	p. 127
The EFM for the Yamanouchi Basis (I)	p. 132
The CSCO-III of the Permutation Group	p. 138
The Quasi-Standard Basis of the Permutation Group	p. 142
The EFM for the Yamanouchi Basis (II)	p. 149
The Inner Product and the CG Series of Permutation Groups	p. 152
Calculation of the CG Coefficients of Permutation Groups	p. 153
Properties of the CG Coefficients of Permutation Groups	p. 158
Tables of the CG Coefficients for S[subscript 3]--S[subscript 5]	p. 159
Outer-Product of the Permutation Group and the Littlewood Rule	p. 165
The Calculation of the Induction Coefficients (IDC) of S[subscript n]	p. 168
Properties of the IDC	p. 170
Tables of the IDC for S[subscript 3]-S[subscript 5]	p. 173
The S[subscript n1+n2] [superset or implies] S[subscript n1] [multiply sign in circle] S[subscript n2] Irreducible Basis	p. 181
The S[subscript n] [superset or implies] S[subscript n1] [multiply sign in circle] S[subscript n2] Isoscalar Factors	p. 191
Appendix: Derivation of the Yamanouchi Matrix Elements by the EFM	p. 202
Lie Groups
Tensors	p. 205
Definition of a Lie Group; With Examples	p. 208
Lie Algebras	p. 210
Finite Transformations	p. 213
Correspondence between Lie Groups and Lie Algebras	p. 214
Linear Transformation Groups	p. 216
Infinitesimal Operators for Linear Transformation Groups	p. 218
The Metric Tensor in n-Dimensional Space and Infinitesimal Operators	p. 221
The Groups U[subscript 2j+1], SO[subscript 2l+1] and Sp[subscript 2j+1]	p. 228
Infinitesimal Operators in Group Parameter Space	p. 232
Isomorphism and Anti-Isomorphism of Lie Groups and Lie Algebras	p. 233
Invariant Integration	p. 235
Representations of Compact Lie Groups	p. 236
The Invariants and Casimir Operators of Lie Groups	p. 239
Intrinsic Lie Groups	p. 241
The CSCO Approach to the Rep Theory of Lie Groups	p. 243
Irreducible Tensors of Lie Groups and Intrinsic Lie Groups	p. 245
The Cartan-Weyl Basis	p. 247
Theorems on Roots	p. 251
Root Diagrams	p. 252
The Dynkin Diagram and the Simple Root Representation	p. 254
The Cartan Matrix	p. 255
Theorems on Weights	p. 256
The Dynkin Representation and the Chevalley Basis	p. 259
Algorithms for Computing the Roots and Weights	p. 266
The Fundamental Weight System	p. 272
The Fundamental Weight System Rep and the Cartesian Rep	p. 273
Comparing the Different Representations	p. 284
The Characters and CG Series of Lie Algebras	p. 286
The Rotation Group
The Differential Operators J[subscript x,y,x] and J[subscript x,y,z] in Group Parameter Space	p. 289
Irreps of the Group SO[subscript 2]	p. 292
The CSCO-I and Characters of SO[subscript 3]	p. 293
The CSCO-III and Irreducible Matrix Elements of SO[subscript 3]	p. 297
The CSCO-II and Irreducible Bases of SO[subscript 3]	p. 299
The Intrinsic State of SO[subscript 3]	p. 300
The Projection State of SO[subscript 3]	p. 301
Irreducible Tensors of SO[subscript 3] and SO[subscript 3]	p. 302
The Unitary Groups
Unitary Groups in Coordinate Space and State Space	p. 306
Relations between Unitary and Permutation Groups	p. 308
The CSCO-II and CSCO-III of U[subscript n] and SU[subscript n]	p. 312
The Gel'fand Basis and Gel'fand Matrix Elements	p. 315
The Gel'fand Basis of Unitary Groups and the Quasi-Standard Basis of Permutation Groups	p. 316
The Contragredient Representation	p. 326
The CG Coefficients of SU[subscript n] Group	p. 328
The CG Coefficients of SU[subscript n] and the S[subscript f] [superset or implies] S[subscript f1] [multiply sign in circle] S[subscript f2] Irreducible Basis	p. 333
The SU[subscript mn] [superset or implies] SU[subscript m] x SU[subscript n] Irreducible Basis	p. 334
The SU[subscript n1n2n3] [superset or implies] SU[subscript n1] x SU[subscript n2] x SU[subscript n3] Irreducible Bases and the Racah Coefficients of Permutation Groups	p. 339
The SU[subscript n1n2n3n4] [superset or implies] SU[subscript n1] x SU[subscript n2] x SU[subscript n3] x SU[subscript n4] Irreducible Basis and the 9v Coefficients of the Permutation Group	p. 341
The SU[subscript m+n] [superset or implies] SU[subscript m] [multiply sign in circle] SU[subscript n] Irreducible Basis	p. 342
The Isoscalar Factors and the Fractional Parentage Coefficients	p. 346
The S[subscript f] [superset or implies] S[subscript f1] [multiply sign in circle] S[subscript f2] [multiply sign in circle] S[subscript f3] Irreducible Basis SU[subscript n] Racah Coefficients	p. 358
The S[subscript f] [superset or implies] S[subscript f1] [multiply sign in circle] S[subscript f2] [multiply sign in circle] S[subscript f3] [multiply sign in circle] S[subscript f4] irreducible basis and the 9v coefficients of SU[subscript n]	p. 360
SU[subscript mn] [superset or implies] SU[subscript m] x SU[subscript n] CFP	p. 364
The SU[subscript m+n] [superset or implies] SU[subscript m] [multiply sign in circle] SU[subscript n] CFP	p. 372
The SU[subscript n] Singlet Factor	p. 379
Second Quantized Expressions for the CFP	p. 380
Generalized Quantized Expressions for the CFP	p. 385
The Point Groups
Basic Operations of Point Groups and Their Faithful Representations	p. 389
Some Commonly Used Point Groups	p. 394
Character Tables of Point Groups	p. 400
The CSCO-I and CSCO-II of Point Groups	p. 400
Algebraic Solutions for the Dihedral Groups D[subscript n]	p. 407
Numerical solutions for T [superset or implies] D[subscript 2] and O [superset or implies] D[subscript 4] [superset or implies] D[subscript 2]	p. 417
Algebraic Solutions for Cubic Groups	p. 420
The CG Coefficients of Point Groups	p. 429
Molecular Orbital Theory	p. 431
Single Electron SALC	p. 433
Double Point Groups for d-v Representations	p. 441
The Representation Group and Its Applications	p. 447
The Time Reversal Symmetry	p. 457
Applications of Group Theory to Many-Body Systems
Nuclear Shell Model: Single-Shell	p. 464
Nuclear Shell Model: Multi-Shell	p. 468
Anti-Symmetric Wave Functions for an A+B System	p. 470
Transformations between Symmetry Bases and Physical Bases in the Quark Model	p. 473
The Dynamical Symmetry Models of Nuclei	p. 474
The Quasispin Model	p. 475
The Proton-Neutron Quasispin Model	p. 477
The Groups Sp[subscript N], SO[subscript N] and the Pairing Interaction	p. 480
The Elliott Model	p. 483
The Interacting Boson Model	p. 487
The Molecular Shell Model	p. 492
The Space Groups
The Euclidean Group	p. 495
The Lattice Group	p. 497
The Space Group	p. 498
The Point Group P and the Crystal System	p. 499
The Bravais Lattice	p. 500
Operators of the Space Group	p. 502
The Reciprocal Lattice Vectors	p. 504
Irreps of the Lattice Group	p. 505
The Brillouin Zone	p. 507
The Electron State in a Periodic Potential	p. 508
Representation Space of the Space Group	p. 508
The Little Group G(k)	p. 509
The Representation Groups G[subscript k] and G'[subscript k]	p. 510
The Irreducible Basis and Matrices of G'[subscript k]	p. 514
Example: the Point W of O[superscript 7 subscript h]	p. 517
Irreducible Basis and Representations of the Space Group	p. 520
The Irreducible Basis and Matrices of C[superscript 4 subscript 2v]	p. 527
The Clebsch-Gordan Coefficients of Space Groups	p. 530
Examples: Getting Space Group Clebsch-Gordan Coefficients	p. 536
The Double Space Groups	p. 542
Appendix	p. 546
Appendix
Dimensions of irreps of the permutation group S[subscript f](f [less than or equal] 6) and the unitary groups SU[subscript n](n [less than or equal] 6)	p. 549
Phase factors [epsilon subscript 1](v[subscript 1]v[subscript 2]v) for the permutation group IDC and SU[subscript n] CG coefficients	p. 549
References	p. 551
Index	p. 567
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Group Representation Theory for Physicists

9812380655

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