Group Representation Theory for Physicists

Title page
Foreword to the first edition
Preface to the first edition
Preface to the second edition
Addendum
Contents
Glossary
Some tables
Glossary
Introduction
Chapter 1 Elements of Group Theory
1.1. The Definition of a Group
1.2. The Permutation Group S_n
1.2.1. The definition of S_n
1.2.2. Permutations expressed in terms of cycles and transpositions
1.3. Subgroups
1.4. Isomorphism and Homomorphism
1.5. Conjugate Classes
1.6. Cosets and Lagrange's Theorem
1.6.1. Left and right cosets
1.6.2. Lagrange's Theorem
1.6.3. Double cosets
1.7. Invariant Subgroups
1.8. Factor Groups*
1.9. Direct Product and Semi-Direct Product Groups
Chapter 2 Group Representation Theory
2.1. Linear Vector Spaces
2.1.1. Defining linear vector spaces
2.1.2. Covariant and contravariant
2.1.3. The metric tensor
2.2. Linear Operators and their Representations
2.3. Complete Sets of Commuting Operators
2.3.1. The eigenspace of self-adjoint operators
2.3.2. A complete set of commuting operators (CSCO)
2.4. Group Representations
2.5. Unitary Representation
2.6. Regular Representation and the Group Algebra
2.6.1. Definition of the regular representation
2.6.2. The group space
2.6.3. The group algebra
2.7. The Space of Functions on the Group
2.8. Equivalent Representations and Characters
2.9. Reducible and Irreducible Representations
2.10. Subduced and Induced Representations
2.11. Schur's Lemma
2.12. Appendix: Non-Orthogonal Bases
2.12.1. Two definitions of the representation of an operator
2.12.2. Representations of adjoint operators
2.12.3. Representations of unitary operators
2.12.4. Representations under basis transformations
2.12.5. Eigenvectors of a self-adjoint operator
Chapter 3 Representation Theory for Finite Groups
3.1. The Class Space and Class Operators
3.1.1. Class Operators
3.1.2. Class Algebra
3.1.3. The Space of Functions on Classes
3.1.4. The Natural Representation of a Class Algebra
3.2. The First Kind of CSCO of G (CSCO-I)
3.2.1. Reduction of the natural representation of the class algebra
3.2.2. The CSCO-I of G
3.2.3. The CSCO of a direct product group G₁ x G₂
3.2.4. The case of non-self-adjoint class operators
3.2.5. The groups S₃ and C_{6v}
3.3. The Projection Operator pC^{(ν)}
3.3.1. Decomposition of the regular rep into inequivalent reps of G
3.3.2. Labels for irreps
3.3.3. Decomposition of an arbitrary rep space
3.4. The Reduction of the Representations of C_{3v} , S₂ and S₃
3.4.1. The group C_{3v}
3.4.2. The group S₂
3.4.3. The group S₃ in the configuration α²β
3.4.4. The group S₃ in the configuration αβγ
3.5. The State Permutation Group
3.6. Reduction of the Regular Rep of S₃
3.7. The Intrinsic Group
3.7.1. Definition of the intrinsic group
3.7.2. The intrinsic state (regular rep case)
3.7.3. The regular representation of intrinsic group
3.7.4. Action of intrinsic group elements on functions on the group
3.7.5. Properties of the intrinsic group
3.7.6. Some remarks
3.7.7. Intrinsic permutation group and state permutation group
3.8. CSCO-II and CSCO-III of G
3.9. Full Reduction of the Regular Representation
3.9.1. Eigenvectors of the CSCO-III of G
3.9.2. The representations D^{(ν)k}(G) and D^{(ν)m}(G)
3.9.3. The standard phase choice for U_{νmk}
3.9.4. The irreducibility of D^{(ν)}(G)
3.9.5. The EFM for G ⊃ G(s) irreducible matrices
3.9.6. Reduction of the regular representation in configuration space
3.9.7. Example: the group S₃
3.10. The Projection Operator P^{(ν)}_{mk} and the Generalized Projection Operator P^{(ν)}_{mk}
3.10.1. Properties of the P^{(ν)}_{mk}
3.10.2. A recursive method for obtaining the P^{(ν)}_{mk}
3.10.3. Generalized irreducible matrices and generalized projection operator
3.10.4. Coset factored projection operator
3.11. The Eigenfunction Method for Characters
3.12. Applications of Simple Characters
3.13. Reduction of Non-Regular Reps (EFM for Irreducible Bases)
3.13.1. Multiplicity free case (τ_ν = 1)
3.13.2. Canonical subgroup chains with τ_ν > 1
3.13.3. Non-canonical subgroup chains
3.13.4. The projection operator method
3.14. Irreducible Basis Vectors in a Non-Orthogonal Reducible Basis
3.15. Kronecker Product of Representations
3.15.1. Clebsch-Gordan series
3.15.2. Symmetrized and anti-symmetrized squares
3.16. The Clebsch-Gordan (CG) Coefficients
3.16.1. Definition and properties of the CG coefficients
3.16.2. The EFM for CG coefficients
3.17. Isoscalar Factors
3.18. Irreducible Tensors for a Group G
3.18.1. The definition of an irreducible tensor
3.18.2. Two kinds of invariants
3.18.3. The Wigner-Eckart theorem
3.19. Symmetries of the CG Coefficients and Isoscalar Factors
3.20. Applications of Group Theory in Quantum Mechanics
3.20.1. When G is the symmetry group of the Hamiltonian
3.20.2. Splitting of the energy level due to a perturbation
3.20.3. Dynamical symmetry
3.20.4. The general case
3.20.5. Selection rules
3.21. Summary
Chapter 4 Representation Theory of the Permutation Group
4.1. Partitions, Young Diagrams and Eigenvalues of CSCO-I
4.2. Characters of Permutation Group
4.3. Branching Laws, the Young-Yamanouchi Basis and Young Tableaux
4.4. Yamanouchi Matrix Elements
4.5. The CSCO-II of Permutation Groups
4.6. The EFM for the Yamanouchi Basis (1)
4.7. The CSCO-III of the Permutation Group
4.7.1. CSCO-III
4.7.2. The labeling for Yamanouchi basis of S_n and S_n
4.7.3. Phase convention and the principal term
4.7.4. The matrix elements of conjugate irreps
4.7.5. The symmetrizer and anti-symmetrizer
4.8 The Quasi-Standard Basis of the Permutation Group
4.8.1. The state permutation group (for the case with repeated state labels)
4.8.2. The quasi-standard basis of the permutation group
4.8.3. Projection operator and quasi-standard basis
4.8.4. The labelling of the quasi-standard basis
4.9 The EFM for the Yamanouchi Basis (II)
4.10. The Inner Product and the CG Series of Permutation Groups
4.11. Calculation of the CG Coefficients of Permutation Groups
4.12. Properties of the CG Coefficients of Permutation Groups
4.13. Tables of the CG Coefficients for S₃-S₅
4.14. Outer-Product of the Permutation Group and the Littlewood Rule
4.15. The Calculation of the Induction Coefficients (IDC) of S_n
4.16. Properties of the IDC
4.17. Tables of the IDC for S₃-S₅
4.18. The S_{n_l+n₂} ⊃ S_{n_l} ⊗ S_{n₂} Irreducible Basis
4.18.1 The Frobenious reciprocity theorem and the S_{n_l+n₂} ⊃ S_{n_l} ⊗ S_{n₂} subduced basis
4.18.2. Transformations between the standard basis and the non-standard bases of S_n
4.18.3. The calculation of the subduction coefficients (SDC)
4.18.4. Tables of the SDC for S₃-S₆
4.19. The S_n ⊃ S_{n_l} ⊗ S_{n₂} Isoscalar Factors*
4.19.1. The S_n ⊃ S_{n-1} ISF
4.19.2. Phase convention
4.19.3. The properties of the S_n ⊃ S_{n-1} ISF
4.19.4. A special case
4.19.5. Tables of the S_n ⊃ S_{n-1} ISF
4.19.6. The S_{n_l+n₂} ⊃ S_{n_l} ⊗ S_{n₂} ISF*
4.20. Appendix: Derivation of the Yamanouchi Matrix Elements by the EFM
Chapter 5 Lie Groups
5.1. Tensors
5.1.1. Vectors (rank one tensors)
5.1.2. Tensors with rank higher than one
5.1.3. The metric tensor
5.1.4. Metric spaces
5.2. Definition of a Lie Group; With Examples
5.3. Lie Algebras
5.4. Finite Transformations
5.5. Correspondence between Lie Groups and Lie Algebras
5.6. Linear Transformation Groups
5.7. Infinitesimal Operators for Linear Transformation Groups
5.8. The Metric Tensor in n-Dimensional Space and Infinitesimal Operators
5.8.1. Unitary groups
5.8.2. Infinitesimal operators of SU_n
5.8.3. The group U(n, m)
5.8.4. The orthogonal group O_n
5.8.5. The real orthogonal group O(n, m)
5.8.6. Symplectic groups
5.9. The Groups U_{2j+1}, S0_{2l+1} and SP_{2j+1}
5.10. Infinitesimal Operators in Group Parameter Space
5.11. Isomorphism and Anti-Isomorphism of Lie Groups and Lie Algebras
5.12. Invariant Integration
5.13. Representations of Compact Lie Groups
5.13.1. The fundamental representation
5.13.2. Adjoint representations
5.13.3. The metric tensor in the r-dimensional vector space
5.14. The Invariants and Casimir Operators of Lie Groups
5.15. lntrinsic Lie Groups
5.15.1. Definition and interpretation of the intrinsic Lie group
5.15.2. Infinitesimal operators of intrinsic groups in group parameter space
5.16. The CSCO Approach to the Rep Theory of Lie Groups
5.17. Irreducible Tensors of Lie Groups and Intrinsic Lie Groups
5.18. The Cartan-Weyl Basis
5.19. Theorems on Roots
5.20. Root Diagrams
5.21. The Dynkin Diagram and the Simple Root Representation
5.22. The Cartan Matrix
5.23. Theorems on Weights
5.24. The Dynkin Representation and the Chevalley Basis
5.24.1. The Dynkin representation
5.24.2. The eigenvalues of the Casimir operators
5.24.3. The Chevalley basis
5.25. Algorithms for Computing the Roots and Weights
5.26. The Fundamental Weight System
5.27. The Fundamental Weight System Rep and the Cartesian Rep
5.28. Comparing the Different Representations
5.29. The Characters and CG Series of Lie Algebras
5.29.1. The Characters of Lie Groups
5.29.2. The CG Series of Lie Groups
Chapter 6 The Rotation Group
6.1. The DifferentiaI Operators J_{x,y,z} and J_{x,y,z} in Group Parameter Space
6.2. Irreps of the Group S0₂
6.3. The CSCO-I and Characters of SO₃
6.4. The CSCO-III and Irreduclble Matrix Elements of SO₃
6.5. The CSCO-II and Irreducible Bases of SO₃
6.6. The Intrinsic State of SO₃
6.7. The Projection State of SO₃
6.8. Irreducible Tensors of SO₃ and SO₃
6.8.1. The irreducible tensor TP) of the adjoint rep of SO₃ and SO₃
6.8.2. Irreducible tensors T) of SO₃ and SO₃ in general cases
Chapter 7 The Unitary Groups
7.1. Unitary Groups in Coordinate Space and State Space
7.2. Relations between Unitary and Permutation Groups
7.2.1. The Gel'fand invariants
7.2.2. The relation between CSCO-I's of permutation and unitary groups
7.2.3. Relations between the generators of unitary and permutation groups
7.3. The CSCO-II and CSCO-III of U_n and SU_n
7.4. The Gel'fand Basis and Gel'fand Matrix Elements
7.5. The Gel'fand Basis of Unitary Groups and the Quasi-Standard Basis of Permutation Groups
7.5.1. The CSCO-II of unitary groups and CSCO of the broken chains of permutation groups
7.5.2. The labeling and finding of the Gel'fand basis
7.6. The Contragredient Representation
7.7. The CG Coefficients of SU_n Group
7.7.1. The CG coefficients of U_n and the IDC of the permutation group
7.7.2. The procedure for evaluating the SU_n CG coefficients
7.7.3. Phase conventions
7.8. The CG Coefficients of SU_n and the S_f ⊃ S_{f₁} ⊗ S_{f₂} Irreducible Basis
7.9. The SU_{mn} ⊃ SU_m X SU_n Irreducible Basis
7.9.1. The CG coefficients of S_f and the SU_{mn} ⊃ SU_m X SU_n irreducible basis
7.9.2. The irreps ([ν₁],[ν₂]) of the groups SU_m and SU_n contained in the irrep [ν] of SU_{mn}
7.9.3. Representation transformation between the SU_{mn} ⊃ SU_m X SU_n irreducible basis and the SU_{mn} Gel'fand basis
7.10. The SU_{n_ln₂n₃} ⊃ SU_{n_l} X SU_{n₂} X SU_{n₃} Irreducible Bases and the Racah Coefficients of Permutation Groups*
7.11. The SU_{n_ln₂n₃n₄} ⊃ SU_{n_l} X SU_{n₂} X SU_{n₃} X SU_{n₄} Irreducible Basis and the 9ν Coefficients of the Permutation Group*
7.12. The SU_{mn} ⊃ SU_m ⊗ SU_n Irreducible Basis
7.12.1. The IDC of permutation groups and the SU_{mn} ⊃ SU_m ⊗ SU_n irreducible basis
7.12.2. The content of irreps ([ν₁, [ν₂]) of SU_m ⊗ SU_n in the irrep of SU_{m + n}
7.12.3. The representation transformation between the irreducible basis SU_{mn} ⊃ SU_m ⊗ SU_n and the Gel'fand basis of SU_{m+n}
7.13. The Isoscalar Factors and the Fractional Parentage Coefficients
7.13.1. Isoscalar factors
7.13.2. The orbital fractional parentage coefficients (CFP)
7.13.3. The spin-isospin CFP
7.13.4. The total CFP
7.13.5. The CFP for j-j coupling
7.13.6. The eigenfunction method for evaluating the CFP
7.14. The S_f ⊃ S{f₁} ⊗Sh S_{f₂} ⊗ S_{f₃} Irreducible Basis and SUn Racah Coefficients*
7.15. The S_f ⊃ S{f₁} ⊗Sh S_{f₂} ⊗ S_{f₃} ⊗ S_{f₄}irreducible basis and the 9ν coefficients of SU_n *
7.15.1. The 9ν coefficients of SU_n
7.15.2. Evaluation of the Racah coefficients and 9ν coefficients of SU_n
7.16. SU_{mn} ⊃ SU_m X SU_n CFP*
7.16.1. SU_{mn} ⊃ SU_m X SU_n CFP and S_{f₁+f₂} ⊃ S_{f₁} ⊗ S_{f₂} ISF
7.16.2. The evaluation of the SU_{mn} ⊃ SU_m X SU_n any-particle CFP
7.16.3. Symmetries of the SU_{mn} ⊃ SU_m X SU_n ISF
7.16.4. More examples
7.16.5. SU_{4(2l+1)} ⊃ (SU_{2l+1} ⊃ SO₃) X (SU₄ ⊃ SU₂ X SU₂) ISF and total CFP
7.17. The SU_{mn} ⊃ SU_m ⊗ SU_n CFP*
7.17.1. The S_f ⊃ S_{f-1} outer-product ISF (The S_f ⊃ S_{f-1} ⊗ U₁ ISF)
7.17.2. The S_f ⊃ S_{f_{12}} ⊗ S_{f_{34}} outer-product ISF (SU_f ⊃ SU_{f_{12}} ⊗ SU_{f_{34}} ISF)
7.17.3. The SU_{m+n} ⊃ SU_m ⊗ SU_n ISF and S_f ⊃ S_{f_{12}} ⊗ S_{f_{34}} outer-product ISF
7.17.4. The evaluation of SU_{m+n} ⊃ SU_m ⊗ SU_n ISF
7.17.5. Symmetries of the SU_{m+n} ⊃ SU_m ⊗ SU_n ISF
7.18. The SU_n Singlet Factor
7.19. Second Quantized Expressions for the CFP
7.19.1. One-particle CFP
7.19.2. Two-particle CFP
7.19.3. CFP in the interacting boson model
7.20. Generalized Quantized Expressions for the CFP
7.20.1. The generalized quantization
7.20.2. The CFP in the generalized quantization
7.20.3. The relation between the generalized and second quantization
Chapter 8 The Point Groups
8.1. Basic Operations of Point Groups and Their Faithful Representations
8.2. Some Commonly Used Point Groups
8.3. Character Tables of Point Groups
8.3.1. The conventional labelling for point group irreps (Mullikan notation)
8.3.2. Character tables of point groups
8.4. The CSCO-I and CSCO-II of Point Groups
8.4.1. The CSCO-I of point groups
8.4.2. The CSCO-II of point groups
8.4.3. The codes for point groups
8.4.4. The point group tables
8.5. Algebraic Solutions for the Dihedral Groups Dn
8.5.1. Factorization lemma for the projection operators
8.5.2. The D_n ⊃ C_{2x} generalized projection operators of Dn
8.5.3. The D_n ⊃ C_n generalized projection operators of Dn
8.5.4. The Dn ⊃ C_n projection operators
8.5.5. The Dn ⊃ C =_{2x} projection operators
8.5.6. The characters and irreducible matrices of D_n
8.5.7. The symmetry adapted functions
8.5.8. The group D_∞
8.5.9. Improper dihedral groups C_{nv} , D_{nd} (even n) and D_{nh} (odd n)
8.6. Numerical solutions for T ⊃ D₂ and O ⊃ D₄ ⊃ D₂
8.7. Algebraic Solutions for Cubic Groups
8.7.1. Double-coset factored projection operators and its application
8.7.2. Algebraic expressions for the T ⊃ C₃ projection operator
8.7.3. The T ⊃ C₃ irreducible matrices
8.7.4. The algebraic expressions for T ⊃ C₃ SAF's
8.7.5. SAFs for the group chain O ⊃ T ⊃ C₃
8.7.6. The splitting of atomic levels in the 0₃ ⊃ G ⊃ G(s) basis
8.8. The CG Coefficients of Point Groups
8.8.1. The CG series of point groups
8.8.2. The CG coefficients of point groups
8.9. Molecular Orbital Theory
8.10. Single Electron SALC
8.11. Double Point Groups for d-v Representations
8.11.1. The double point group method
8.11.2. Euler angles and group tables
8.11.3. Some basic relations between point group operators
8.11.4. The Opechowski rule for classes
8.11.5. The double-group method for d-v representations
8.12. The Representation Group and Its Applications
8.12.1. The representation group
8.12.2. Characters of d-v irreps of point groups
8.12.3. Algebraic solutions for cyclic groups C_n, S_{2n} and C_{nh}
8.12.4. Algebraic solutions for dihedral groups D_n În d-v reps
8.13. The Time ReversaI Symmetry
8.13.1. The time reversaI operator
8.13.2. The time reversaI group
8.13.3. Three types of irreps
8.13.4. Degeneracy due to time reversaI symmetry
8.13.5. The transformation of irreducible basis under time reversaI
Chapter 9 Applications of Group Theory to Many-Body Systems*
9.1. Nuclear Shell Model: Single-Shell
9.1.1. First quantization
9.1.2. Generalized quantization
9.2. Nuclear Shell Model: Multi-Shell
9.3. Anti-Symmetric Wave Functions for an A+B System
9.4. Transformations between Symmetry Bases and Physical Bases in the Quark Model
9.5. The Dynamical Symmetry Models of Nuclei
9.6. The Quasispin Model
9.7. The Proton-Neutron Quasispin Model
9.8. The Groups S_{p_N}, SO_N and the Pairing Interaction
9.8.1. Pairing interaction for identical particles
9.8.2. Pairing interaction for electrons and non-identical nucleons
9.9. The Elliott Model
9.10. The Interacting Boson Model*
9.11. The Molecular Shell Model
9.11.1. The Hamiltonian as a function of infinitesimal operators of the unitary group
9.11.2. Spin-free approximation
Chapter 10 The Space Groups
10.1. The Euclidean Group
10.1.1. Definition of the Euclidean Group
10.1.2. Properties of the Euclidean Group Operators
10.2. The Lattice Group
10.3. The Space Group
10.4. The Point Group P and the Crystal System
10.5. The Bravais Lattice
10.6. Operators of the Space Group
10.6.1. The properties of group operators
10.6.2. Example: Group D^{14}_{4h}
10.7. The Reciprocal Lattice Vectors
10.8. Irreps of the Lattice Group
10.9. The Brillouin Zone
10.10. The Electron State in a Periodic Potential
10.1l.Representation Space of the Space Group
10.12. The Little Group G(k)
10.13.The Representation Groups G_k and G'_k
10.13.1. The rep group G_k
10.13.2. The rep group G'_k
10.13.3. Special cases of the rep group G'_k
10.14. The Irreducible Basis and Matrices of G'_k
10.14.1. The group table of G'_k
10.14.2. The CSCO-II and CSCO-III of G'_k
10.14.3. The irreps of G'_k and the projective irreps of G₀(k)
10.14.4. The irreducible basis of G(k)
10.15. Example: the Point W of O⁷_h
10.15.1. Seeking the CSCO and the characters of the point W of the space group O⁷_h
10.15.2. Obtaining the CSCO-I from the existing character table
10.15.3. Constructing irreps of the rep group G'_k . The point W of O⁷_h
10.16.Irreducible Basis and Representations of the Space Group
10.16.1. The k star
10.16.2. The induced rep
10.16.3. A simple algorithm for full rep matrices
10.16.4. The G ⊃ G(k_σ) ⊃ G(s_σ) ⊃ T irreducible basis
10.17. The Irreducible Basis and Matrices of C⁴_{2v}
10.17.1. A general star: p = (p_l,p₂,p₃)
10.17.2. The star Γ : p = (0,0,0)
10.17.3. The star Σ : p=(p_l,0,0)
10.17.4. The star X : p=(1/2,0,0)
10.18. The Clebsch-Gordan Coefficients of Space Groups*
10.18.1. The CG series
10.18.2. The calculation of the CG coefficients
10.18.3. Relative phase of the CG coefficients
10.18.4. The full CG coefficients of space groups
10.18.5. A summary of the eigenfunction method for space group CG coefficients
10.19. Examples: Getting Space Group Clebsch-Gordan Coefficients*
10.19.1. The CG coefficients of O⁷_h for *X(l) ⊗ *X(2) --> *X(v")
10.19.2. The CG coefficients of O⁷_h for *X(l) ⊗ *W(l) --> *Δ(v")
10.20. The Double Space Groups
10.21. Appendix
Appendix
Table Al. Dimensions of irreps of the permutation group S_f (f<7) and the unitary groups SU_n (n<7)
Table A2. Phase factors ε₁(ν₁ν₂ν) for the permutation group IDC and SU_n CG coefficients
References
Index