Group Theory and its Application to Physical Problem

Title page
INTRODUCTION
CHAPTER 1. ELEMENTS OF GROUP THEORY
1-1 Correspondences and transformations
1-2 Groups. Definitions and examples
1-3 Subgroups. Cayley's theorem
1-4 Cosets. Lagrange's theorem
1-5 Conjugate classes
1-6 Invariant subgroups. Factor groups. Homomorphism
1-7 Direct products
CHAPTER 2. SYMMETRY GROUPS
2-1 Symmetry elements. Pole figures
2-2 Equivalent axes and planes. Two-sided axes
2-3 Groups whose elements are pure rotations: uniaxial groups, dihedral groups
2-4 The law of rational indices
2-5 Groups whose elements are pure rotations. Regular polyhedra
2-6 Symmetry groups containing rotation reflections. Adjunction of reflections to C_n
2-7 Adjunction of reflections to the groups D_n
2-8 The complete symmetry groups of the regular polyhedra
2-9 Summary of point groups. Other systems of notation
2-10 Magnetic symmetry groups (color groups)
CHAPTER 3. GROUP REPRESENTATIONS
3-1 Linear vector spaces
3-2 Linear dependence; dimensionality
3-3 Basis vectors (coordinate axes); coordinates
3-4 Mappings; linear operators; matrix representations; equivalence
3-5 Group representations
3-6 Equivalent representations; characters
3-7 Construction of representations. Addition of representations
3-8 Invariance of functions and operators. Classification of eigenfunctions
3-9 Unitary spaces; scalar product; unitary matrices; Hermitian matrices
3-10 Operators: adjoint, self-adjoint, unitary
3-11 Unitary representations
3-12 Hilbert space
3-13 Analysis of representations; reducibility; irreducible representations
3-14 Schur's lemmas
3-15 The orthogonality relations
3-16 Criteria for irreducibility. Analysis of representations
3-17 The general theorems. Group algebra
3-18 Expansion of functions in basis functions of irreducible representations
3-19 Representations of direct products
CHAPTER 4. IRREDUCIBLE REPRESENTATIONS OF THE POINT SYMMETRY GROUPS
4-1 Abelian groups
4-2 Nonabelian groups
4-3 Character tables for the crystal point groups
CHAPTER 5. MISCELLANEOUS OPERATIONS WITH GROUP REPRESENTATIONS
5-1 Product representations (Kronecker products)
5-2 Symmetrized and antisymmetrized products
5-3 The adjoint representation. The complex conjugate representation
5-4 Conditions for existence of invariants
5-5 Real representations
5-6 The reduction of Kronecker products. The Clebsch-Gordan series
5-7 Clebsch-Gordan coefficients
5-8 Simply reducible groups
5-9 Three-j symbols
CHAPTER 6. PHYSICAL APPLICATIONS
6-1 Classification of spectral terms
6-2 Perturbation theory
6-3 Selection rules
6-4 Coupled systems
CHAPTER 7. THE SYMMETRIC GROUP
7-1 The deduction of the characters of a group from those of a subgroup
7-2 Frobenius' formula for the characters of the symmetric group
7-3 Graphical methods. Lattice permutations. Young patterns. Young tableaux
7-4 Graphical method for determining characters
7-5 Recursion formulas for characters. Branching laws
7-6 Calculation of characters by means of the Frobenius formula
7-7 The matrices of the irreducible representations of S_n. Yamanouchi symbols
7-8 Hund's method
7-9 Group algebra
7-10 Young operators
7-11 The construction of product wave functions of a given symmetry. Fock's cyclic symmetry conditions
7-12 Outer products of representations of the symmetric group
7-13 Inner products. Clebsch-Gordan series for the symmetric group
7-14 Clebsch-Gordan (CG) coefficients for the symmetric groupe. Symmetry properties. Recursion formulas
CHAPTER 8. CONTINUOUS GROUPS
8-1 Summary of results for finite groups
8-2 Infinite discrete groups
8-3 Continuous groups. Lie groups
8-4 Examples of Lie groups
8-5 Isomorphism. Subgroups. Mixed continuous groups
8-6 One-parameter groups. Infinitesimal transformations
8-7 Structure constants
8-8 Lie algebras
8-9 Structure of Lie algebras
8-10 Structure of compact semisimple Lie groups and their algebras
8-11 Linear representations of Lie groups
8-12 Invariant integration
8-13 Irreducible representations of Lie groups and Lie algebras. The Casimir operator
8-14 Multiple-valued representations. Universal covering group
CHAPTER 9. AXIAL AND SPHERICAL SYMMETRY
9-1 The rotation group in two dimensions
9-2 The rotation group in three dimensions
9-3 Continuous single-valued representations of the three-dimensional rotation group
9-4 Splitting of atomic levels in crystalline fields (single-valued representations)
9-5 Construction of crystal eigenfunctions
9-6 Two-valued representations of the rotation groupe The unitary unimodular group in two dimensions
9-7 Splitting of atomic levels in crystalline fields. Double-valued representations of the crystal point groups
9-8 Coupled systems. Addition of angular momenta. Clebsch-Gordan coefficients
CHAPTER 10. LINEAR GROUPS lN n-DIMENSIONAL SPACE. IRREDUCIBLE TENSORS
10-1 Tensors with respect to GL(n)
10-2 The construction of irreducible tensors with respect to GL(n)
10-3 The dimensionality of the irreducible representations of GL(n)
10-4 Irreducible representations of subgroups of GL(n): SL(n), U(n), SU(n)
10-5 The orthogonal group in n dimensions. Contraction. Traceless tensors
10-6 The irreducible representations of O(n)
10-7 Decomposition of irreducible representations of U(n) with respect to 0^+(n)
10-8 The symplectic group Sp(n). Contraction. Traceless Tensors
10-9 The irreducible representations of Sp(n). Decomposition of irreducible representations of U(n) with respect to its symplectic subgroup
CHAPTER 11. APPLICATIONS TO ATOMIC AND NUCLEAR PROBLEMS
11-1 The classification of states of systems of identical particles according to SU(n)
11-2 Angular momentum analysis. Decomposition of representations of SU(n) into representations of 0^+(3)
11-3 The Pauli principle. Atomic spectra in Russell-Saunders coupling
11-4 Seniority in atomic spectra
11-5 Atomic spectra in jj-coupling
11-6 Nuclear structure. lsotopic spin
11-7 Nuclear spectra in L-S coupling. Supermultiplets
11-8 The L-S coupling shell model. Seniority
11-9 The jj-coupling shell model. Seniority in jj-coupling
CHAPTER 12. RAY REPRESENTATIONS. LITTLE GROUPS
12-1 Projective representations of finite groups
12-2 Examples of projective representations of finite groups
12-3 Ray representations of Lie groups
12-4 Ray representations of the pseudo-orthogonal groups
12-5 Ray representations of the Galilean group
12-6 Irreducible representations of translation groups
12-7 Little groups
BIBLIOGRAPHY AND NOTES
INDEX