Applications of Finite Groups

1st Edition - January 1, 1959
Author: J. S. Lomont
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 6 8 9 6 - 5

Applications of Finite Groups focuses on the applications of finite groups to problems of physics, including representation theory, crystals, wave equations, and nuclear and… Read more

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Applications of Finite Groups focuses on the applications of finite groups to problems of physics, including representation theory, crystals, wave equations, and nuclear and molecular structures. The book first elaborates on matrices, groups, and representations. Topics include abstract properties, applications, matrix groups, key theorem of representation theory, properties of character tables, simply reducible groups, tensors and invariants, and representations generated by functions. The text then examines applications and subgroups and representations, as well as subduced and induced representations, fermion annihilation and creation operators, crystallographic point groups, proportionality tensors in crystals, and nonrelativistic wave equations. The publication takes a look at space group representations and energy bands, symmetric groups, and applications. Topics include molecular and nuclear structures, multiplet splitting in crystalline electric fields, construction of irreducible representations of the symmetric groups, and reality of representations. The manuscript is a dependable source of data for physicists and researchers interested in the applications of finite groups.

PrefaceList of SymbolsI. MatricesII. Groups 1. Abstract Properties 2. Applications A. Thermodynamics B. The Dirac Equation C. Fermion Annihilation and Creation OperatorsIII. Representations 1. Matrix Groups 2. The Key Theorem of Representation Theory 3. Character Tables 4. Computation of Character Tables 5. Properties of Character Tables 6. Faithful Representations 7. Kronecker Products 8. Simply Reducible Groups 9. Reduction by Idempotents 10. Groups of Mathematical Physics A. Cyclic Groups B. Dihedral Groups C. Tetrahedral Group D. Octahedral Group E. Icosahedral Group 11. Tensors and Invariants 12. Representations Generated by Functions 13. Subduced RepresentationsIV. Applications 1. Fermion Annihilation and Creation Operators 2. Molecular Vibrations (Classical) 3. Symmetric Waveguide Junctions 4. Crystallographic Point Groups 5. Proportionality Tensors in Crystals 6. The Three-Dimensional Rotation Group 7. Double Point Groups 8. Nonrelativistic Wave Equations 9. Stationary Perturbation Theory 10. Lattice Harmonics 11. Molecular Orbitals 12. Crystallographic Lattices 13. Crystallographic Space Groups 14. Wave Functions in CrystalsV. Subgroups and Representations 1. Subduced Representations 2. Induced Representations 3. Induced and Subduced Representations 4. Projective Representations 5. Little GroupsVI. Space Group Representations and Energy Bands 1. Representation Theory 2. Example—Two-Dimensional Square Lattice 3. Reality of Representations 4. Analysis 5. Compatibility 6. PhysicsVII. Symmetric Groups 1. Abstract Properties of ℓ(n) 2. Representations of ℓ(n) 3. Miscellany and the Full Linear Groups 4. Construction of Irreducible Representations of the Symmetric GroupsVIII. Applications 1. Permutation Degeneracy and the Pauli Exclusion Principle 2. Atomic Structure A. The Central Field Approximation B. LS Coupling 3. Multiplet Splitting in Crystalline Electric Fields 4. Molecular Structure 5. Nuclear Structure A. Spatial Coordinate Approximation B. Spin Approximation 6. Selection RulesReferencesAppendix I: Proof of the Key Theorem of Representation TheoryAppendix II: Irreducible Representations of D3, D4, D6, T, O, and IAppendix III: The Lorentz GroupsSubject Index