Applications of Finite Groups

Applications of Finite Groups

1st Edition - January 1, 1959

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  • Author: J. S. Lomont
  • eBook ISBN: 9781483268965

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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.

Table of Contents

  • Preface

    List of Symbols

    I. Matrices

    II. Groups

    1. Abstract Properties

    2. Applications

    A. Thermodynamics

    B. The Dirac Equation

    C. Fermion Annihilation and Creation Operators

    III. 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 Representations

    IV. 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 Crystals

    V. Subgroups and Representations

    1. Subduced Representations

    2. Induced Representations

    3. Induced and Subduced Representations

    4. Projective Representations

    5. Little Groups

    VI. Space Group Representations and Energy Bands

    1. Representation Theory

    2. Example—Two-Dimensional Square Lattice

    3. Reality of Representations

    4. Analysis

    5. Compatibility

    6. Physics

    VII. 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 Groups

    VIII. 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 Rules


    Appendix I: Proof of the Key Theorem of Representation Theory

    Appendix II: Irreducible Representations of D3, D4, D6, T, O, and I

    Appendix III: The Lorentz Groups

    Subject Index

Product details

  • No. of pages: 358
  • Language: English
  • Copyright: © Academic Press 1959
  • Published: January 1, 1959
  • Imprint: Academic Press
  • eBook ISBN: 9781483268965

About the Author

J. S. Lomont

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