Applications of Finite Groups - 1st Edition - ISBN: 9781483231327, 9781483268965

Applications of Finite Groups

1st Edition

Authors: J. S. Lomont
eBook ISBN: 9781483268965
Imprint: Academic Press
Published Date: 1st January 1959
Page Count: 358
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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


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


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© Academic Press 1959
Academic Press
eBook ISBN:

About the Author

J. S. Lomont

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