Group Theoretical Methods and Their Applications

1 Preliminaries.- 1.1 The Concept of Groups.- 1.1.1 Transformation Groups.- 1.2 Price Index in Economics.- 1.3 The Realization of Groups.- 1.4 Representation of Groups.- 1.5 Equivalence of Representations.- 1.6 Reducibility of Representations.- 1.7 Complete Reducibility.- 1.8 Basic Conclusions.- 1.9 Representations of Special Finite Groups.- 1.9.1 The Cyclic Group Cg.- 1.9.2 The Dihedral Group Dh.- 1.10 Kronecker Products.- 1.11 Unitary Representations.- 1.11.1 Unitary Representations and Unitary Matrices.- Problems.- 2 Linear Operators with Symmetries.- 2.1 Schur's Lemma.- 2.2 Symmetry of a Matrix.- 2.2.1 Representations of Abelian Groups.- 2.3 The Fundamental Theorem.- Problems.- 3 Symmetry Adapted Basis Functions.- 3.1 Illustration by Dihedral Groups.- 3.1.1 The Representation ?per.- 3.1.2 The Notion of the Orbit.- 3.1.3 Instructions on How to Use Table (3.7).- 3.1.4 Real Orthonormal Symmetry Adapted Basis.- 3.2 Application in Quantum Physics.- 3.3 Application to Finite Element Method.- 3.3.1 Discretization and Symmetry of the Problem.- 3.3.2 Elliptic Boundary Value Problem.- 3.3.3 Heat conduction.- 3.4 Perturbed Problems with Symmetry.- 3.5 Fast Fourier Transform on Finite Groups.- 3.5.1 Definitions and Properties.- 3.5.2 Direct and Fast Algorithm.- 3.5.3 The Classical Fast Fourier Transform (FFT).- 3.5.4 Applications and Remarks.- 4 Continuous Groups And Representations.- 4.1 Continuous Matrix Groups.- 4.1.1 Comments on U(n).- 4.1.2 Comments on SU(n).- 4.1.3 Connectedness of Continuous Groups.- 4.2 Relationship Between Some Groups.- 4.2.1 Relationship between SL(2, C) and the Lorentz group.- 4.2.2 Relationship Between SU(2) and SO(3).- 4.3 Constructing Representations.- 4.3.1 Irreducible Representations of SU(2).- 4.3.2 Irreducible Representations of SO(3).- 4.3.3 Complete Reduction of Representations of SU(2) and SO(3).- 4.4 Clebsch-Gordan Coefficients.- 4.4.1 Spherical Functions.- 4.4.2 The Kronecker Product ?l??m.- 4.4.3 Spherical Functions and Laplace Operator.- 4.5 The Lorentz group and SL(2,C).- 4.5.1 Irreducible Representations.- 4.5.2 Complete Reduction.- Problems.- 5 Symmetry Ad. Vectors, Characters.- 5.1 Orthogonality of Representations.- 5.2 Algorithm for Symmetry Adapted Bases.- 5.3 Applications.- 5.4 Similarity Classes of Groups.- 5.4.1 Subgroups of SO(3).- 5.4.2 Permutation Groups.- 5.5 Characters.- 5.5.1 General Properties.- 5.5.2 Orthogonality Relations of Characters.- 5.5.3 A Composition Formula for Characters.- 5.5.4 Fundamental Theorems about Characters.- 5.5.5 Projectors.- 5.5.6 Summary.- 5.6 Representation Theory of Finite Groups.- 5.6.1 The Regular Representation.- 5.6.2 The Character Matrix.- 5.6.3 Completeness.- 5.7 Extension to Compact Lie Groups.- Problems.- 6 Various Topics of Application.- 6.1 Bifurcation and A New Technique.- 6.1.1 Introduction.- 6.1.2 The Hopf Bifurcation.- 6.1.3 Symmetry-Breaking.- 6.1.4 A New Approach.- 6.1.5 The Brusselator.- 6.2 A Diffusion Model in Probability Theory.- 6.2.1 Introduction.- 6.2.2 General Considerations.- 6.2.3 The Bernoulli-Laplace Model.- 6.2.4 Discussion and Applications.- Problems.- 7 Lie Algebras.- 7.1 Infinitesimal Operator and Exponential Map.- 7.1.1 Infinitesimal Operators.- 7.1.2 The Exponential Mapping.- 7.2 Lie Algebra of a Continuous Group.- 7.3 Representation of Lie Algebras.- 7.4 Representations of SU(2) and SO(3).- 7.4.1 Infinitesimal Aspects of the Kronecker Product.- 7.4.2 Clebsch-Gordan Coefficients.- 7.5 Examples from Quantum Mechanics.- 7.5.1 Energy and Angular Momentum.- 7.5.2 Spin-Orbit Coupling.- Problems.- 8 Applications to Solid State Physics.- 8.1 Lattices.- 8.1.1 Reciprocal Lattice.- 8.1.2 Brillouin Zone, Stabilizer.- 8.2 Point Groups and Representations.- 8.2.1 The List of All (+)-Groups.- 8.2.2 Representations and Characters of (+)-Groups.- 8.2.3 (+-)-Groups of the First Kind.- 8.2.4 (+-)-Groups of the Second Kind.- 8.2.5 Review.- 8.3 The 32 Crystal Classes.- 8.4 Symmetries and the Ritz Method.- 8.5 Examples of Applications.- 8.6 Crystallographic Space Groups.- Problems.- 9 Unitary and Orthogonal Groups.- 9.1 The Groups U(n) and SU(n).- 9.1.1 Similarity Classes, Diagram.- 9.1.2 Characters of Irreducible Representations.- 9.1.3 Division Algorithm.- 9.1.4 Weights.- 9.1.5 Algorithm for the Complete Reduction.- 9.2 The Special Orthogonal Group SO(n).- 9.2.1 Weyl Group and Diagram of SO(n).- 9.2.2 Algorithm for the Computation of Weights.- 9.3 Subspaces of Representations of SU(3).- A.- Answers to Selected Problems.