Group Theoretical Methods and Their Applications

x system Ib-TEX. I wish to thank her for the beautiful work and the numerous discussions on the contents of this book. I am indebted to Peter Fassler, Neu-Technikum Buchs, Switzerland, for drafting the figures, to my students Kurt Rothermann and Stefan Strahl for computer enhancing and labeling the graphics, to Pascal Felder and Markus Wittwer for a simulation program that generated the figures in the stochastics sections. My thanks go to my new colleague at work, Daniel Neuenschwander, for the inspiring discussions related to the section in stochastics and for reading the manuscript to it. I am also grateful to Dacfey Dzung for reading the whole manuscript. Thanks go especially to Professor \Valter Gander of ETH, Zurich, who at the finishing stage and as an expert of 'JEXgenerously invested numerous hours to assist us in solving software as well as hardware problems; thanks go also to Martin Muller, Ingenieurschule Biel, who made the final layout of this book on the NeXT computer. Thanks are also due to Helmut Kopka of the Max Planck Institute, for solving software problems, and to Professor Burchard Kaup of the Uni- versity of Fribourg, Switzerland for adding some useful software; also to Birkhauser Boston Inc. for the pleasant co-operation. Finally, let me be reminiscent of Professor E. Stiefel (deceased 1978) with whom I had many interesting discussions and true co-operation when writing the book in German.

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.