Group Theoretical Methods and Their Applications

1 Preliminaries.- 1.1 The Concept of 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.- 1.10 Kronecker Products.- 1.11 Unitary Representations.- Problems.- 2 Linear Operators with Symmetries.- 2.1 Schur’s Lemma.- 2.2 Symmetry of a Matrix.- 2.3 The Fundamental Theorem.- Problems.- 3 Symmetry Adapted Basis Functions.- 3.1 Illustration by Dihedral Groups.- 3.2 Application in Quantum Physics.- 3.3 Application to Finite Element Method.- 3.4 Perturbed Problems with Symmetry.- 3.5 Fast Fourier Transform on Finite Groups.- 4 Continuous Groups And Representations.- 4.1 Continuous Matrix Groups.- 4.2 Relationship Between Some Groups.- 4.3 Constructing Representations.- 4.4 Clebsch-Gordan Coefficients.- 4.5 The Lorentz group and SL(2,C).- 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.- 5.5 Characters.- 5.6 Representation Theory of Finite Groups.- 5.7 Extension to Compact Lie Groups.- Problems.- 6 Various Topics of Application.- 6.1 Bifurcation and A New Technique.- 6.2 A Diffusion Model in Probability Theory.- Problems.- 7 Lie Algebras.- 7.1 Infinitesimal Operator and Exponential Map.- 7.2 Lie Algebra of a Continuous Group.- 7.3 Representation of Lie Algebras.- 7.4 Representations of SU(2) and SO(3).- 7.5 Examples from Quantum Mechanics.- Problems.- 8 Applications to Solid State Physics.- 8.1 Lattices.- 8.2 Point Groups and Representations.- 8.3 The 32 Crystal Classes.- 8.4 Symmetries and the Ritz Method.- 8.5 Examples ofApplications.- 8.6 Crystallographic Space Groups.- Problems.- 9 Unitary and Orthogonal Groups.- 9.1 The Groups U(n) and SU(n).- 9.2 The Special Orthogonal Group SO(n).- 9.3 Subspaces of Representations of SU(3).- A.- Answers to Selected Problems.