Group Theoretical Methods and Their Applications

Inhaltsverzeichnis

• 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.
• 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).
• 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.
• 6 Various Topics of Application.
• 6.1 Bifurcation and A New Technique.
• 6.2 A Diffusion Model in Probability Theory.
• 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.
• 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.
• 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.