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(2C). - 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. Language: English