]> January 8 Predrag Cvitanović 1. History. Finite groups intro Tinkham Chapter 2 Abstract group theory homework #1: Tinkham (2.1), (2.2); optional (2.6) - due Tue January 15 [solutions to exercises] January 10 2. Finite groups Cosets, classes, normal divisors and factor subgroups January 15 3. Group representations Matrix representations are unitary. Schur's lemma. schur Tinkham Chapter 3 Theory of group representations homework #2: Tinkham (3.1), (3.3); optional (3,7), (3.8) - due Tue January 22 January 17 4. Characters The great orthogonality theorem. Character orthogonality. Character tables. January 21 MLK holiday January 22 5. Characters Hard work builds character. January 24 6. Decomposition of reducible representations Regular representation. Transformation operators. Representations. schur Tinkham sections 3.5-3.8 Harter1-2Bd Harter Sect. 1.2Bd Commuting matrices homework #3: Harter (1.2.1), (1.2.6); optional (1.2.2) - due Tue January 29. [bra, ket refers to left/right eigenvectors. Sect. 1.2Bd is the same as my Appendix C, section 2.2] January 29 7. Projection operators All eigenvalues distinct. Complex eigenvalues in real representation. Degenerate eigenvalues: hermitian case, Jordan case. January 31 8. Irreducible reps of abelian groups Projection operators for abelian groups from character tables. D_2 example: Harter's propeller. appendStability ChaosBook.org Appendix B Linear stability (ver. Feb 1, 2008) This appendix is continuously updated - wisest not to print it on paper yet. homework #4: Appendix B exercise B.1, Appendix C exercise C.1 - due Tue January 29. (You are in luck - class secretary is too exhausted to type yet another problem.) February 5 9. Irreducible reps of abelian groups Projection operators for abelian groups from character tables. C_2, C_3 coupled harmonic oscillators reduction to normal modes. Harter1-2Bd Lecture notes Abelian groups reduction (ver. Feb 7 2008) February 7 10. Irreducible reps of abelian groups Irreps for C_n. Discrete Fourier transforms from character tables. appendSymm ChaosBook.org Appendix C Discrete symmetries of dynamics (ver. Feb 8 2008) Read sections C.3-C.5. This appendix is continuously updated - wisest not to print it on paper yet. homework #5: Appendix C exercises C.2, C.3, C.5 - due Tue February 12. February 12 11. Fourier transforms If the symmetry group is the group of translations on a line of rotations/shifts on a circle, the reduction to 1-dimensional irreps is known as the Fourier transform. It trades in nonlocal operators, such as the Laplace operator for pure numbers, such as the momentum^2. February 14 Valentine's day February 14 12. Irreducible reps decomposition Worked out problem C.2: 3 pendulums on a line, with mirror C2 symmetry. Reduce by symmetry first. Harter3-2 Harter 3.2 Nonabelian symmetry analysis Work through section 3.3. appendSymm Harter Double group theory on the half-shell (1978) Read appendices B and C on spectral decomposition and class algebras. Article works out some interesting examples. homework #6: Appendix C exercise C.4 - due Tue February 19. February 19 13. Irreducible reps of nonabelian groups Projection operators for C_3v nonabelian group from character tables. February 21 14. Continuous symmetries / back to triangulating C_3v Rotations in a plane. Equilateral 3-mass spring system, not pinned down. stability ChaosBook.org Chapter 4 Local stability (ver. Feb 21, 2008) Read sects. 4.2.2, 4.3.1 - how SO(2) Lie algebra generates rotations in a plane. This chapter is continuously updated - wisest not to print it on paper. Porter1 Frank Porter CalTech Physics 129b Read chapter "Representation theory," most of it for pleasure. Focus in particular on sect. 3.10. homework #7, Problem 1, due February 26: Work through Porter sect. 3.10. (a) Derive (3.95), matrix U in terms of the 1/3 turn [2x2] rotation matrices (3.103), keep it in that format. (b) Verify that the matrix U is C_3v invariant. (c) evaluate (3.15), (3.16) using your invariant form of U (rather than the explicit [6x6] bunch of square roots of 3). (d) Compute explicitely \lambda_31=0 and its eigenvactors, show that they correspond to translations, rotations. (e) optional for everybody EXCEPT Jonathan and Vaggelis (for them it is required): quotient out T^2 and O(2), ie. rewrite dynamics so quotiented dynamics has no zero eigenvalues. homework #7, Problem 2, due February 26: The relation of irreducible representations and the invariant subspaces of a vector space: Do problem 11 (click here). This problem takes some thought. Also, there many different, equally good ways to solve it. [Porter solution to problem 11, now called 18] February 26 15. Continuous groups Lie groups defined. Examples. Lie algebras, first try. Chen1 Chen, Ping and Wang Group Representation Theory for Physicists Sect 5.2 Definition of a Lie group, with examples February 28 16. Lie algebras Groups, vector spaces, tensors, invariant tensors, invariance groups. Chapter3 birdtracks.eu Chapter 3 Invariants and reducibility appendSymm C K Wong 1-D continuos groups (power point notes) Wong is entirely optional, not covered in the lectures, but completes discussion of Fourier analysis as continuom limit of cyclic groups C_n: Read chapter 6 on representations of SO(2), O(2) and translational group. homework #8, due March 4: Same as homework #7 - sing it until you get it right. March 4 17. So many indices, so little time Indices. Tensors. Invariant tensors. Indices. March 6 18. Birdtracks Goodbye to indices. Clebsch-Gordan coefficients. Infinitesimal transformations. Lie algebras. Chapter4 birdtracks.eu Chapter 4 Diagrammatic notation homework #9: Derive the Lie algebra commutator and the Jacobi identity as particular examples of the invariance condition on invariant tensor, using both index and birdtracks notations. Due March 27. March 11 John Wood 19. The nature and use of dynamical groups March 13 Evangelos Siminos and Jonathan Halcrow 20. Trading in a dogeared Lorenz for a cute Van Gogh Quotienting symmetries of nonlinear dynamical systems, or: How Lorenz lost one ear. VanGogh ChaosBook.org limbo Desymmetrization of the Lorenz flow (rev. 459 03/27/2008) GolStu1 Golubitsky and Stewart Chapter 1 Steady-state bifurcation [optional: this chapter was not used in the course] discrete ChaosBook.org Chapter 9 World in a mirror [optional: this chapter was not used in the course] March 18 spring break Alex has read no Dyson, so here is a fun sample: Dyson Freeman J. Dyson in NYRB The World on a String March 20 spring break A fun read on group theory we definitely will not cover: moonshine Marcus du Sautoy Finding Moonshine: A Mathematician's Journey Through Symmetry March 25 21. Birdtracks refresher VanGogh ChaosBook.org limbo Desymmetrization of the Lorenz flow (rev. 459 03/27/2008) homework #10, due April 1: Exercise 5.1 in "Desymmetrization of the Lorenz flow" March 27 22. Mutiny in the class Chapter5 birdtracks.eu Chapter 5 Recouplings AbelPrize Abel Prize J. G. Thompson and J. Tits You doubt group theory is good for anything? How does $1.2 million sound to you? April 1 23. Symmetrizations. Antisymmetrizations Chapter6 birdtracks.eu Chapter 6 Permutations April 3 24. Unitary representations, Young tableaux Chapter9 birdtracks.eu Chapter 9 Unitary groups Read sects. 9.1, 9.2, 9.11 and 9.12. Optional: sects. 9.3, 9.4. homework #11, due April 8: Derive projection operators and dimensions listed in Table 9.3. (Ignore "indices," we have not defined them). April 8 25. Orthogonal groups Chapter10 birdtracks.eu Chapter 10 Orthogonal groups Read sects. 10.1, 10.2, 10.4 and 10.5 homework #12, due April 15: Decompose the Riemann-Christoffel curvature tensor into its SO(n) irreducible tensors: curvature scalar, traceless Ricci tensor and Weyl tensor, equations (10.57) to (10.59). How many components does each irreducible tensor have in n=4 dimensions? You do not need to know general relativity or worry about SO(1,3) Lorenz group for this exercise - this is a question only of the reduction of V^4 tensor representations of SO(n). April 10 26. Symplectic groups. SU(2), SU(3) as invariance groups Chapter12 birdtracks.eu Chapter 12 Symplectic groups Chapter15 birdtracks.eu Chapter 15 SU(n) family of invariance groups Read sects. 15.1 and 15.2. April 15 27. Invariance group of a cubic invariant A quick overview of the construction of exceptional Lie algebras. dyson03 birdtracks.eu lite the webbook in 20 minutes April 16 Fall registration starts April 17 Jogia Bandyopadhyay 28. Group theory made coherent Representation of SU(1,1) and the construction of coherent states. JogiaVer2 J. Bandyopadhyay Optimal Concentration for SU(1; 1) Coherent State Transforms and An Analogue of the Lieb-Wehrl Conjecture for SU(1; 1) April 22 29. Exceptional group E_6 E_6 family of invariance groups of a symmetric cubic invariant. April 24 30. Exceptional magic A summary of the continous Lie groups part of the course. ExcMagic P Cvitanovic The webbook at a cyclist pace, in 50 overheads takehome final: Do any part of problems 1, 2 and 5 in the order you find most convenient. 5 is straightforward, for 1 and/or 2 a partial solution is good enough. Sorry for few illegible lines of problem 1, I am learning how to use a CyberPad. April 25 GT classes end May 1 10:50 take-home final exam due, Predrag's office notes solutions to the final exam to May 2 Course opinion survey CETL web link May 5 GT grades due at noon May 6 have a good summer! The rest has yet to be worked out.