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January 8
Predrag CvitanoviÄ‡
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.