This page lists what is covered in lectures, reading assignments, and also archives handouts.
To bottom (most recent lectures)| Lecture | Date | Covered in lecture | Reading (not covered or partially covered) |
| 1.1 | 3/31 |
Class organization (brief). Definition of groups Examples: group multiplication tables, cyclic groups, dihedral group D3, permutation groups. |
Read class web page. First chapter or two of your favorite finite group theory text (e.g. Chesnut)! |
| 1.2 | 4/2 |
Subgroups, cosets and conjugacy classes
Begin representation theory |
|
| 2.1 | 4/7 |
More on representations: non-faithful reps, reducibility, equivalence,
characters Unitary representations and complete reducibility |
Convince yourself that the "fibers" of a mapping all have the same size |
| 2.2 | 4/9 |
Summary of results needed to determine irreps of finite
groups (character orthogonality, ...)
Examples of the use of these relations Begin derivation of these results by stating and proving Schur's Lemma's |
|
| 3.1 | 4/14 |
Proving Schur's second Lemma Demonstrating orthogonality of representation matrices (and thus characters) using Schur's Lemmas Showing sum of n^2=[G] using regular representation Begin discussion of algebra of classes |
|
| 3.2 | 4/16 |
Complete discussion of class algebra and implications Representations of symmetric group and Young tableaux |
|
| 4.1 | 4/21 |
Applications of finite groups: QM selection rules and tensor product representations |
For further discussion see, e.g., Tinkham in Chapters 3 and 4. |
| 4.2 | 4/23 |
Constraints on form of conductivity tensor Point groups of crystals Direct products of groups and their representations Summary of other applications (if time) |
|
| 5.1 | 4/28 |
Begin discussion of continuous groups General introduction and examples SO(2)=U(1), including characters and classes |
Georgi is now the text. Read Ch. 2 of Georgi for a more formal introduction |
| 5.2 | 4/30 |
Lie algebra, structure constants Examples of SO(2) and SO(3) Intro to representations of algebras; adjoint rep |
You should be conversant with the simple properties of the exponentials of matrices. Some practice will be given in the exercises. |
| 6.1 | 5/5 |
Representations of the Lie algebra of SO(3) (done
without the Casimir operator) SU(2) vs SO(3) Generators of SU(3) Introduction to Cartan subalgebra and weights |
Read Georgi Ch. 3.1-3 for a slightly different discussion
of SO(3)/SU(2) irreps For SU(3) see Georgi Ch. 7 We are going to skip over 3.4, 3.5 and Chs. 4 and 5 (addition of angular method, tensor methods, and applications of SU(2)). |
| 6.2 | 5/7 |
Connected and simply connected and implications General discussion of implication of compactness Definition of semi-simplicity and simplicity More on weights and roots of SU(3) and in general (from Georgi Chs. 6 and 7) |
Results stated in class that are demonstrated in Georgi: Diagonalization of Tr(T_aT_b) in adjoint (2.4) Irreducibility of adjoint rep (2.5) Uniqueness of root vector and "+alpha,-alpha only" (6.5) |
| 7.1 | 5/12 |
A little more on U(1)'s and U(2) vs SU(2)xU(1) Schur's Lemmas hold for finite-dim unitary irreps SU(2) subgroups of general Lie groups and their implications, including constraints on angles between roots Decomposing SU(3) adjoint into SU(2) irreps Definition of simple roots |
|
| 7.2 | 5/14 |
Properties of simple roots Reconstructing full root diagram from simple roots |
|
| 8.1 | 5/19 |
Reconstructing full Lie algebra from simple roots Classification of simple, compact Lie algebras |
Georgi, 8.10, discusses the reconstruction of the
Lie algebra of G2 For the classificaiton, we are following Georgi, Ch.20 |
| 8.2 | 5/21 |
Complete classification of simple, compact Lie algebras What are the algebras we have found? (We discussed SU(N) and SO(2N).) |
For further reading on the classical (non-exceptional) Lie algebras, see Georgi, Ch. 19. At present I do not plan to work through $SO(2N+1)$ and $Sp(2N)$ in lectures. |
| 9.1 | 5/26 |
Fundamental weights Constructing irreps of simple compact Lie algebras using "highest weight" method Examples from SU(3) Begin decomposition of tensor products |
Read Georgi Ch. 9 for more examples. |
| 9.2 | 5/28 |
Decomposition of tensor products in general Tensor methods for SU(3) |
Read Georgi Ch. 10 for more examples of the use of tensor methods |
| 10.1 | 6/2 |
Young tableaux as a tool for SU(3) Application: isospin and strangeness and baryon properties (we got through the introductory part) |
Young Tableaux for SU(3) are discussed in Ch. 12 Strangeness and baryon properties in Ch. 11 |
| 10.2 | 6/4 |
Using SU(3) tensor methods to describe baryon properties Decomposing irreps under subalgebras using Young Tab. |
Georgi Ch. 11 (continued) and 12.3 |
| 11 | 6/11 | Final HW due Noon. |