(This class also has a web page on Courseworks.)
Time: MW 1:10-2:25 p.m.
Place: Math 520
Required Textbook: Teleman's Group Representation Theory notes
Recommended Textbook: Serre's Linear Representations of Finite Groups
Office hours: M 3:30-4:30pm and Tu 1-2pm in 626.
Graduate teaching assistant: Alex Perry
Help room hours: MW 12-1.
Final exam date: Projected to be Monday December 20th 1:10-4:00pm
Announcements:
| Syllabus | Problem sets | Policies |
|---|
Prerequisites.
The main prerequisite for this course is a very solid understanding of linear algebra. You should be completely at home with abstract vector spaces and abstract linear transformations, you should be comfortable changing bases, and you should understand eigenvalues and eigenvectors. Furthermore, you should be familiar with the basics of group theory (subgroups, conjugation, normal subgroups, quotients). But you do not need to remember advanced topics in group theory (for example, the proofs of Sylow theorems). Finally, this should not be your first proof-based course.
Typically these prerequisites are fulfilled by taking Math W4041 and one of Math V2010, V2020, or V1207-V1208. However, if you took Math V2010 and struggled or do not remember the material well, then your linear algebra background may not be strong enough.
Description and goals.
Group representation theory is the study of symmetries of space. More precisely it is the study of linear actions of groups on vector spaces (after choosing a basis, this assigns a matrix to every element of the group in a way that's compatible with the group multiplication). This subject is interesting in its own right but it also allows applications of the powerful tools of linear algebra to the study of groups.
The two main results in this course are complete reducibility (which reduces the study of all representations of finite groups to certain irreducible ones) and the classification of irreducible representations via their traces. The latter is called character theory and associates to every group a beautiful combinatorial object called the character table which encodes much of the information about the groups representations. Additional topics include tensor products of representations, induced representations, and the representation theory of SU(2)
Policies.
Grading
| Homework | 40% |
| Midterm exam | 25% |
| Final exam | 35% |
The lowest homework score will be dropped.
Homework
Problem sets are due on Wednesday at the beginning of class, except as noted below. If you can't make it to class, put it in my mailbox before class.
You're welcome to work on the homework together. However, you must write up your final answers by yourself. I consider writing them up together cheating.
You are also generally welcome to use any resources you like to solve the problems. However, any resource you use other than the course texts (Teleman's notes and Serre's book) must be cited in your homework. This includes electronic resources (including Wikipedia and Google) and human resources (including the help room and your classmates). Failure to cite sources constitutes academic misconduct.
Students with disabilities
Students with disabilities requiring special accommodation should contact Office of Disability Services (ODS) promptly to discuss appropriate arrangements. I am not authorized to make such accommodations myself. This includes both long-term disabilities and temporary disabilities (e.g. repetitive stress injuries or broken bones).
Missed exams
If you have a conflict with any of the exam dates, you must contact me ahead of time so we can make arrangements. (At least a week ahead is preferable.) If you are unable to take the exam because of a medical problem, you must go to the health center and get a note from them -- and contact me as soon as you can. Exam dates do not conflict
Syllabus and schedule.
| Date | Material | Textbook | Announcements |
|---|---|---|---|
| 9/8 | What is a representation? | § 1 | Welcome! |
| 9/13 | Maps of representations | § 2.9 and §4 | |
| 9/15 | All reps of Z, Z/p, and S_3 | § 2 and Example 5.1 | Problem Set 1 due. |
| 9/20 | Complete reducibility | § 3 | |
| 9/22 | Complete reducibility | § | Problem Set 2 due. |
| 9/27 | Duals & orthogonality | None | |
| 9/29 | Tensor products | § 6 | Problem Set 3 due. |
| 10/4 | Tensor products | § 6 | |
| 10/6 | Character theory | § 8 | Problem Set 4 due. |
| 10/11 | The regular representation | § 9 | |
| 10/13 | Some character tables | § 10 & 12 | Problem Set 5 due. |
| 10/18 | Review | ||
| 10/20 | MIDTERM | ||
| 10/25 | Induced representations | § 14 | |
| 10/27 | Characters & induction | § 15 | Problem Set 6 due. |
| 11/1 | Mackey theory | § 16 | |
| 11/3 | No class | Go vote! | |
| 11/8 | Supersolvable groups | § 17 | |
| 11/10 | Integrality | § 13 | Problem Set 7 due |
| 11/15 | More integrality | § 18 | |
| 11/17 | Burnside's theorem | ||
| 11/22 | Representations over R | § 19 | |
| 11/24 | No class, thanksgiving break | ||
| 11/29 | Compact groups and U(1) | § 19 | . |
| 12/1 | SU(2) | § 20 | Problem set 9 due |
| 12/6 | SU(2) | § 20 | |
| 12/8 | Temperley-Lieb algebra | Problem Set 10 due | |
| 12/13 | Temperley-Lieb algebra |
Problem sets.
- Problem set 1 (PDF). Due September 15th.
- Problem set 2 (PDF). Due September 22th.
- Problem set 3 (PDF). Due September 29th.
- Problem set 4 (PDF). Due October 6th.
- Problem set 5 (PDF). Due October 13th.
- Solutions to selected problems (PDF).
- Solutions to the midterm (PDF).
- Problem Set 6 (PDF). Due November 3.
- Problem Set 7 (PDF). Due November 10.
- Problem Set 8 (PDF). Due November 17.
- Problem Set 9 (PDF). Due December 1.
- Problem Set 10 (PDF). Due December 8.
For questions or comments, please contact Noah Snyder.