Representations of Finite Groups

Professor A.J. de Jong, Columbia university, Department of Mathematics.

This semester I am teaching the undergraduate course on the representations of finite groups. Besides the beautiful standard theory we will discuss: an application to the theory of finite groups, the structure of invariant rings and complex reflection groups, and Jordan's theorem on subgroups of GL_n(C).

Lectures: Tuesday and Thursday 11:40 PM -- 12:55 PM in Room 307 in the mathematics building. It is very important to attend the lectures and participate in the discussion.

My office hours: Wednesday 9 - 10:30 AM in my office Room 523 in math. If you want to come by another time during the week or zoom with me, please email me. Also, feel free to email math questions any time.

TA: Ryan Xu (kx2186). You can email him math questions and questions about the problem sets and you can ask for zoom sessions with him.

Prerequisites: MATH UN2010 and MATH GU4041 or the equivalent. For Linear Algebra, some topics are matrices, vector spaces, direct sums, tensor products, linear transformations, eigenvalues and eigenvectors, and canonical forms. For Modern Algebra, some topics are groups, homomorphisms, normal subgroups, the isomorphism theorems, symmetric groups, group actions, the Sylow theorems, and the structure of finitely generated abelian groups.

Exams: There will be a midterm and there will be a final exam. The midterm will be Thursday, Oct 16, in class. According to the projected exam schedule, the final will be on Thursday, Dec 18. List of dates:

• September 12: Last day to add/drop via SSOL
• October 7: Drop date
• October 16: Midterm
• November 3--4: Election break
• November 13: Course withdrawal and pass/fail deadline
• November 26--28: Thanksgiving break
• December 8: Last day of class, last day to choose pass/fail
• December 18: Final exam

Grading: Grades will be computed using scores on weekly problem sets, a midterm, and a final exam. The course grade will be computed as follows:

• The final exam will be worth about 40%.
• The midterm will be worth about 20%.
• The remaining 40% will be from weekly problem sets.

Lectures and notes: It is very important to be present during the lectures and to take notes! The material covered in the lectures is the material for the course. From the book by Serre below, we have roughly covered:

• Sections 1.1, 1.2, 1.3, 1.4, 1.5
• Sections 2.1, 2.2 (not the corolaries), 2.3, 2.4 (we used 1/g times the formula for rho(f) in Proposition 6), 2.6
• Sections 3.1, 3.2, 3.3
• We've discussed parts of Chapter 5, especially Sections 5.1, 5.3, 5.7, 5.8.
• Section 6.1, 6.2, 6.3, 6.5.
• Exercise 6.8 (statement only), Exercise 6.9, exercise 6.10, and exercise 8.6 (i) and (ii)
• Sections 7.1, 7.2, 7.3, 7.4.
• Theorem 1.4 from Notes on Induced Representations II
• Sections 8.1.
• Jordan's theorem and its proof; see links in "Additional topics" below.
• Invariant rings of a finite group G acting on the symmetric algebra of a representation V of G. In particular the characterization of complex reflection groups as those finite G ⊂ GL(V) whose invariant ring is a polynomial algebra. For reading see "Additional topics" below.

Also, I've started a pdf file with a list of material I intend to cover: representations-of-finite-groups.pdf. I will update this file as the semester goes along.

Problem sets: It is very important to work on the problem sets to keep up with the course. The problem sets are due either in class on Thursday or by email to Ryan by midnight of the due day. No late homework accepted. Just hand in whatever part of the work you were able to complete by the deadline!

1. First problem set due Thursday, September 11: problem-set-1.pdf. If a lot of people have trouble with this set, then we'll adjust the level of abstraction and/or difficulty in future sets. Please let me and Ryan know if you are having trouble and which parts you have trouble with!
2. Second problem set due Thursday, September 18: problem-set-2.pdf.
3. Third problem set due Thursday, September 25: problem-set-3.pdf.
4. Fourth problem set due Thursday, October 2: problem-set-4.pdf.
5. Fifth problem set due Thursday, October 9: problem-set-5.pdf.
6. Sixth problem set due Thursday, October 16: problem-set-6.pdf.
7. Seventh problem set due Thursday, October 23: problem-set-7.pdf.
8. Eigth problem set due Thursday, October 29: problem-set-8.pdf.
9. Ninth problem set due Thursday, November 6: problem-set-9.pdf.

Material: Online and offline texts to use:

• Two good books:
  1. J.-P. Serre, Linear representations of finite groups. This book is available through Spring Link for Columbia students. It covers a different set of topics, but Part I is a very good and succinct introduction to most of the material in the course.
  2. G. Lehrer and D. Taylor, Unitary reflection groups. This has a wealth of information on what we will call complex reflection groups. We will only discuss a tiny part of this, namely, some of the material from Chapters 3 and 4.
• Additional topics.
  1. Reading on Jordan's theorem. The proof I gave is taken from Terry Tao's page. It is very close to the (more detailed) proof of Theorem A in Robinson and very close to the discussion in Section 2 of Breuillard and Green. There is an interesting paper by Breulliard discussing the original proof of Jordan which has a discussion of the history of proofs of Jordan's theorem in the introduction, see Breulliard.
  2. Reading on invariant rings. I have not yet found a really good exposition along the lines of what I will talk about in class. You can take a look at Section 1 of this foundational paper by Cohen. There are also these course notes by Heckman where in section 3 you can find material very close to what I am talking about but his reflection groups are real reflection groups (so he is working over the real numbers occasionally). If you find an online text you like about this, let me know and I will put it here.
• Course webpages and course notes
  1. Michael Harris webpage for his version of 4044.
  2. Mikhail Khovanov webpage for his version of 4044.
  3. Robert Friedman webpage for his version of 4044. Here are the notes from Professor Friedman's course: Notes on Linear Algebra, Notes on Inner Products, Notes on Group Theory, Notes on Representations, Notes on Characters I, Notes on Permutation Representations, Notes on Characters II, Notes on the Fourier Transform, Notes on Characters III, Notes on Induced Representations I, Notes on Induced Representations II, Notes on Real Representations, Notes on Representations of the Symmetric Group, and Notes on Representations of GL_2.
  4. Drew Armstrong's year-long course in reflection groups at the University of Miami. His course notes are available here and here.
• Online textbooks (there are many others). Unfortunately, I have not found one that uses exactly the same language as I will be using.
  1. P. Webb, Representation Theory Book
  2. A. Baker, Representations of finite groups
  3. A.N. Sengupta, Representations of algebras and finite groups: An Introduction
  4. D.M. Jackson, Notes on the representation theory of finite groups
  5. K. Christianson, Representations of Finite Groups Course Notes
• For more online resources, see the end of this older webpage of Mikhail Khovanov.
• Other textbooks on the subject:
  1. B. Sagan, The symmetric group.
  2. B. Simon, Representations of finite and compact groups.
  3. G.James and M.Liebeck, Representations and characters of groups.
• I have a number of books on this topic myself in my office. If you come by, I will happily lend you one.