Math 668: Combinatorics of GLn representation theory

Professor: David E Speyer

Fall 2022


Course meets: Monday, Wednesday, Friday 11-12; 4088 East Hall

Office Hours: I will have office hours just for this class on Wednesdays, 9:30-10:50 AM in my office, East Hall 2844. I also have office hours scheduled for Math 214 on Thursday 1:00-2:20 PM (Zoom) and Friday 9:30-10:50 AM (East Hall 2844). You may come during those times as well, but you may find a crowd.

Webpage: http://www.math.lsa.umich.edu/~speyer/668

Level: Graduate students who are comfortable with abstract linear algebra (vector spaces, tensor products, symmetric and wedge product), who have some familiarity with groups and representation theory, and a high level of mathematical maturity. There will be a few manifolds and integrals. Towards the end of the term, I will start using the language of categories.

Anticipated topics: We will definitely cover: Classical combinatorics of symmetric polynomials, Young tableaux, generalities on representation theory of compact and of reductive groups, construction of the irreducible representations of GLn. We will likely cover: RSK, jeau de tacquin, crystals, the Littlewood-Richardson rule. Additional topics I'd like to fit in as possible: Representation theory of GLn, webs, standard monomial theory, connections to cluster algebras and total positivity, connections to quiver representation theory, honeycombs, puzzles.

I previously taught this course in 2012; here is what I covered then.

Student work expected: I will give problem sets every week, due Wednesday evenings on Gradescope. I will also require students to take turns serving as scribe for the course, meaning taking TeXed notes on what we have covered that day. Finally, I will require you all to either write an expository 10-15 page paper, or to prepare a 30-50 minute talk on some subject in combinatorial representation theory that interests you.
Here is a list of possible topics and references; and I am glad to discuss other options. You can satisfy this last requirement by giving a talk on an appropriate topic in the student combinatorics seminar. If you are interested in speaking, please contact the organizers, Will Dana (willdana@umich.edu) and Scott Neville (nevilles@umich.edu). The student combinatorics seminar meets Mondays at 4 PM in East Hall 3866. I encourage you to attend the student combinatorics seminar, and the regular combinatorics seminar, in general.

Problem Sets


Homework Policy: You are welcome to consult each other provided (1) you list all people and sources who aided you, or whom you aided and (2) you write-up the solutions independently, in your own language. If you seek help from mathematicians/math students outside the course, you should be seeking general advice, not specific solutions, and must disclose this help. I am, of course, glad to provide help!

I don't intend for you to need to consult books and papers outside your notes. If you do consult such, you should be looking for better/other understanding of the definitions and concepts, not solutions to the problems.

You MAY NOT post homework problems to internet fora seeking solutions. Although I participate in some such fora, I feel that they have a major tendency to be too explicit in their help; you can read further thoughts of mine here. You may post questions asking for clarifications and alternate perspectives on concepts and results we have covered.

All problem sets should be turned in through Gradescope. You should have gotten a notification that you were enrolled in the Gradescope course; if you didn't, please write me.

Course Notes

With your aid, I will be compiling a set of notes for the course in the Course Notes file.

All students will be required to take turns scribing notes for this file. When it is your turn to scribe, download the template file and write in a summary of what happened in class that day. Then e-mail it to me. The deadline for editing the notes is 24 hours after the lecture. I will, in turn, proofread and edit your entries in the next 24 hours and post them back to this webpage, so that the class always has a good record of what we have covered. You are welcome to download and read the source of the notes but please do your writing in the template file; my experience is that it is easier for me to resolve merge conflicts when I copy your text into the master file than if you edit the master file directly.

If you do not know LaTeX, you should learn! I can suggest sources; I also find TeX.stackexchange incredibly useful for specific questions.

If you have forgetten when you are scheduled to write the notes, you can check here.

In the week of October 3-7, we took a break from lectures and solved problems about the basics of finite dimensional representation theory. The problems can be found here. We also had an in class problem solving session on the "unitary trick" on October 21; those problems can be found here.