Math 665: Combinatorics of GL representation theory

Professor: David E Speyer

Fall 2012


Course meets: Monday, Wednesday, Friday 11-12; 3096 East Hall

Office Hours: 2844 East Hall, by drop in or appointment. If there is demand for it, I will schedule a regular time.

Webpage: http://www.math.lsa.umich.edu/~speyer/665.html

Level: Graduate students who are very 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. Towards the end of the term, I may start using the language of categories.

Student work expected: I will give problem sets every week, due Mondays in class. I will also require students to take turns serving as scribe for the course, meaning taking TeXed notes on what I have said that day.

Scribing: I will require students to take turns signing up to record, in TeXed format, what I said in lecture. The due date for turning notes is one week after the lecture, sent to me by e-mail. I highly encourage you to be faster, as this will both be easier for you and more helpful for you fellow students. You may want to bring a laptop to class and try live TeXing (some useful tips here). I will edit these notes before posting them.

If you are scribing and are confused about some point from the lecture, please let me know so that I can work with you. Of course, this also applies if you are confused and are not scribing!

There are currently 13 students registered for a 40 lecture class, and I will scribe the first lecture. If all of you remain for the whole term, then you will each scribe 3 times. I expect a more likely estimate is that you will wind up scribing 4–5 times.

Please use this LaTeX template, which provides some convenient formatting and macros. I will add useful macros throughout the term as I think of them.

If you have never learned how to use LaTeX, then you really need to, as it is the standard tool of all mathematical publishing and will be crucial to anyone pursuing a career as a mathematician. Please alert me to your situation, and I will help you. A useful website for getting help with LaTeX difficulties is tex.stackexchange.com.

Note that these notes will be posted publicly online, in both TeX and PDF form, with your name attached to them, and that I may edit them to remove errors in my lectures or your understanding of them. You give me permission to do this, but otherwise retain copyright to your notes.


Problem Sets

I reserve the right to edit these problem sets up until the previous one is due, so you might not want to work too far ahead.

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 know of cases where such fora are valuable, and I participate in some, I feel that they have a major tendency to be too explicit in their help. You may post questions asking for clarifications and alternate perspectives on concepts and results we have covered.

Problem Set 1, due Sept 17
Problem Set 2, due Sept 24
Problem Set 3, due Oct 1
Problem Set 4, due Oct 8
Problem Set 5, due Oct 17 (Wednesday) because of Fall break
Problem Set 6, due Oct 22
Problem Set 7, due Nov 5 (two week problem set)
Problem Set 8, due Nov 12
Problem Set 9, due Nov 19
Problem Set 10, due Dec 3 (two week problem set)
Problem Set 11, due Dec 10. This problem set works you through a beautiful application of crystals.

Rough Syllabus

As the term goes on, this will get more precise.

Sept 5: Overview (TeX) Scribe: David Speyer

September: Introduction to symmetric polynomials

Here is a symmetric function cheat sheet.

Early October: Representation theory of compact groups

Mid October: Construction of the representations of GLn

Late October: Kevin Carde lectures on web diagrams

David is at MSRI this week. Kevin has graciously agreed to replace me.

Early November: Relations between the general linear group and the symmetric group

Mid November: Lie algebras (an extremely brief introduction)

November 23: THANKSGIVING.

Late November, December: Crystals