Welcome to Math 681, Representation Theory. The syllabus is here.
This website will be used to make announcements, post homework assignments, etcetera.
There will be homework assignments, to help you acquire the main tools in the class. I will try to assign homeworks on Wednesdays and one should aim to hand things in on Wednesdays. To accommodate grad student schedules, I will only require some of the homework to be handed in, and not necessarily on time, but completing some of the homework will be required, and completing it on time is the best way to stay on top of things.
This course is a standard year-long graduate course in representation theory. It assumes basic knowledge of algebra: modules, linear algebra, representation theory of finite groups. Our main textbooks for the course will be:
Introduction to Lie Algebras and Representation Theory, by James Humphreys.
Lie Groups, by Daniel Bump.
Lie Groups, Lie algebras, and Representations, by Brian Hall. arXiv
It has come to my attention that Brian Hall has now expanded his notes into a new textbook. This is a different exposition than the arXiv link above.
Also, Mikhail Khovanov taught an excellent course at Columbia on Lie Groups, and beautiful notes were taken by You Qi, whose handwriting should be a font. Here is a link to the Khovanov notes.
If you need assistance purchasing any textbooks, please let me know.
We will begin by studying Lie groups, and segue to studying Lie algebras. While the representation theory of (semisimple) Lie algebras is the main focus of the course, and essentially governs the representation theory of Lie groups, I believe it will be better to begin with groups and pass to algebras, lest groups be underemphasized. The textbook by Brian Hall is a fairly gentle and thoughtful introduction to the theory, and is more readable than the other books. The textbook by Bump is a far more thorough resource on the topic, though it is a difficult book to read from the start. The textbook by Humphreys is excellent but focuses entirely on the Lie algebra side of the story.
Once the representation theory of Lie groups is understood, we will transition to another large and interesting story: Coxeter groups and their representation theory, as well as the special features of the representation theory of symmetric groups.
After this standard curriculum is complete, there are a large number of topics to pursue, and our true path will be determined later.
In the beginning of the class, I will begin with a side topic, the McKay Correspondence, which will serve as a reminder of the representation theory of finite groups, but with a very interesting new flavor.
On this website, I will post exercises and suggested reading. I should also mention this excellent list of links collected by Mikhail Khovanov.
Topic I: the McKay correspondence.
Suggested reading: You Qi has taken some beautiful notes from a presentation of the McKay correspondence in a class given by Mikhail Khovanov. A slightly more advanced presentation can be found in Chapter 1 of the Khovanov notes above.
My lecture notes are posted here.
Here is the second sheet of exercises. For time reasons I did a crappy job on the profinite section (exercises 10, 11) and will try to improve on it soon, so maybe wait on that.
For week 2 the assignment is:
From the the McKay exercises:
Important: 9, 14.
Additional: 10, 11, 12.
From the second exercise sheet:
Important: 1, 2, 3, 7, 8, 9 (do four of them, the rest will be on the following week).
Additional: 4, 6, 10, 11.
Topic II: Topological groups and Lie groups.
Reading: Chapter 2 of the Hall notes.
My notes on topological groups are posted here. I still haven't corrected a few errors.
My notes on manifolds and computing tangent spaces are posted here.
Here is the third sheet of exercises.
For week three the assignment is:
Any of the important exercises from last week you didn't do.
Additional: 5, 6.
Topic III: Compactness, integration, Schur orthogonality, and the Peter-Weyl theorem.
Reading: Chapter 1, 2, 4 of Bump.
Here is the fourth sheet of exercises.
For week four the assignment is:
Important: 3, 6, 8, 12.
Warmup: 1, 2, 5.
Additional: 4, 7, 9, 10, 11.
Topic IV: Matrix exponentials and lie algebras.
Reading: Chapter 3 of Hall up through 3.7, Chapter 5 of Bump. Then Chapter 4 of Hall (skim) for the BCH formula.
Notes are here and here.
Crappily written notes on lie algebra basics and complexification are here.
Finally, notes on how to read Bump Chapters 6-8 are here.
Here is the fifth sheet of exercises. In addition, here is one more exercise:
a) Show that diagonalizable matrices are dense in all matrices.
b) Show that diagonalizable invertible matrices are dense in invertible matrices.
c) Show that any diagonal matrix is in the image of the exponential map.
d) Show that any upper triangular matrix with ones on the diagonal is in the image of the exponential map.
e) Show that any invertible matrix is in the image of the exponential map. (Hint: Look up the multiplicative Jordan-Chevalley decomposition.)
Here is the sixth sheet of exercises.
Important: 1, 3, 5, 7, 11, 13.
Warmup: 2, 9, 12.
Additional: 4, 6, 8, 10(harder).
Topic V: Representations of sl_2. Interwoven with:
Topic VI: General theory of representations of lie algebras.
My notes are posted here and here.
Reading: Hall Chapter 5.3, 5.4 for reps of sl_2, 5.6 for tensor products, 5.8 for SO(3).
Bump Ch 12 very good for sl_2, Casimir element. Chapter 10 for universal enveloping algebra, Chapter 9 for tensor products etcetera.
Humphreys Chapter 7 for sl_2, Chapter 17 for PBW theorem.
Bergman, "The diamond lemma for ring theory" was the source of the Bergman diamond lemma. No need to read it unless you want it.
Hopf algebras: don't have a source in mind, unfortunately. The internet is good though. If you find a favorite source, suggest it.
Here is an exercise sheet.
Here is another.
Topic VII: Representations of sl_3, towards the general case.
Reading: Hall Chapter 6.
My notes are here.
Topic VIII: More on Lie algebras: nilpotent, solvable, simple, semisimple, Killing form, Casimir element, complete reducibility, Jordan form.
Reading: Humphreys Chapter 1-6.
Notes are posted. These are the notes through the next topic too.
Here is the first exercise sheet of quarter 2.
Here is the second exercise sheet of quarter 2. It scanned terribly, I will rescan on monday. Should be sufficient for most purposes.
Here is the third exercise sheet of quarter 2.
Topic IX: Semisimple theory, root systems.
Reading: Humphreys Chapter 8-13. And the Khovanov notes for supplemental stuff on Weyl groups.
My notes were posted above, and more are here.
The fourth homework is: every exercise in Humphreys Ch 8 (1 is a warmup) - there are many but several are easy. When considering classical lie algebras (type ABCD), you are welcome to use my description from class rather than the one in Humphreys' book.
The fifth homework is here.
Topic X: Serre's theorem.
Reading: Khovanov's notes.
The previous notes pdf covers this topic.
Topic XI: Finite dimensional representations, Verma modules, characters, Weyl character and dimension formulas, Steinberg tensor product formula.
Reading: Humphreys chapter 20-24, Khovanov's notes.
Notes are here and here.
The sixth homework is here.
The seventh is here.
Topic XII: The Harish-Chandra isomorphism, category O, projective functors, and the BGG resolution.
Reading: Humphreys Chapter 23, and Humphreys book on category O, chapter 1 and 6.1-6.5.
Notes are here.
The eighth homework is not here. There was no week 8 homework... I was off by a week, so it should have been week 9 anyway. But I forgot, so it is now part of week 10.
The last homework is here.
Topic XIII: Projective functors, translation functors, projectives in category O, and the Kazhdan-Lusztig conjectures.
Reading: Humphreys book "Representations of Semisimple Lie Algebras in the BGG Category O", chapter 7 and 8.1 through 8.4.
Also, watching lecture 1 of this workshop or this workshop will help.
Lecture notes will be posted soon. Perhaps even homework, one day.
Topic XIV: The double centralizer theorem, and Schur-Weyl duality.
Reading: a reasonable reference is in Chapter 4 of Etingof's representation theory notes .
Lecture notes will be posted soon.
Topic XV: Representations of the symmetric group.
This is a big one. Several parts.
Part a: overview. Lecture notes posted soon.
Part b: Okounkov-Vershik approach, Young-Jucys-Murphy operators. Reading: this article.
Part c: Standard approach. Reading: most likely Fulton-Harris.