Welcome to Math 681, Representation Theory.
This website is primarily for posterity, and will eventually have the readings, exercises, lecture notes, etcetera for the course. However, day to day activities and stuff only of interest to folks at UO (announcements, regular updates of exercise files, syllabus, etcetera) will be managed from canvas. The syllabus is on Canvas.
This course the first term of a standard year-long graduate course in representation theory. It assumes basic knowledge of algebra: modules, linear algebra, representation theory of finite groups. This particular iteration is somewhat experimental. I am only teaching the first and third term, and the third term is meant to be partially a topics course on categorification and the Hecke algebra. When I taught the sequence previously, the first two terms of the course would cover representations of Lie groups and lie algebras with some additional topics, while the third term covers category O, representations of symmetric groups, Schur-Weyl duality, and web algebras. The third term will address many of the same topics, ultimately, but perhaps with a different spin, and (possibly) more accessibly to those who didn't take the whole sequence.
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.
After this, we will begin by studying (semisimple or compact) 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. This approach does involve more elementary topology (computing tangent spaces) and algebraic topology (thinking about paths and homotopies) than the purely algebraic analysis of Lie algebras. I expect that by the end of the first term, we will understand the representation theory of SL3 and its lie algebra, and will have prepared the ground for an in-depth study of semisimple lie algebras in the second quarter.
During the first term I expect to follow roughly the textbook by Brian C Hall, linked here. I think this textbook takes a unique path through the material, one which I feel is well-thought out, though it purposely limits its scope. Perhaps because of the popularity of this textbook, Hall was encouraged to expand it into a full fledged textbook, and this expansion lost the unique path and tight feeling of the original. It is very different, and I do not recommend it above the many other similar textbooks around. I'm not sure it is possible to buy a copy of the original book, but thankfully we have the arXiv link above.
We will supplement the Hall textbook with Lie Groups by Daniel Bump. This is a more thorough resource on the topic and is also quite nice, though more difficult to read from the start. Later on, we will reach the canonical textbook, Introduction to Lie Algebras and Representation Theory by Humphreys. This book is excellent but focuses entirely on the Lie algebra side of the story.
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 You Qi's webpage; item [5] is the course in question. We may also use other notes from this page.
If you need assistance purchasing any textbooks, please let me know.
I should also mention this excellent list of links collected by Mikhail Khovanov.