# A second course in the representation theory

I've read Etingof's and then Fulton-Harris' books about the representation theory ("Intrdouction to representation theory" and "Representation theory. A first course" respectively) and found their subject very exciting!

Can someone, please, recommend me a textbook containing a kind of the "second course" in this branch of the mathematics? (It'd be great if someone recommended a "russian-style" book.)

(I heard a lot about the "Representation theory and complex geometry" written by Chriss and Ginzburg. Is it really so wonderful? And what other good references do you know?)

I'm going to learn the advanced representation theory for its own sake. But I'll be glad to see interesting intersections with the algebraic or complex geometry!

UPD: User Vincent's answer was really great! But it was only about the Lie groups. I'm also interested in such topics as Kac-Moody algebras, (double affine) Hecke algebras, category $$\mathcal O$$, Soergel (bi)modules, quantum groups and other things like those (maybe, quivers)... Can some of them be a part of the second (not third, fourth, etc.) course? And are there (less or more) introductory textbooks covering a part of this material?

• If you can say what is your purpose, people can help... what do you want to learn after reading some advanced representation theory? DO you want to use it for some thing or just learn it for its own sake? – Praphulla Koushik Aug 20 at 10:49
• There are some good suggestions in the first question appearing in the sidebar ('learning roadmap') – Vincent Aug 20 at 11:24
• @Vincent The problem with this question is that in many answers the first and the second courses aren't differentiated. – Richard Hopkins Aug 20 at 11:30
• I don't know if it is because of our comments, but the other question now disappeared from the sidebar! – Vincent Aug 20 at 11:37
• You question is already answered, doc: mathoverflow.net/questions/2755/… – Bugs Bunny Aug 20 at 12:19

The best textbook that covers a wide range of subjects is for me "A Tour of Representation Theory " by Lorenz. Of course it is close to a first course but I would say a little more advanced than the two books that you mentioned. Another similar one is "A Journey Through Representation Theory: From Finite Groups to Quivers via Algebras" by Gruson and Serganova.

Here is a small collection in specific directions:

Modern (modular) group representation theory: "The Block Theory of Finite Group Algebras" Volume 1 and 2 by Linckelmann .

Representation theory of finite dimensional algebras with some indepth look into some special classes such as hereditary, Hecke or Hopf algebras: "Frobenius Algebras " part 1 and 2 (3 will come soon) by Skowronski and Yamagata.

Representation theory of the symmetric group (including modular representation theory): "The Representation Theory of the Symmetric Group" by James and Kerber.

Homological methods in representation theory (mostly with application for group algebras): "Representation Theory: A Homological Algebra Point of View " by Zimmermann.

For an application of representation theory (of quivers) to data science: "Persistence Theory: From Quiver Representations to Data Analysis " by Oudot

I think when it comes to second courses, the question OF WHAT you take your representations becomes more relevant. I will say something about representations of Lie groups because that is where I ended up, but sometimes I had the feeling that people working on representations of other stuff (quantum groups, double affine Hecke algebras, braid groups, Kac-Moody algebras, Lie super algebras, etc etc) were more happy than I was. But perhaps that is only how they appear when other people are present, it is hard to tell.

Anyway, in the representation theory of real Lie groups, the thing that distinguishes the stuff for first and second courses (or put more opinion-based: distinguishing the most beautiful theory in the world from a more complicated, less intuitive and much, much more technical version of it) is a simple one: whether you work with finite dimensional or infinite dimensional representations.

When your group is compact all irreducible representations are finite dimensional and every representation decomposes as a direct sum of irreducible ones. Hence no need to look at anything infinite dimensional there. For non-compact groups however, even ones whose Lie algebra is semi-simple, none of these two statements is true anymore!

At the same time, of course, we would still like to know what are the irreducible representations and how do they combine into other, non-irreducible but still natural representations (e.g. spaces of functions on geometric objects whose symmetry group is a quotient of your Lie group) if not by simply taking their direct sum?

I think a good 'second course' type book on this kind of questions is 'Representation theory of real Lie groups, an overview based on examples' by Anthony Knapp (1986).

A specific problem (or family of problems) that is really beautiful in finite dimensions (and discussed very well in Fulton and Harris) and only partially understood (while still very beautiful) in the infinite dimensional case is the branching problem: given an irreducible representation of a big group $$G$$, what does it look like (i.e. what subrepresentations appear) when we we restrict the action of $$G$$ to a smaller subgroup $$G' \subset G$$?

There is really a big zoo of different types of answers to this depending on the nature of $$G$$ and $$G'$$. Someone who has written a lot about this (and really clearly at that) is Toshi Kobayashi. You can check his website to see if he has written any good introductory texts (I am too lazy to do it now) about the subject but I think the first few sections of this may qualify (the rest is probably more like 3rd-or-higher course material): https://link.springer.com/content/pdf/10.1007%2F978-3-319-23443-4_10.pdf (which is a chapter in this: https://link.springer.com/book/10.1007/978-3-319-23443-4)

• Thank you very much for your answer! I've updated my question according to it. – Richard Hopkins Aug 20 at 14:26
• Another book by Knapp that I like is Lie Groups: Beyond an Introduction. – Timothy Chow Aug 20 at 21:53
• O yes I really like that one too, perhaps even more than the other one. But, as the title says, it is more about Lie groups and less about representation theory. – Vincent Aug 20 at 22:35