A learning roadmap for Representation Theory - MathOverflow

A learning roadmap for Representation Theory

2009-10-27T03:28:36Z

As Akhil had great success with his question, I'm going to ask one in a similar vein. So representation theory has kind of an intimidating feel to it for an outsider. Say someone is familiar with algebraic geometry enough to care about things like G-bundles, and wants to talk about vector bundles with structure group G, and so needs to know representation theory, but wants to do it as geometrically as possible.

So, in addition to the algebraic geometry, lets assume some familiarity with representations of finite groups (particularly symmetric groups) going forward. What path should be taken to learn some serious representation theory?

Answer by Andy Putman for A learning roadmap for Representation Theory

2009-10-27T03:38:44Z

The best "first course" in representation theory is Fulton and Harris's book. I've only skimmed it, but Ginzburg and Chriss's book "Representation Theory and Complex Geometry" looks like a wonderful second course.

Answer by Dinakar Muthiah for A learning roadmap for Representation Theory

2009-10-27T04:12:28Z

I really like "Lie Groups and Lie Algebras" by Kirillov Jr. It's available here for free. It covers a lot of material in a relatively short book, so I recommend it if you're trying to get a good overview of what Lie theory is about.

Fulton and Harris is all right, but I found the book to be too drawn out for its own good. They have a lot of worked out examples, which are good to look at after you've learned the material elsewhere, but they are difficult to follow if you've never seen them before.

I second Ginzburg and Chriss. It's full of tons of interesting stuff.

Answer by Ben Webster for A learning roadmap for Representation Theory

2009-10-27T04:18:07Z

I second the suggestion of Fulton and Harris. It's a funny book, and definitely you want to keep going after you finish it, but it's a good introduction to the basic ideas.

You specifically might be happier reading a book on algebraic groups.

While I third the suggestion of Ginzburg and Chriss, I wouldn't call it a "second course." Maybe if what you really wanted to do was serious, Russian-style geometric representation theory, but otherwise you might want to try something a little less focused, like Knapp's "Lie Groups Beyond an Introduction."

If you want Langlandsy stuff, then Ginzburg and Chriss is actually a bit of a tangent; good enrichment, but not directly what you want, since it skips over all the good stuff with D-modules. Look in the background reading for the graduate student seminar we're having in Boston this year: http://www.math.harvard.edu/~gaitsgde/grad_2009/

Answer by Fran Burstall for A learning roadmap for Representation Theory

2009-10-27T23:49:04Z

Sounds like Goodman and Wallach's Representations and invariants of the classical groups might be worth looking at. It has a rather algebro-geometric perspective, is beautifully written and goes further than many texts: for example, both the analytic and cohomological proofs of the Weyl character formula; branching laws; spinors and more.

Answer by Jon Yard for A learning roadmap for Representation Theory

2009-10-28T05:01:54Z

My favorite book right now on representation theory is Claudio Procesi's Lie groups: an approach through invariants and representations. It is one of those rare books that manages to be just about as formal as needed without being overburdened by excessive pedantry. He gives a rather complete picture of both compact and algebraic groups and how they interplay, while doing a nice job of explaining the necessary background in algebraic geometry and functional analysis. He covers all the "standard" material on Young symmetrizers, Schur duality, representations of GL_n, semisimple Lie groups & algebras, as well as more advanced stuff like Schubert calculus and some basic geometric invariant theory. This book was the first place I started to feel like I was "getting" the big picture, after picking up bits and pieces from different places.

If your institution has a subscription to SpringerLink, you can probably download this book for free (legally) and purchase an on-demand print version for around $25 USD.

Since this question was about a "learning roadmap," and not just for a single textbook, let me mention my favorite book that fits in your back pocket: "Lectures on Lie groups and Lie algebras" by Carter, Segal and Macdonald. The section by Segal is especially nice.

Answer by Chirag Lakhani for A learning roadmap for Representation Theory

2010-03-13T19:17:08Z

I ran across an excellent book by Lakshmibai and Brown called Flag Varieties: an Interplay of Geometry, Combinatorics, and Representation Theory. It seems like an excellent book for an algebraic geometer who is interested in representation theory and algebraic groups.

Answer by Chuck Hague for A learning roadmap for Representation Theory

2010-03-14T18:28:15Z

All of these recommendations are very good, and I'd like to add that the book D-Modules, Perverse Sheaves, and Representation Theory (which you can download at the provided link if you have institutional access; otherwise you can get it from, say, Amazon) contains some very good introductory chapters (chapters 9, 10, and 11) on the various sorts of things one would want to know in representation theory and algebraic geometry. The whole book is quite good if you're interested in the D-modules/perverse sheaves side of the story, but even if you're not interested in that, those particular chapters might be of interest.