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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?

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7 Answers

up vote 18 down vote accepted

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/

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Well, right now, my interest is largely related to geometric Langlands type questions, so Ginzburg is a name I'm getting rather familiar with... –  Charles Siegel Oct 27 '09 at 4:22
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Unfortunately, the correct answer for how to learn a lot of this stuff is "hang out with the right Russians." –  Ben Webster Oct 27 '09 at 4:36
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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.

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Procesi looks rather good, from a quick skim of the contents. Does he do generalized Schubert calculus for homogeneous varieties, or just for the usual Grassmannian of linear subspaces? –  Charles Siegel Oct 28 '09 at 5:19
    
He does the usual Grassmannian, as well as the Bruhat decomposition in algebraic groups. –  Jon Yard Oct 28 '09 at 7:14
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I second Carter, Segal and Macdonald. A great little book. –  Dinakar Muthiah Oct 28 '09 at 14:46
    
I have just bought this and it is now my new favourite book on Lie theory too. It is utterly fabulous. A good place to see how classical invariant theory (Cayley, Capelli etc) fits into the modern picture. –  Fran Burstall Dec 13 '09 at 11:39
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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.

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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.

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Pretty much everyone I know speaks super-highly of Fulton and Harris, but I've always found the organization somewhat confusing, and the style hard to learn from. But perhaps there is really no other (textbook) source for some of the material. –  Akhil Mathew Oct 27 '09 at 23:04
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I agree with Akhil here. I never really enjoyed Fulton and Harris - the writing and organization just feels too "messy." On the other hand, the book does contain a fair amount of material and has many interesting comments and remarks scattered throughout the text. –  Faisal Oct 28 '09 at 16:26
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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.

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Oooh, thanks for letting me know about that. I've been looking for something that covers pretty much these exact topics. –  Charles Siegel Mar 14 '10 at 2:40
    
No problem. I really enjoyed their treatment of representations of algebraic groups. –  Chirag Lakhani Mar 14 '10 at 17:33
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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.

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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.

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I've been meaning to check that out...learned most of what I know about Lie groups from Goodman using Rossman's intro book while that one was undergoing revisions...I think the revised version got published as "Symmetry, Representations and Invariants" by Springer (math.rutgers.edu/~goodman/symmetry.html) –  Charles Siegel Oct 28 '09 at 2:14
    
I've only looked at the new revised version Charles mentions, I think it's fantastic! –  Grétar Amazeen Dec 10 '09 at 4:49
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