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So my question is pretty much summed up by the summary - basically I've run into a need to teach myself some of the basics of invariant theory and was looking for a good place to start. I'd prefer online / freeish resources if anyone knows of any that can be trusted - otherwise if someone can recommend a good introductory book on the subject that would be most appreciated.

(Currently considering getting or perhaps convincing my library to order for me P. Olivers Classical Invariant Theory - but I thought I'd ask around first).

As a second question - two in one if you will - and perhaps this will help to let you know how little I know about invariant theory - can anyone quickly summarize the difference between Classical and Geometric Invariant Theory? Presently I am thinking what I need is the classical kind - but again more information is always helpful.

P.S. Geometric invariant theory may not have been the best tag for this question, but apparently new users can't create new tags, so I picked the closest one - anyone that has the power to change the tag to something more appropriate please do so.

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The book Algorithms in Invariant Theory of Bernd Sturmfels should be a nice introduction. –  Dung Duong Nov 20 '12 at 11:37

8 Answers 8

up vote 5 down vote accepted

Usual invariant theory is dedicated to studying rings; a good example of a result from classical invariant theory is that the ring of invariant polynomials on any representation of a reductive group is finitely generated.

Geometric invariant theory is about constructing and studying the properties of certain kinds of quotients; a good example would be the moduli space of semi-stable vector bundles on an algebraic variety.

In my mind, the difference is this: Classical invariant theory is a collection of results about the interaction between group actions and commutative algebra. Geometric invariant theory is a technique for constructing interesting spaces.

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I think one of the best books for GIT is the book by Mumford himself, Mumford - "Geometric Invariant Theory". It is very interesting, starts off with some things including a section about strata, and then in the core of the book has two constructions to get these quotients. If you are interested in homology of the spaces constructed in GIT, maybe the following paper by Woolf is very helpful: arxiv.org/PS_cache/math/pdf/0110/0110137v1.pdf . –  Vinoth Dec 20 '09 at 6:41

I can also recommend Dolgachev's book. It used be freely available on his home page, but now it is published. It treats both some of the classical invariant theory and GIT.

The difference between the two is the following. In classical IT you are interested in finding the invariants of a ring under the action of a group. The prototypical example is the description of the algebra of symmetric polynomials as the polynomial algebra on elementary symmetric polynomials.

In GIT you do the following. Let $A$ be a ring, maybe the function ring of some affine algebraic variety, with an action of $G$. Then $A^G$ should be the ring of functions on the quotient of your variety by $G$. So invariant theory is viewed as a tool to describe the function rings of quotient varieties. The problem is that not all varieties are affine, and so GIT goes on to study what does it mean to take the quotient by the action of a group of a more general variety (or scheme), tipically in the projective case. One of the main difference is that it turns out that there are bad points that you have to discard altogether before taking a quotient.

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With some algebraic geometry background, the classic book by Mumford (and helpers) and the recent informal account by Dolgachev are both useful for relative beginners. For the classical-groups-and-rings treatment using algebra, books by Goodman-Wallach and Procesi I cited in another comment are thorough and readable. –  Jim Humphreys Jul 20 '10 at 19:50

Well, since no-one has had an answer yet, I thought I'd provide a possible answer to myself. I did find one pdf which seems to be a reasonable introduction to the theory online and appears to be freely available at this url: http://www.math.unibas.ch/~kraft/Papers/KP-Primer.pdf

Again - I'm clearly not an expert on this subject and have only started reading up on it, so if someone knows a better reference than this it's appreciated.

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Aside articles by Kraft, Dieudonné-Carrol's "Invariant Theory old and new" is nice to read. –  Thomas Riepe Dec 20 '09 at 12:14
There is no better introductory reference! –  Q.Q.J. Feb 27 '10 at 12:15

I think this book by Goodman&Wallach, "Representations and Invariants for Classical Groups", (you can find a google preview if you wish) is very good. Only a few chapters in this book are 100% relevant to the topic, but if you focus on them you should get a good basic picture.

Else if you are interested in invariant theory for unipotent groups (often if you try computing invariant theory for direct sums of representations of reductive groups, you will end up with something involving unipotent groups as well, for instance), there is a good paper, (it is somewhat algorithmic though) : http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=57495 .

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For invariant theory (but not GIT), two good modern texts emphasizing classical groups among other Lie groups are (1) MR2522486. Goodman, Roe (1-RTG); Wallach, Nolan R. (1-UCSD), Symmetry, representations, and invariants. Graduate Texts in Mathematics, 255. Springer, Dordrecht, 2009. (Second edition of book mentioned.) (2) MR2265844. Procesi, Claudio (I-ROME), Lie groups. An approach through invariants and representations. Universitext. Springer, New York, 2007. –  Jim Humphreys Jul 20 '10 at 19:44

Maybe you would enjoy Olver: Classical Invariant Theory.

Here is the Google Books preview: http://tiny.cc/ntpgK

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In addition to the already mentioned

Procesi, Lie groups. An approach through invariants and representations

which provides an excellent update on both Weyl's "Classical groups. Their invariants and representations" and the relevant parts of Hodge and Pedoe's "Methods of algebraic geometry", I'd like to mention an amazing book by Kraft, unortunately not translated into English yet (there is a Russian translation):

Kraft, Geometrische Methoden in der Invariantentheorie

For a bird's eye view, I recommend the following survey in Russian (Yellow Springer) Math Encyclopaedia:

Vinberg and Popov, Invariant theory, Algebraic geometry IV

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Streklin has already provided a link to the best introductory reference, but two more worth noting are

  • some classical invariant theory material in an appendix (E or F?) at the end of Fulton and Harris.

  • "Lie groups: an approach through invariants and representations" by Procesi also contains a wealth of knowledge. I believe this is available on Springerlink if you have access.

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In addition to the mentioned very good references there is also the book by Shigeru Mukai "An introduction to invariants and moduli" Cambridge University Press, Cambridge, 2003. ISBN: 0-521-80906-1.

Also, there is not much difference between classical invariant theory (as done by the classics themselves) and GIT, they both are about studying quotients in algebraic geometry. If you need to work with explicit coordinates on your quotient then you are doing CIT, if not you are doing GIT.

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I disagree about "there is no difference": even a cursory look at Grace and Young, or another comparable classical book shows that the "classics", if one can indeed lump together so many different strands, were mostly interested in combinatorial and algebraic aspects (e.g. explicit generators and relations for invariants and covariants). If anything, classical invariant theory that I am familiar with is closer to modern representation theory than to algebraic geometry. –  Victor Protsak Jul 20 '10 at 22:19
@Victor: Not many moderns read Grace and Young and just for that your comment deserves an up vote! However GY is not enough to get the complete picture of CIT. Some of the key players like Clebsch, Cayley, Sylvester,...were as much algebraic geometers as one can get. In fact, CIT as practiced by the classics is inseparable from elimination theory. Maybe a good way to see this would be to look at the two books by Faa di Bruno available on google books: the one on binary forms, and the one on elimiation theory. See also Gordan's papers on symbolic forms of resultants. –  Abdelmalek Abdesselam Jul 21 '10 at 0:01

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