Customer Reviews

Representation Theory: A First Course (Graduate Texts in Mathematics / Readings in Mathematics)

5 star:		(3)
4 star:		(2)
3 star:		(2)
2 star:		(0)
1 star:		(0)

 

 

An excellent book. The approach, working toward the general theory via examples, has some great pedagogical virtues but also drawbacks. It also means the book has drawbacks as a reference, as important general theorems can be hard to locate (often they are in an appendix, but relevant definitions or lemmas are in several places in the text). Despite the example-oriented style, the level of mathematical sophistication assumed is reasonably high (so some physicists, for example, may find some of the explanations require boning up on certain ideas found more in pure mathematics than physics). However, many things are given very nice explanations that are lacking in some dryer texts (e.g. Varadarajan, or even Humphreys). Particularly nice is the discussion of relations between the representation theory of finite groups and Lie groups. Many mathematicians might find this book an enjoyable read to see connections made and examples worked out at a high level of sophistication, after learning the general theory. Some may also find it useful primarily as a repository of worked-out examples. I found Humphreys book "Introduction to Lie algebras and representation theory (Springer GTM series) to be an essential companion for getting the general theory with full proofs in a somewhat more logical order, if somewhat terse and a tad dry; Knapp's book "Lie groups beyond an introduction" could also serve this purpose, perhaps even somewhat better. If teaching a course, I would probably use this as supplemental reading rather than a primary text (though it could also turn out that gradually-generalizing-from-examples approach works better in a course than for self-teaching). It has been a useful book for me to own, and I recommend it, with the caution that you will probably want to supplement it with a book like Knapp's. (If you want to use only one book, and are reasonably mathematically sophisticated and already know basically what Lie groups and algebras are, use Knapp's.) I am a math-oriented physicist, who recently learned much of this material, using this and other books, in order to use it in my research.

This is an absolutely delightful introduction to the theory of Lie groups and their representations. The style is informal but informative, with some of the important proofs hidden in the appendex or even omitted (i.e. existance of the finite dimensional representations for all lie algebras). However, this is a fully rigorous text, and all the important theorems are stated, and most are proved. Mathematicians should suppliment this book with Humphries standard text on Lie algebras. However, this book provides motivation and intuitive insight that Humphries is missing. Additional enjoyment may be derived from the sampling of other unusual topics, such as Schur functors and applications to algebraic geometry. Of course, these can also be omitted as the reader desires. Read a lecture every few nights before bedtime, and soon Lie theory will seem beautiful and almost intuitive.

An excellent companion for anybody learning lie algebras or representation theory. Also good for physics folk needing to pick up more than the basics of lie algebras; a nice followup to a "lie algebras in physics" book (and there are many of those.)

In particular, some people really need to buy this book.

This is the book that seems to have everything. Written by two of the masters with such ambition as to cover representations of finite groups, representations of Lie algebras, together with countless detailed examples (and many pictures to boot!), what could go wrong?

Within the first few pages, though, you should begin to feel that something is amiss. Proofs and arguments are almost always incomplete. Details are never provided under any circumstances. Example computations are beautiful and swift, but usually rely on an understanding that is deeper than actually presented in the text, using lemmas not present anywhere in the entire volume. They are the sorts of computations which, if included on a homework assignment and graded by someone well-versed in the subject, would get at most half the marks with several copies of the comment "yes, but you need to explain why." Nearly half the subject, you will realize after close analysis, is just left to the reader. The authors also supplement the instruction with an annoying delusion that the entire book is trivial; they will repeatedly tell you that everything here is trivial, easy, or immediate, but they will never acknowledge anything as being hard. Not only is this of course wrong, but it's disrespectful to their brilliant predecessors who toiled day and night to bring to them these apparently trivial truths.

This is an exceptionally dangerous book to learn from. It's the sort of book that makes you think you understand the details when in fact you have no idea what you're talking about. It makes you think something's trivial or simple when it actually requires some clever thinking. Given the book's length, it is clear that the authors were simply too ambitious. One (or, evidently, two!) cannot cover this range of material in appropriate detail and with due care to the reader without violating all reasonable restrictions on how fat and bloated any single volume should permit itself to become before giving into gluttonous sin.

This book isn't all bad, though. It makes a decent reference due to its ambition. There are some nice pictures. And the methods of computation really are nice--just don't think you understand them if you haven't written pages of extra notes filling in the gaps.

Vinberg's Linear Representations of Groups is a much superior treatment of the basics of the subject. After using Fulton and Harris's book, you may be surprised to see how much more space it takes Vinberb to cover what Fulton and Harris annihilate in a few pages or even paragraphs here. And then you will realize how frail and weak the treatment of individual topics actually is in the present book.

This book is an excellent introduction to representation theory of finite groups, Lie groups and Lie algebras. It is easy to read, not too dense, contains many exercises, and spends a lot of time on examples before exposing the general theory. Probably my favorite intro to repn theory book.

I'm using this book as the text for one graduate course representation theory. This book is written in a modern fashion. Very good to take a survey of modern treatment of group representation. Futon and Harris use notations from category theory. At some place, they also use vector bundle. They assume readers have been familiar with those things. Means the author assume a high start point. Read Michael Artin's algebra as well as S. Lang's Algebra before you start this one would help a lot.

The book I received is very new - newer than I expected.

The authors seem pretty good on Young's diagrams,
but mostly as far as Cartan algebra, Lie algebra
and representation theory they are pretty clueless.
I spent way too much money buying this book
for it to be this useless as a self-teaching tool.
Since this is my 4th representation theory book
I have to say that these guys make
Jean-Pierre Serre's book Linear Representations of Finite Groups (Graduate Texts in Mathematics) (v. 42)
look better
and makes a hero out of James E. Humphreys Introduction to Lie Algebras and Representation Theory (Graduate Texts in Mathematics).
If you are buying a book by Fulton stick to algebraic geometry
or intersection theory, maybe?