Customer Reviews

Linear Representations of Finite Groups (Graduate Texts in Mathematics) (v. 42)

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This is a an excellent introduction to the subject. The book really breaks into 3 distinct parts. The first 5 chapters are a rapid introduction to the basics, similar to what one would get from any indroductory text. They are most notable for actually going through the details on D_n, S_n cyclic groups... The second section (chapters 6-13) gives a more graduate level presentation of the material. Starting with a discussion of group algebras, moving onto inducted representations Artin's theorem (the existence of virtual characters) The third section is Brauer Theory. The book is by Serre so it goes without saying it one of the best if not the best book on the market. His failure to deal with the additional complexities of the infinite group case (which he indicates in the title) is a small problem. He could have spent at least 1 chapter addressing how the results of the book could be extended. The index of notation is a fantastic asset for a subject where notation plays such a large role.

Jean-Pierre Serre is a famous fellow: that is why I bought his book.
I have as contrast a book by Willem Brouwer and one by Coexter which are clean
concise and useful. If you can get the Coexter and Moser classic:Generators and Relations for Discrete Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge) ,also found under ISBN-10: 0387092129.
Both versions are out of print.Representation Theory of Finite Groups has the virtue of being cheap and available and somewhat more readable than the Serre book. The Brouwer book of tables is a Rice university press book from the library without a ISBN and isn't listed at amazon.
The French method of presenting Math goes back to the Nicolas Bourbaki books that were written by groups
of people and are "formally correct", but just not how you want to teach math.
Unless you already have a PH.D. in group theory, try something cheaper and easier to start.
There are a lot of books on the theory of representations of groups, but none are very easy.

I'm using this book as an undergraduate, so my rating is clearly skewed, as evidenced by the huge "Graduate Texts in Mathematics" on the cover. We've only covered the first five chapters so far, and while the overarching ideas are quite clear, I find the notation confusing. No (even small) reviews of the linear algebra you studied years ago; it just dives in. Perhaps graduate students can follow it all quickly with no concrete examples, but it takes me a few readings through each section to begin to understand what is being said. By concrete I mean a real-world see-it put-your-hands-on-it example, or at least an example involving a few numbers as elements.

Here is an excerpt (so you can judge for yourself how helpful the first chapter will be for you) of a representation example from section 1.2 entitled 'Basic Examples:' "Leg g be the order of G, and let V be a vector space of dimension g, with a basis (e-sub-t)sub-t-in-g indexed by the elements t of G. For s-in-G, let rho-sub-s be the linear map of V into V which sends e-sub-t to e-sub-st; this defines a linear representation, which is called the regular representation of G. Its degree is equal to the order of G. Note that e-sub-s = rho-sub-s(e-sub1); hence note that the images of e-sub1 form a basis of V. Conversely, let W be a representation of G containing a vector w such that the rho-sub-s(w), s-in-G, form a basis of W; then W is isomorphic to the regular representation (an isomorphism tau: V --> W is defined by putting tau(e-sub-s) = rho-sub-s(w))."

The language is very concise and usually quite clear, and I suppose for someone with a sophisticated math background it could be a preferred book. For someone like me who has had only one semester of introductory linear algebra two years ago, I would prefer a more "bridging" text -- that is, one which often and quickly reviewed basic concepts from linear algebra and was less concise in its explanations of definitions and examples.