Local Representation Theory: Modular Representations as an Introduction to the Local Representation Theory of Finite Groups

Cambridge Studies in Advanced Mathematics #11

by J.L. Alperin

Representation theory has applications to number theory, combinatorics and many areas of algebra. The aim of this text is to present some of the key results in the representation theory of finite groups. Professor Alperin concentrates on local representation theory, emphasizing module theory throughout. In this way many deep results can be obtained rather quickly. After two introductory chapters, the basic results of Green are proved, which in turn lead in due course to Brauer's First Main Theorem. A proof of the module form of Brauer's Second Main Theorem is then presented, followed by a discussion of Feit's work connecting maps and the Green correspondence. The work concludes with a treatment, new in part, of the Brauer-Dade theory. Exercises are provided at the end of most sections; the results of some are used later in the text.

Genres: Mathematics

192 pages, Hardcover

First published June 27, 1986

Reviews

[Steve Dalton · Rating 5 out of 5]

This is a great book on modular representation theory, focusing on the basics of the theory and how the (projective, indecomposable, simple, etc.) modules over a local subgroup correspond to similar modules over the whole group. The books main goal (and the unifying theme of the last chapter) is studying the projective indecomposables over a block with cyclic defect group, which is essentially the only well-understood case in block theory, aside from a few exceptional cases.

The book is organized in an excellent way, going from the general to the specific, with each result being built upon the ones from the previous section(s). The author also takes the time to explain what certain theorems are about, and why one might care; this was especially nice when the details became a little too much to process all at once. Right then, the author would include a paragraph or two describing the important case to pay attention to, or the general "idea" of the theorem and how it is used.

Many key topics are treated here, with surprising clarity and conciseness, and just the right level of sophistication: the reader still gets her hands dirty, but very rarely does she have to resort to mindless computation. The sections on the Wedderburn theorem and the Krull-Schmidt theorem are great, and the later sections on the Heller operator, the Green correspondence, and Brauer's Main Theorems are all very clear. The discussion in the last few sections on Brauer trees and graphs is also very insightful and well-done.

There are a few complaints though. I find the exposition get a little too involved at points, going through lines of calculation that are easy to gloss over (and I definitely did a few times), but sometimes those calculations are the only way to go. The notation also leaves a lot to be desired, but that's characteristic of representation theory as a whole it seems. My only real complaint - without a companion excuse - is that the exercises are a little bit all over the place: some are easy, some are difficult, some are interesting, and some seem very dull. Unfortunately, the only ones that get used later in the book are the dull, difficult ones.

When I was thinking how many stars to give this book (or any group theory book), I always compare to my favorite: Isaacs's Finite Group Theory. This is an unfair comparison for most books: FGT is clear and engaging, with complete proofs that somehow avoid messy details; the exposition flows naturally and many times I would find myself thinking "of course!" or "I see how to finish this proof"; and the exercises are very challenging, yet extremely rewarding. Alperin's book holds up on most fronts, except it is a little computational at points, but the explanations and writing style are truly top-notch. Highly recommended.