# Good books in Modular Representation Theory

Hi every one!

I am reading some paper and it uses Modular Representation Theory. I even dont really know about Representation Theory and I am looking for a good book for beginner. Could you please give me some suggestion for beginning Modular Representation Theory?

-
Everyone seems to be assuming that you mean modular representations of finite groups. There are other kinds of modular representation theory: can you be more specific? – Sheikraisinrollbank Feb 19 '11 at 16:17
I would also recommend Navarro's book and the book by Collins. – Steve D Feb 20 '11 at 3:37

If I understand you correctly, you're saying that you're not familar with ordinary (ie char $0$) representation theory. If this is correct, then you'll find books on modular representation theory tough going. I recommend starting with Serre's short book "Linear representations of finite groups". The first 2 sections are a very elegant and efficient treatment of the char $0$ theory, and the last section is a nice start on the char $p$ theory. After that, you'll be much better prepared for one of the other books people mention (and I'll second Alperin's "Local representation theory").

-

The question strikes me as too vague (what is that paper you're reading, by the way?). Like others, I'm assuming the subject is finite groups, rather than algebraic groups or quantum groups or something connected with modular forms. Usually the modular theory of finite groups is approached via the classical Frobenius-Schur theory, which is especially slanted toward characters but can be recast in terms of the Wedderburn structure theory of group algebras. Modular theory allows for a wide variety of approaches, including the narrow but fruitful one in Alperin's book. Serre's concise book starts out in a fairly elementary style (for an audience of chemists), but by the third part on modular representations the sophistication has escalated a lot. It's an important short treatment but scary for newcomers.

There's a lot to be said for the somewhat old-fashioned, concrete approach taken in the original Curtis & Reiner book Representations of Finite Groups and Associative Algebras, toward the end of the book. It does presuppose some of the earlier material, slanted toward ring theory rather than characters.

For a really short and relatively elementary expository treatment of both "ordinary" (characteristic 0) and modular representation theory, motivated by a single family of almost-simple groups, there's an old paper of mine in American Mathematican Monthly (1975), "Representations of $SL(2,p)$".

The bottom line is that you need to focus on which specific area of representation theory is essential for your needs. It's a very big subject.

-

Hi, a very elementary written book is Local Representation Theory by Alperin.The second book on finite groups by Huppert has also a big part about modular representation theory.(you should read the first book too,with an long introduction to representation theory in the semisimple case) a more advanced book is that of feit and a recent (2010) book is "Representations of Groups: A Computational Approach" by Lux and Pahlings.It has many nice examples with gap and the sporadic groups.You should also take a look for the books of curtis and reiner.The one from 1962 is easy to read and might be the best introduction.

-
How can a book be very hard to read "since it covers many other topics"? You must be hiding the real reason. – darij grinberg Feb 19 '11 at 16:04
Ok,that sounds strange so i changed that.I dont know how to explain.Maybe just try to read it. – trew Feb 19 '11 at 16:08
Alperin's book is not just "elementary written", it's beautifully written. – Chris Godsil Feb 19 '11 at 16:14
But one has to say that Alperin covers only a small part of modular representation theory, in particular block theory. There is nothing about decomposition theory. Also everything is written in the language of modules, but some results become more clearer if expressed for example in terms of central idempotents. After reading Alperin's book for example you may have a hard time understanding what Brauers main theorems as listed on wikipedia have to do with the three theorems you learned from Alperin... – Johannes Hahn Feb 19 '11 at 16:52

David Benson: Modular Representation Theory. New Trends and Methods. Lecture Notes in Mathematics 1081, 1984 (2nd print 2006)

is also a good book on the topic.

-

Daniel Bump's "Automorphic forms and representations" is pretty nice if by your question you mean something about modular forms and representation theory.

I agree with J.Humphreys that your question isn't specific enough...

-