Best books on Representation theory

Question

What are some of the best books on Representation theory for a beginner? I would prefer a book which gives motivation behind definitions and theory.

Answer 1

I would recommend Representations and Characters of Groups by Liebeck and James (a word to the wise though, the notation is all backwards for some reason! e.g if memory serves correctly, $\phi(x)$ is written $(x)\phi$ etc). I found what I read in Linear Representations of Finite Groups by Serre to be nice too, if not harder. When I was studying Group Rep I found these two sets of notes to be useful also:

https://www.homepages.ucl.ac.uk/~ucaheps/papers/Group%20Representation%20theory%202014.pdf

http://tartarus.org/gareth/maths/notes/ii/Representation_Theory.pdf

Again the second recommendation is harder than the first.

Answer 2

I studied Representation theory for the first time 3 months ago. I had two books in hand, firstly ''Representation theory of finite groups, An introductory Approach'' by Benjamin Steinberg, and secondly Serre's ''Linear Representations of Finite Groups.''

I definitely recommend Serre's book (where you should read the first part only, the second and third parts are more advanced). Steinberg's book is not so elegant, but the exercices set is better. The subjects which are described within 50 pages in serre are explained within 100 pages in Steinberg's book, but I find Serre's exposure clearer and more efficient.

Answer 3

There are really no "best" books in any field. It depends on how everyone decide which material best caters to their individuals needs and taste. So the answer to your question is largely influenced by your personal preference. For example, a rough classification can be: are you a physicist? (then studying Lie groups is necessary); are you a number theorist(then you must be interested in finite fields)? are you a group theorist(then you must be interested in the geometric picture)? are you interested in harmonic analysis? Or the classification can be: I am interested in a textbook of some kind of style(geometric, algebraic, concise, terse, detailed, etc). The list goes very large because representation theory associated with many areas of mathematics.

Some personal recommendations (inclined to Lie algbra side) are:

Fulton&Harris, Brian Hall, Serre(both linear representations and Lie algebras), Humphreys(Lie algebra), Daniel Bump(Lie groups), Adams(Lie groups), Sholomo Sternberg(Lie algebra), and any paper written by Bott.

Answer 4

I am doing Representation theory course, I like the book by James on Representation theory Burow on representation theory on finite groups.

Answer 5

Group Representation Theory for Physicists by Jin-Quan Chen is the best I have read and he explains things with a minimum of requiring external references. In other words fairly self- contained and understandable but still is able to explain Young diagrams,root spaces,Cartan-Weyl basis, Dynkin diagrams etc. with concrete examples.

Answer 6

Definitely NOT Fulton&Harris as it is very difficult, not self-contained, too much assumed prior knowledge even if you are not a beginner. It does have somewhat brief appendix but still requires too much assumed prior knowledge. Would recommend much more for a beginner: 1) Group Representation Theory for Physicists by Jin-Quan Chen which also first starts with finite groups including Young diagrams. 2) Lie Groups and Lie Algebras for Physicists by Ashok Das and Okubo. Both of these more self contained and much more understandable. Prior to study of Lie groups best to first learn finite groups and Jin-Quan Chen above has finite groups which includes Young diagrams and treats permutation groups which is especially important. Probably even a little easier to understand is 'Theory of Finite Groups' by Jansen and Boon especially good but omits Young diagrams.

Answer 7

Representation Theory is one of those topics which are not easy to approach. One needs a significant background to start being a "decent beginner". It is kind of related to the level you need to understand topics like Measure Theory.

I come from non-relativistic Quantum Mechanics and "Classical" Random Matrix Theory where groups are important, but not too important as in QFT and not too hardcore as in Phase Space + Quantum Optics approaches.

Having said this, there are I think two approaches to learn the subject: the "general approach" and the "applied one". I will suggest you books from the second approach and some that are in between; in the applied case representation theory is seen in action but "is not needed to understand too deeply what is going on".

1. P. Woit. Quantum Theory, Groups and Representations: An Introduction. Very oriented to Physics.

2. Vinberg. Linear Representations of Groups. Math.

3. Lie Groups and Lie Algebras for Physicists by Ashok Das and Okubo. I haven't read this book, but from Ashok I learned tons of stuff myself from his books on QFT.

Last but not least, Fulton and Harris is not too bad. If one sees many examples then the discussion in the book is understandable, even if everything is not proved.