# Best books on Representation theory

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.

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.

What I really can recommend to read are two brilliant papers of the eminent mathematician T.Y. Lam (UCLA), Representations of Finite Groups: A Hundred Years, Part I and Part II, which appeared in the Notices of the American Math. Soc. March 1998, 45, 3, and April 1998, 45, 4, respectively.

Part I recounts the story of how Dedekind proposed to Frobenius the problem of factoring a certain homogeneous polynomial arising from a determinant (called the “group determinant”) associated with a finite group G. In the case when G is abelian, Dedekind was able to factor the group determinant into linear factors using the characters of G (namely, homomorphisms of G into the group of nonzero complex numbers). In a stroke of genius, Frobenius invented a general character theory for arbitrary finite groups, and used it to give a complete solution to Dedekind’s group determinant problem.

In Part II, William Burnside enters the scene.

If you Google the above title, you will easily find a pdf of the papers.

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.

• Definitely NOT Fulton&Harris as it is very difficult and 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)Lie Groups and Lie Algebras for Physicists by Ashok Das and Susumu Okubo and 2)Group Representation Theory for Physicists by Jin-Quan Chen .Both of thse 2 are much more self contained and understandable Apr 8, 2021 at 9:44

I would totally recommend the notes by Etingof et al called "Introduction to Representation theory"!

I think this is the best introduction to Representation theory I've read. They start from basics, and they give a lot of motivation and nice examples. These notes also have one of the best exercise sets I've seen. All exercises are very interesting (not just boring "check-the-details"), and they often show non-trivial and surprising applications of the subject.

• I like the first sentence of its introduction, but then I can't understand the rest Nov 15, 2017 at 0:52

One book I like a lot is A Course in Finite Group Representation Theory by Peter Webb. The preprint is freely available in the author's webpage. It may differ from other more classical books in some aspects. Probably the termonology is more modern. There are good amount of representation theory books that goes towards the representation theory of Lie algebras after some ordinary representation theory. This book does finite group representation theory and goes quite in depth with it (including some mention of the case where Maschke's theorem does not hold).

I believe it is intended for a graduate course but I personally feel like it is a book an undergraduate can also grow into.

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

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.

I think M.A.Naimark 'Linear Representations of the Lorentz Group' its one of the books to start with. In this book (maybe this is the only one except H Weyl ofcourse:))you can find a motivation to get into the modern representation theory. And btw Naimark's book its also a good math book. No SF physics.

Another classic book which has been left out in the other answers so far:

I. Martin Isaacs: Character Theory of Finite Groups

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.

I can also recommend the two volumes of Yakov Berkovich, Lev Kazarin and Emmanuel Zmud' Characters of Finite Groups as well as the excellent three volumes written by Gregory Karpilovsky, Group Representations.

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.

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.