# 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.

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I would recommend Reprentations 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 correct, $\phi(x)$ is written $(x)\phi$ etc). I found what I read of Linear Representations of Finite Groups by Serre to be nice to, if not harder. When I was studying Group Rep I found these two sets of notes to be useful also:

http://www2.imperial.ac.uk/~epsegal/repthy/Group%20representation%20theory.pdf

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

Again the second recommendation being harder than the first.

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Must have been a Forth fan! – Snowball Sep 9 '12 at 15:26

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.

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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.

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

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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.

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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.

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