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.

Solution 1:

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.

Solution 2:

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.

Solution 3:

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.

Solution 4:

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.