Prerequisites for representation theory

Question

Currently, most of what I know in algebra is contained in the first four chapters of Hungerford's Algebra, which covers basics of groups, rings and modules. For linear algebra, I've read Hoffman & Kunze, up to Jordan forms.

Since I know some group theory, I would like to read texts like Character Theory of Finite Groups by Isaacs, and Linear Representations of Finite Groups by Serre. But my interest is not limited to representation theory of finite groups only. I know there is, e.g., representation theory of algebras, which I also want to study, although presently I know nothing about algebras.

So what are the prerequisites for studying representation theory? By prerequisites I mean just everything (if that's possible to list) one might need for a thorough and detailed study of the subject. I'm in no hurry, so I plan to spend a lot of time preparing. I just don't know what to read. I've searched through the site but surprisingly, it seems that this question has not been asked yet.

Answer 1

Elements of Representation Theory of Associative Algebras by Assem, Simson and Skowronski (LMS Student Texts 65) state no prerequisites, however in their introductory chapter they cover:

• basic ring theory
• quotient vector spaces
• ideals in an algebra, radical of an algebra
• modules
• basic category theory (categories, opposite categories, functors, Hom categories)

In Chapter II they start working with quivers (directed graphs) which require some very basic ideas from graph theory.

In Quiver representations by Schiffler it is stated that "only familiarity with basic notions of linear algebra" is a prerequisite.

However I would add that both books require some mathematical maturity on at least an undergraduate level.

Answer 2

I present a portion of the preface of A Tour of Representation Theory, by Martin Lorenz of Temple University:

The AMS link for the book is here. The MAA review is here. The Amazon link is here, from which one can view said preface.

Answer 3

There are many on this site better qualified to answer this than I, but I'll take a shot at it. Representation theory is a quite broad subject and depending on what exactly your interests are, you may need more or less background.

An afterthought: representation theory is such an enormous subject that it is simply impossible to say what the prerequisite topics are for "all of representation theory." There is much more to be said than what I have written, and there are a lot of major topics I know little about. However, the following things will get you started comfortably. Of course, you should not spend too much time "learning prerequisites" as there will always be another prerequisite. You should start learning something you are interested in, and learn the other things as you need them. Of course, a solid grasp of linear algebra is absolutely essential, so that is perhaps one fundamental you should not neglect.

• Finite Groups:

1. Linear Algebra: Jordan Decomposition, Inner Product Spaces, Dualization

2. Multilinear Algebra: Tensor product, Exterior Product, Symmetric Product, and the various identities relating them are quite important. This is not so much a field unto itself, but you will need to familiarize yourself at some point with identities like $V^*\otimes W\cong \operatorname{Hom}(V,W).$

3. Finite Group Theory: Lagrange's Theorem and other basic properties of finite groups, Conjugacy Classes, Orbit-Stabilizer theorem, examples of finite groups like $S_n,D_n, A_n$ etc. The Sylow theorems might also be useful.

4. Basic Ring Theory: There are some simple ring theoretic constructions that naturally pop up, like group rings $\Bbb{C}[G]$ and so on. Just having some familiarity with rings should suffice.

• Compact Lie Groups: (Lie algebra theory is needed in some parts here, and is arguably part of the theory.)

1. Manifolds: you should know what an abstract manifold is, smoothness, examples of manifolds, differentials of maps, and subsequently examples of Lie groups like $GL(n,k),SL(n,k),SO(n), SU(n),O(n),U(n)$, etc. where $k=\Bbb{R,C}$.

2. Linear Algebra: as above, also you should know about bilinear forms and what it means to preserve a bilinear form e.g. that $O(n)$ preserves an inner product

3. "Modern" Analysis: The Peter-Weyl Theorem requires some analysis background, so you want to learn about Haar measures, function spaces, etc. It might also help here to know a little bit about the Fourier transform I believe.

4. Riemannian Geometry: This can help you understand things like the exponential map on Lie groups in a different way, and there are some elegant proofs of results using notions of curvature. (Probably more optional than the others.)

5. Algebraic Geometry: If you want to learn results like the Borel-Weil-Bott Theorem, then knowing about sheaf cohomology on $\Bbb{P}^n$ (over $\Bbb{C})$ is essential. (This is also optional in some sense.)

• Finite Dimensional Lie Algebras: I'll suppose here you work over an algebraically closed field of characteristic $0$, otherwise there are more things to say.

1. Linear Algebra: Jordan Decomposition, Inner Product Spaces, Bilinear Forms, examples of matrix Lie algebras like $\mathfrak{sl}(n,k)$, etc.

2. Algebraic Knowledge: Actually Lie algebras have fairly light pre-requisites, but the beginning of the theory is mostly linear algebra in conjunction with structure-type theorems reminiscent of those from Groups and Rings. So, having some knowledge there would not hurt.

3. Basic Lie Groups: Knowing how Lie Algebras arise as tangent spaces of Lie Groups might make things feel better motivated, and also clarify some of the constructions.