## Lecture notes on representations of finite groups

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Next term I am supposed to teach a course on representation of finite groups. This is a third year course for undegrads. I was thinking to use the book of Grodon James and Martin Liebeck "Representations and characters of groups", but also looking for other references.

The question is: could you advise some other books (or lecture notes)? Maybe you had a nice experience of teaching or listening to a course with a similar title? It would be really nice if this book (notes) has also exercises.

ADDED. I would like to thank everybody who answered the question, very helpful answers!!! The answer of John Mangual below contains a "universal" reference. For the moment my favourites are Serre (very clear and short introduction of main ideas), some bits from notes of Teleman and Martin, and Etingof for beautiful exposition. My last problem is to have enough of exercises, in particular to write down a good exam. So I would like to ask if there are some additional references for exercises (with or without solutions)?

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Since you are planning a course in representation theory could you share if you found a reference/notes which shows a lot of actual examples of computing roots and weights and Dynkin diagrams?    <Apart from Fulton and Harris's book>    – Anirbit Dec 27 2009 at 19:28
Anirbit, this is not planned in my course :)), this is too advanced. Have you seen below the reference to the page of Khovanov that John Mangual gave? This seem to contain everything you can imagine!  – Dmitri Dec 27 2009 at 21:27
I'm wikifying this question. See meta.mathoverflow.net/discussion/6 – Anton Geraschenko♦ Dec 30 2009 at 20:08

## 11 Answers

Some material from the undergrad rep theory course in Cambridge: Example sheets, A recent set of notes (by Martin), and a less recent (but very nice) set of notes (by Teleman).

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Finally, I decided to use the notes of Teleman for the course :) Hope it will go fine. – Dmitri Jan 13 2010 at 21:45
Since Dmitri just edited, I figured now might be a good time to comment that the first link above is broken. Perhaps those example sheets were taken down? – David White Jun 28 2011 at 17:24

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While I'm at it, I agree with Dmitri's choice, and gave this answer +1. I think representation theory is one of the hardest subjects an undergrad can learn. I recall how painful it was for me to try to learn as a 3rd year undergrad out of Isaac's Character Theory of Finite Groups. That book (and Serre's below) are great for grad students but I think Teleman's notes are some of the best I've seen at the undergrad level. I also retagged above to reflect that it's a request for a textbook-recommendation – David White Jun 28 2011 at 17:28

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The middle third of Serre's "Linear Representations of Finite Groups" is excellent. It's in 3 totally seperate sections, the first third is ok but very elementary and the last third is tough going. But the middle is "just right."

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+1. I am now imagining Noah as Goldilocks. – S. Carnahan♦ Nov 22 2009 at 1:22

I enjoyed Pavel Etingof's lecture notes for his representation theory class, which can be found here: http://www-math.mit.edu/~etingof/replect.pdf (there is a link to it on his website) They move fast, but without skipping too much and still providing insightful proofs of results.

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Thanks a lot! This looks cool indeed! May be the level is a bit advanced, but it could be that I'll need only first 50 pages, even less – Dmitri Nov 21 2009 at 20:08
Since I can't figure out a better place to ask this: Can anyone help me with two exercises from the above notes? In Problem 1.26 (c) I would like to know whether the field $k$ is assumed to be algebraically closed (no hints, please; just an answer to this question). In Problem 1.34 (d), am I right in assuming that the grading on $P_Q$ is given by $\mathrm{deg}p_i=0$ and $\mathrm{deg}a_h=1$ ? I really enjoy this text, by the way - it's concise and straight to the point (and not analysis-biased as Fulton-Harris). – darij grinberg Dec 22 2009 at 10:57

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@darij, it's better to post a new question -- many more people will see it. – Ilya Nikokoshev Dec 31 2009 at 18:20

Mikhail Khovanov lists a bunch of materials for his course Representations of Finite Groups. Of course, he would be more interested in Hopf Algebras, their Representations, Applications, and Categorifications...

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This answer seem to contain everything!!! Amazing :) Huge thanks! – Dmitri Dec 27 2009 at 19:11

Artin's Algebra has a good chapter on representations of finite groups. The exercises are nice.

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Thanks, I'll have a look! – Dmitri Nov 21 2009 at 20:09

The first section of Representation Theory by Fulton and Harris is a great introduction to representations of finite groups (about a quarter of the book, if I remember correctly). There are lots of examples and exercises. The rest of the book is devoted to Lie theory.

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This is nice! I have the book :)) – Dmitri Nov 21 2009 at 20:28

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I second Fulton and Harris: I taught such a course several times initially with James and Liebeck but with a lot of changes after I read F&H. – Fran Burstall Nov 21 2009 at 21:22

I was surprised to see Dummit and Foote's "Abstract algebra" book has a decent amount of basics on representations of finite groups. I believe this was our 2nd year algebra text, when I was an undergraduate. If your university uses this book, it might be cost effective for your students.

When I took a course on representations we used Serre's book. It's quite nice though it sounds like you want something that's a little more rich in examples and exercises.

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Thanks for this too! – Dmitri Nov 21 2009 at 20:10

Alperin and Bell's book is where I got my introduction to the representation theory of finite groups. I didn't understand much at the time, but it helped a lot with getting my bearings as I delved deeper into representation theory in general.

If you're interested (as I am) specifically in representations of finite groups, there's a book by Digne and Michel; but this is probably far too much for undergraduates.

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I like M. Isaacs, "Representation theory of finite groups" since it has lots of exercises.

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Do you mean "Character theory of finite groups"? If so, it should be pointed out that the book is much more focused on character theory than on representation theory per se, both in terms of the language/terminology used and the kinds of question it discusses – Yemon Choi Jul 1 2011 at 22:30

Here is one more reference (kindly communicated to me by one of my colleagues), the course of Iain Gordon. You can find the photos of all blackboards (I still have to go through this course, but it seem a bit similar in spirit to the courses of Teleman and Martin).

http://www.maths.ed.ac.uk/~igordon/4rt/rt2008.htm

ADDED.

I just read the article "Representation Theory" of Ian Grojnowski in the book "Princeton companion of mathematics". I find it really vivid, inspiring and amazingly well written. The first 6 pages are on finite groups, then it proceeds to compact Lie groups and non-compact Lie groups, and smoothly finishes with Langlands correspondence :). The beginning will serve perfectly for the introductory lecture of my course :).

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I think it's a really good introduction, but unfortunately one of the photos is broken. If you are reading Iain... ;) – Grétar Amazeen Dec 27 2009 at 18:54

At the risk of tooting my own horn, I have a new book that will be published by Springer which is a course on representation theory of groups intended for undergrads and beginning grads. It assume only linear algebra, group theory and basic ring theory. It assumes no module theory. Included are applications to combinatorics and probability. The link is

http://www.springer.com/mathematics/algebra/book/978-1-4614-0775-1?detailsPage=authorsAndEditors

and it should be out by the end of the year.

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