## Group representation theory notes

May 1, 2009 by dberenstein

I just finished a quick primer for my class on Advanced QFT on Representations of Lie algebras and useful facts about them. Here is a link to the lecture notes. It’s a hands on approach with no proofs, but at least it makes various concepts a little bit more accessible and can serve for a quick reminder of basic stuff like what is a Dynkin diagram and labels of various representations. The notes cover mostly SU(N), but various concepts are quite general.

My first serious encounter with Dynkin diagrams was for a paper I wrote together with Richard Corrado and Jacques Distler in 97, when I was working on Matrix theory in graduate school. The difference is that the Dynkin diagrams that were appearing there where those of the Affine Lie algebras (they have an extra node). There is some fun discrete group theory and geometry there. I was reminded of this stuff when I was preparing my notes.

The main issue for this class was to get enough group theory so that most computations that we are going to do for Grand Unified Theories become simple. I remember also reading various sections of the book on group theory by Hammermesh when I was an undergraduate. Although a lot of that was on Crystallographic groups. Nowadays I recommend the Fulton and Harris book on representation theory. However, I have not read other books recently that might contain more information on Dynkin diagrams (the book of Fulton and Harris does not treat these in a lot of detail).

Group theory appears all of the time in theoretical physics, both discrete and continuous. It’s probably one of the most useful tools to approach physical problems in general and any student who is seriously considering becoming a theorist should be well advised to learn Lie theory (continuous group theory). The one sound bite I can give about this is that if you understand well the simplest non-abelian group ( SU(2) angular momentum), you are 50% of the way there already.

I’m providing these notes as a service to whoever can use them. After all the work that goes into typing them, it would be wasteful if they do not become more widely available. readers beware: I can not guarantee that they are free of typos. Fortunately, there are not that many formulae.

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on May 2, 2009 at 8:37 amcarlbrannenThey’re beautiful notes and I’ve downloaded them. I have this problem with Georgi that I keep buying copies and losing them.

But over the years, I’ve come to see symmetry as more of a tool and less of a fundamental description of reality. The geometric algebra (Clifford algebra) guys converted me. The big problem is that there’s so few people working in it.

on May 3, 2009 at 6:24 pmLuboš MotlDear David, your group representation is cool. I am sure it must be enough to find a simple explanation of the following fact.

Two weeks ago, you asked about the average number of throws to get N heads in a row. The answers were

2, 6, 14, 30, 62, 126

for 1,2,3,4,5,6 heads in a row. Well, it’s 2(2^N-1). Now, I ask you a simple question. Find all dimensions of manifolds whose Kervaire invariant is nonzero. The correct answer must be

2, 6, 14, 30, 62, and maybe 126.

That’s it. You should only found the proof. Feel free to use coins – but some simple new anomalies in an engineered QFT or CFT could be more useful. ;-)

See more information here:

http://motls.blogspot.com/2009/05/kervaire-invariant-math-homework.html

on May 4, 2009 at 7:41 amCurtisThe notes are great. Thanks!

on May 4, 2009 at 8:30 pmHaelfixThe notes are pretty good. Particularly the Dynkin diagram and Young Tableaux segments.

One thing that always bothered me about physics presentations on group theory is they always settle on SU(N) groups. Only later did I learn that you can actually do many of the same things like Young Tableaus for different lie groups (with some generalization) but thats a little hard to find.

on May 4, 2009 at 8:44 pmdberensteinHi Haelfix

If you learn of a good reference on generalized tableaux for SO(N), Sp(N), please let me know.

The highest weight stuff works pretty well for most applications one ever encounters.

Unfortunately, I am constrained by time to not include these in the class I am teaching. It could take a whole quarter just to get through these constructions.

on May 4, 2009 at 10:38 pmHaelfixYea, it seems they’re kinda hard to find, at least a presentation that isn’t geared for mathematicians. I had a colleague explain it to me rather than finding a book.

I had these in my bookmarks on the subject, but I don’t have them handy, and its been awhile so not sure about relevance or readability.

Group Theory for Physicists By Zhong-Qi Ma

and

Lie Groups by Daniel Bump