Group Theory for Unified Model Building

[ClubDogo]

Hi everyone, I want to ask everybody if someone knows a book, or some lecture notes available on the net, to lear how to decompose the Lie Groups in irreps in physical notation, like

8_v \otimes 8_v=1+28+35

that can be found everywhere on books like BB&S or Polchinski.
It is really hard to understand what's goin' on without study that in a right way, but it turns out that this "physical" notation isn't so usual for mathematicians.
Thank you

 

[jambaugh]

To be precise your are decomposing the tensor product of two irreducible representations of the Special Linear or Special Unitary Lie groups into a tensor sum of irreducible representations. This is accomplished with Young diagrams. You may want to start with a Google search on Young diagrams.

You can also use young diagram notation to express the decomposition you wrote. See attached image.

The algorithm is a bit involved but the dimension formula is not too bad. I found it helps if you view the dimension formula as a generalized version of the combinatorial binomial coefficient.

Here's a link to a pdf file I just found which gives a good summary of the methods:

http://www.isv.uu.se/~rathsman/grouptheory/Beckman-Loffler-report.pdf"

 

[BenTheMan]

Well, it's funny...there's actually a "book" called "Group Theory for Unified Model Building". In fact, I thought th OP was a joke :)

http://www.slac.stanford.edu/spires/find/hep/www?j=PRPLC,79,1

This is a Physics Reports article by Richard Slansky, and has extensive tables in the back with exactly the information that you want.

RE the Young's Tableaux---I think they only work for SU(N), and not any of the other classical Lie groups. If someone knows the rules for the other groups, please let me know, because I've only ever seen them for the SU(N) case.

 

[ClubDogo]

In fact, I knew there is a rewiev By Slanski about group theory but is not self-contained, so I wrote this title just to let the others know how about to LEARN some group theory appositely to study the structure of multiplets in the case of GUTs. I am not aware of what kind of Young Tableaux rules you need for the others group... expecially for the exceptional groups! Btw, thank you all

 

[BenTheMan]

Ahh sorry. Yeah the review by Slansky is for physicists who don't really care a lot about the maths involved. I use it as a reference, but (outside of SU(N), which I can do with Young's boxes), I don't think I could come up with the decompositions myself.

There's a book by Di Francesco, Mathieu, and Senechal called "Conformal Field Theories". I think chapter 13 and 14 may help you with the decompositions.

 

[George Jones]

There is a close relationship between Lie groups and Lie algebras, and physicists tend to look at representations of Lie algebras.

Semi-Simple Lie Algebras and their Representations,

http://phyweb.lbl.gov/~rncahn/www/liealgebras/book.html,

by Robert Cahn is a free book (wasn't free when I picked it up!) on Lie algebras that was written for physicists.

Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists by Jürgen Fuchs and Christoph Schweigert ,

https://www.amazon.com/dp/0521541190/?tag=pfamazon01-20,

was also written for physicists.

Representation Theory: A First Course by William Fulton and Joe Harris,

https://www.amazon.com/dp/0387974954/?tag=pfamazon01-20,

was written for graduate math students, but, since it is largely a series of example, it might be good for physicists, too. Its notation, while standard for math, is, however a little non-standard for physicists. Physicists tend to refer to complex Lie algebras by specific real forms, so, for example, its treatment of sl\left(2 , \mathbb{C}\right) looks like quantum angular momentum theory of su\left(2\right). It starts off terse, but becomes very readable and (maybe too) expansive in its middle. It's cool to see quark multiplet diagrams (as representations of sl\left(3 , \mathbb{C}\right) \cong \mathbb{C} \otimes su\left(3\right)) appearing in a pure math book, even though the book doesn't identify them as such.

 

[BenTheMan]

I forgot about Cahn's book. It is supposedly very good, although I've never read through it.

 

[tmc]

A great classic is Georgi's "Lie Algebras in Particle Physics"

 

[rntsai]

Hi everyone, I want to ask everybody if someone knows a book, or some lecture notes available on the net, to lear how to decompose the Lie Groups in irreps in physical notation, like

8_v \otimes 8_v=1+28+35

that can be found everywhere on books like BB&S or Polchinski.
It is really hard to understand what's goin' on without study that in a right way, but it turns out that this "physical" notation isn't so usual for mathematicians.
Thank you

As others have noted, this is the decomposition of tensor products of representations.
In this case these are reps of the simple lie algebra d4 (which can be made to correspond to the lie algebra so(8) and its associated lie group).

You could spend years studying the mechanics of how this is done. It's an interesting enough subject if you have the time. If you don't and just want to get an overall picture of what's happening while you think of other things, then you're better off skipping the mechanics and leave the details to a good math program...I use GAP (freely available)

An irrep for a simple lie algebra of rank n is uniquely defined by a n-tuple of
positive integers. In the case of d4, you need 4. Physicists substitue the dimension
of the irrep for these numbers. Most the time they can get away with it because
there's no ambiguity. Sometimes there is and they use subscripts, primes, overbars,...
There's choice of this notation is "folklore" more than anything else​

In the case of d4 here's a "dictionary" :

[1,0,0,0] <-> 8_V
[0,1,0,0] <-> 28
[0,0,1,0] <-> 8_S+
[0,0,0,1] <-> 8_S-
[2,0,0,0] <-> 35 (this really should be 35_V)
[0,0,2,0] <-> 35_S+
[0,0,0,2] <-> 35_S-
[0,0,0,0] <-> 1

so your example :

[1,0,0,0] x [1,0,0,0] = [0,0,0,0] + [0,1,0,0] + [2,0,0,0];

the GAP code that does this is below :

L:=SimpleLieAlgebra("D",4,Rationals);
w1:=[1,0,0,0];
w2:=[1,0,0,0];
w3:=DecomposeTensorProduct(L,w1,w2);
d:=List(w3[1],x->DimensionOfHighestWeightModule(L,x));

 

[BenTheMan]

the GAP code that does this is below :

L:=SimpleLieAlgebra("D",4,Rationals);
w1:=[1,0,0,0];
w2:=[1,0,0,0];
w3:=DecomposeTensorProduct(L,w1,w2);
d:=List(w3[1],x->DimensionOfHighestWeightModule(L,x));

What is GAP?

 

[rntsai]

What is GAP?

GAP - Groups, Algorithms, Programming -
a System for Computational Discrete Algebra

http://www.gap-system.org/

(it's freely available and maintained by world class universities)

 

[bomanfishwow]

A great classic is Georgi's "Lie Algebras in Particle Physics"

I'm just chomping my way through that book, it's very approachable.