Explained: Decomposing Lie Groups in Theoretical Physics

[AlphaNumeric2]

It's common in theoretical physics papers/books to talk about the decomposition of Lie groups, such as the adjoint rep of E_8 decomposing as

\mathbf{248} = (\mathbf{78},\mathbf{1}) + (\mathbf{1},\mathbf{8})+(\mathbf{27},3) + (\overline{\mathbf{27}},\overline{\mathbf{3}})

How is this computed? I'm familiar with working out things like \mathbf{3} \otimes \mathbf{3} using Young Tableaux or weight diagrams but I've suddenly realized I don't know how to do decompositions which aren't tensor products. I can use Dynkin diagrams to limited success but I don't think they apply here. I've tried various Google searches and flicking through a couple of group textbooks I have but they don't cover this method.

Can someone either point me to a book/website which covers this or if they are feeling particularly generous, explain it for me please. Thanks for any help you can provide.

 

[matt grime]

Work out the weight spaces.

 

[AlphaNumeric2]

Thanks Matt, I've had a read around and can see how that leads to the decomposition.

I've been reading through Georgi and it goes into some details about how to work out the SU(n)xSU(m) irreps in both the adjoints of SU(n+m) and SU(nxm) and I've worked out how to do such things, including work out the U(1) charge on any given irrep. I didn't realize that when you give an 'equation' like in my first post, you have to predefine what groups you're breaking your big group into. In the case of my first post, it's E_{8} \to SU(3) \times E_{6}.

I've only got a handle on how to do it for adjoints of SU(N) (give or take a U(1) here and there) but that's demystified a great deal of things! Thanks a lot :)