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(adsbygoogle = window.adsbygoogle || []).push({});

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

How is this computed? I'm familiar with working out things like [tex]\mathbf{3} \otimes \mathbf{3}[/tex] using Young Tableaux or weight diagrams but I've suddenly realised 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.

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# Decomposing Lie groups

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