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I've been following a group theory course which I am struggling with at the moment. I'm from 3 different books (Georgi, Cahn and also Jones). I'm trying to understand section 8.7 and 8.9 in the book by Georgi.

I (think I) understand that any pair of root vectors of a simple Lie algebra has an SU(2) subalgebra (as Georgi explains on page 93).

And due to equation 8.47, 8.48, and 8.49 (sorry I don't know how to make equations in Latex), we can say (I think) that the off-diagonal elements of the 2x2 cartan matrix are equal to 2*m, where m denotes the projection of the total angular momentum (see equation 3.23).

Therefore, if we look at the illustration on page 118 which illustrates how to find all the roots for G_2, we see that the simple roots are represented by the boxes alpha_1 == (2 -1) (thus m_1 = -1/2) and alpha_2 == (-3 2) (thus m_2 = -3/2).

I think that until here I understand it (please correct me if I'm wrong).

And then Georgi writes: "We know that the simple roots alpa_1 and alpha_2 are the highest weights of spin 1 representations for their respectives SU(2)s.

Why is this the case? As far as I know they are simple roots and thus they are the lowest weights? And I don't understand what he means with spin 1 representation? I've look at equation 3.31 but that doesn't clarify it for me.

Now he says m_1 is the bottom of a doublet and m_2 is the bottom of the quartet. I'm not sure why you can just assume that's the case?

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# The Cartan matrix in order to find the roots.

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