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I have the following question regarding orbifold compactifications of the heterotic string:

What is the relation between a certain representation and the weights of the root lattice? I mean: take the following example from Uranga & Ibañez's book:

enter image description here

I understand that the possible weights of the $\left [ E_{6}\times SU(3) \right ]\times E_{8}{}'$ are $0,1$ and $2$ and each of them should correspond to one state $(\mathbf{27},\boldsymbol{3};\mathbf{1}'),(\mathbf{27},\mathbf{1};\mathbf{1}')$ or $(\mathbf{1},\mathbf{\bar{3}};\mathbf{1}')$.

But which is this correspondece (and why)?

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  • $\begingroup$ You should study and understand weights for something simple like SU(3) before trying to understand something this complicated. $\endgroup$
    – G. Smith
    Commented Jan 17, 2019 at 23:46

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Each dominant weight corresponds to an irreducible representation for which it is the highest weight.

https://en.wikipedia.org/wiki/Theorem_of_the_highest_weight

Weights are vectors in the root space, usually written in the basis of the fundamental weights so that all components are integral.

In the case you’re talking about, they aren’t just integers like 0, 1, and 2. The weight these authors are talking about is

$$\frac{1}{3}(1,1,2,0,...,0)\times(0,...,0)$$

The part after the $\times$ is the $E_8’$ part of the weight vector, corresponding to a 1-dimensional irrep. The part before must be a weight vector for $E_6\times SU(3)$ but I don’t know why it is written as a single vector or why it has a prefactor of 1/3.

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