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Say I have some specific irreps of e.g. $SU(3)$ in the case of a Calabi-Yau 3-fold. Now assume you have some section transforming in a specific representation of the holonomy group. Is it possible to infer from the representation alone the target space of the section ?

To give an example, can we say a section transforming in the 3 of $SU(3)$ is an element of $\Omega^{1,0}(M)$ ? And if so, what about elements of $\Omega^{2,1} = \Omega^{1,0} \otimes \Omega^{0,1}$ ?

If we look at the tensor product of the representations, we get: $$ \bar{3} \otimes \bar{3} \rightarrow 3 \oplus \bar{6} $$

Do both of these representations now assemble in $\Omega^{2,1}$ and in what way is this unambigious ?

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