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This is more of a particle physics question than maths. Since $\operatorname{SO}(6)$ and $\operatorname{SU}(4)$ are isomorphic, how are the fields (say for example scalar fields of ${\mathcal{N}}=4$ Super Yang Mills in $4d$) transforming under 6 dimensional vector representation of $\operatorname{SO}(6)$ related to the fields transforming under antisymmetric 6 of $\operatorname{SU}(4)$?

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