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I think my problem is that I misunderstand what "transforms under" really means.

Let's take $SU(3)$, for the $\mathbf{3}$ with Dynkin indices $(1,0)$, a state transforms like : $ψ→gψ$. For the $\mathbf{\bar{3}}$ with Dynkin indices $(0,1)$, a state transforms like : $ϕ→ϕg^{−1}$. And for the adjoint representation $(1,1)$: $\mathcal{O}→g\mathcal{O}g^{−1}$.

But then, if I take $SU(2)$, because the $\mathbf{2}$ and $\mathbf{\bar{2}}$ are equivalent, they should transform in the same way? And what about a representation labeled by $(2,0)$ in $SU(3)$ ? Should a state in this representation transforms like $\Psi→g^2\Psi$?

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