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I'm having trouble figuring out how SO(N) adjoint rep. transforms

under a SO(3) subgroup.

Unlike SU(N), SO(N) fundamental N gives

\begin{equation} N \otimes N = 1 \oplus A \oplus S \end{equation}

So the \begin{equation} S \end{equation} part really bothers.

Can you give a help?

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# SO(N) adjoint rep. under SO(3) subgroup

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