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

[mkgsec]

Hi.

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?

 

[arivero]

What about using indexes for the vector spaces associated to the tensor products? Sometimes it helps to visualize the situation, specially now that you want to separate the symmetric and antisymmetric parts.