A Lagrangian with $SO(N)$ symmetry can be written like:
$${\mathcal{L}} = \frac{1}{2}(\partial_\mu \Phi)^T (\partial^\mu \Phi) - (\frac{1}{2}\mu^2 \Phi^T \Phi + \frac{1}{4}\lambda (\Phi^ T \Phi)^2).$$
where $$\Phi =\begin{bmatrix} \phi_1 \\ \phi_2\\ \vdots \\ \phi_N \end{bmatrix}.$$
What additional terms can be included if Lagrangian need only to be $O(N)$ symmetric? How a Lagrangian of this form will look like in $O(N)$ symmetry? Do we need to replace the "transpose" with "dagger" (due to component fields in $\Phi$ becoming complex and also the "$1/2$" terms in Lagrangian to be corrected following that). Is this the only change it can have?
Now if I want a $SU(3)$ symmetric Lagrangian is the following correct?
$${\mathcal{L}} = (\partial_\mu \Phi)^\dagger (\partial^\mu \Phi) - (\mu^2 \Phi^\dagger \Phi + \lambda (\Phi^\dagger \Phi)^2).$$
where $$\Phi =\begin{bmatrix} \phi_1 \\ \phi_2\\ \phi_3 \end{bmatrix}$$ with all $\phi_i$'s being complex?
