I'm trying to get a good source about the following questions but I didn't find a clear one - if you can help me with it or at least mention some good references it would be great (BTW I'm not a mathematician, I'm a physicist so references in mathematics are less convenient to me).
The question- We have a scalar field with 2 indices which transforms under the group $SO(N)$:
$\phi_{ij} \to R_{ik}R_{jl}\phi_{kl}$
$R$ belongs to the $SO(N)$ group, $\phi$ is a traceless symmetric $N\times N$ matrix. I guess this means that $\phi$ is represented in the bi-fundamental representation, or equivalently it is a rank 2 tensor with 2 indices each represented in the fundamental. For the sake of some calculation I need to evaluate the trace of two generators in this representation, meaning: $Tr(\Lambda^a \Lambda^b)=C\delta^{ab}$ where $\Lambda^a$ is the generator of that group, and C is a constant parameter depends on the representation.
How should I get to this value?
By linearizing the transformation, I want to say that in this representation the generators are in the form of:
$\Lambda^a=t^a \bigotimes 1+ 1\bigotimes t^a$, where $t_a$ is the generator that acts on one index.
Therefore:
$Tr(\Lambda^a \Lambda^b)=2N\delta^{ab}$ (taking into account that the trace of each generator is zero),
But I think I would expect to get something that is proportional to the number of independent $\phi$ fields, or the number of degrees of freedom of the matrix $\phi$.
What is wrong and what is the right way of doing this?