- #1

- 470

- 58

I was wondering if anyone knows what the completeness relation for the fundamental representation of SO(N) is.

For example, in the SU(N) we know that, if [itex]T^a_{ij}[/itex] are the generators of the fundamental representation then we have the following relation

$$

T^a_{ij}T^a_{km}=\frac{1}{2}\left(\delta_{im}\delta_{jk}-\frac{1}{N}\delta_{ij}\delta_{km}\right)

$$

This follows from the fact that the [itex]T^a[/itex], together with the identity form a complete basis for the [itex]N\times N[/itex] complex matrices.

Does anyone know how to find the analogous for SO(N) (if any)?

Thanks a lot!