I am studying this classical paper by Regge and Wheeler Stability of a Schwarzschild singularity.
In the second page they introduce their formalism with spherical harmonics and generalization thereof. In particular in eqns. (7) and (8), they introduce vector spherical harmonics, with the definitions:
$$\psi^M_{L,\mu}\propto \partial_\mu Y_l^M \quad \phi_{L,\mu}^M\propto \epsilon_\mu^\nu \partial_\nu Y_L^M$$
where $\mu,\nu=\{\theta,\phi\}$ and $\epsilon_\theta^\theta=\epsilon_\phi^\phi=0$, $\epsilon_\theta^\phi = -\dfrac{1}{\sin\theta}$ and $\epsilon_\phi^\theta=\sin\theta$.
I do not understand how are vectors decomposed into their components wrt these harmonics.
