What is the "natural" representation i.e. the formula $\pi(g,v)$? I just can't find anywhere a clear statement saying what it is
The representation is supposed to be a map $\pi(g,v):G\times V \to V$ s.t $\pi (g)$ is linear for every $g\in G$. Since these group is composed of rotations my guess is that $\pi(g) $ is the the linear map given by
$$\pi(g):V\to V: v\mapsto \begin{pmatrix} \cos \phi & - \sin \phi \\ \sin \phi & \cos \phi \end{pmatrix}v $$
But still what is the formula for the action $\pi(g,v)$?
Secondly, why is this representation unitary?