1. If $A=diag\{a_1,\ldots,a_n\}$, then $P(\pi)AP(\pi^{-1})e_k=P(\pi)Ae_{\pi^{-1}(k)}=a_{\pi^{-1}(k)}e_k$, i.e. $P(\pi)AP(\pi^{-1})$ is the matrix $diag\{a_{\pi^{-1}(1)},\ldots,a_{\pi^{-1}(n)}\}$. 2. Since $P(\s\pi)AP((\s\pi)^{-1})=P(\s)P(\pi)AP(\pi^{-1})P(\s^{-1})=\Psi(\s)\Psi(\pi)A$, $\Psi$ is a representation. 3. It remains to establish irreducibility: Let $E^j$ be the standard basis of the space of diagonal matrices, then $\Psi(\pi)E^j=E_{\pi(j)}$ and thus the matrices of $\Psi(\pi)$ with respect to this basis coincide with the matrices of the standard representation with respect to the canonical basis $e_1,\ldots,e_n$. Since the space of traceless diagonal matrices corresponds via $E^j\mapsto e_j$ to the space orthogonal to $e_1+\cdots+e_n$, the representation is irreducible.