I have the following proposition which I have to show through an example of the symmetric group $S_3$:
Proposition. Let $G$ be a finite group. Let $\varphi^{(1)}, \ldots, \varphi^{(s)}$ be a complete set of representatives of the equivalence classes of irreducible representations of $G$ and set $d_{i}=\operatorname{deg} \varphi^{(i)}$. Then the functions $$ \left\{\sqrt{d_{k}} \varphi_{i j}^{(k)} \mid 1 \leq k \leq s, 1 \leq i, j \leq d_{k}\right\} $$ form an orthonormal set in $L(G)$ and hence $s \leq d_{1}^{2}+\cdots+d_{s}^{2} \leq|G|$.
I don't know how to prove that this applies on the following example : The trivial Representation $\varphi^{(1)}$ which sends every element to $[1]$;
The Signum $\varphi^{(2)}$ i.e. $(12) \mapsto[-1]$ and $(132) \mapsto[1]$;
The standard representation $S_3$, $\varphi^{(3)}$: $(1) \mapsto\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right],(12) \mapsto\left[\begin{array}{cc}1 & 1 \\ 0 & -1\end{array}\right],(13) \mapsto\left[\begin{array}{cc}0 & -1 \\ -1 & 0\end{array}\right]$, $(23) \mapsto\left[\begin{array}{cr}-1 & 0 \\ 1 & 1\end{array}\right],(123) \mapsto$ $\left[\begin{array}{cc}0 & 1 \\ -1 & -1\end{array}\right]$ and $(132) \mapsto\left[\begin{array}{cc}-1 & -1 \\ 1 & 0\end{array}\right]$
How do I show the orthogonality of the functions of $\varphi^{(3)}$ ?