$G$ is a finite group, $V$ is a dimension $n$ vector space over $\mathbb{C}$.
$\rho: G \to \mathrm{GL} (V)$ is an irreducible representation of $G$. Choose a $G$-invariant metric in $V$ and an orthonormal basis under the metric. Then all $\rho(g)$ are unitary matrices, denoted $(a_{ij}(g))$, $a_{ij} \in C(G)$, complex valued functions on G.
Let $W=\left< a_{ij}\right> \subset C(G)$, try to find all class functions in W.
A function $f \in C(G)$ is called a class function if $f(t^{-1}st)=f(s), \forall t,s \in G$. Obviously, $\chi=\sum_i a_{ii}$is a class function.
I guess $\left< \chi \right>$ are all what I am looking for.
$a_{ij}$ are linearly independent. All is to prove $\rho_t B=B \rho_t$, where B is a matrix of coefficients. By Schur's Lemma, $B=c\delta_{ij}$.