The number of irreducible representations I am reading a textbook "Representation theory" by Fulton and Harris and I have a question.
They proved the following theorem on page 16.
With an Hermitian inner product on a set of class function, the characters of the irreducible representation of a finite group $G$ are orthonormal.
For a corollary of this theorem, they mentions that
Corollary: The number of irreducible representation of $G$ is less than or equal to the number of conjugacy classes.
I don't know how to prove this corollary. Could you give me some advice, please?
 A: You should note that the dimension of the space of class functions is equal to the number of conjugacy classes, and that orthonormal vectors in a Hermitian inner product space are linearly independent.
A: To prove that the characters of the irreducible representations span the centrum of the associated group algebra it is enough to prove that every central function can be spanned by the characters of the irreducible representations (by their orthonormality they are linearly independent). We already know from the Peter Weyl theorem that the functions $\pi_{ij}$, i.e. the coefficients of the matrix representations $\pi$
of the irreducible representations of the group generate the group algebra. Then,
let $f$ be a central function of $\mathbb{C}(G)$. Let $\widetilde{G}$ be the set of irreducible non equivalent representations of $G$. Since the $\pi_{ij}$ for all $\pi\in G$ are a basis $\mathbb{C}(G)$ we can write
    $$
 f(g)=\sum_{\pi\in \widetilde{G}}\sum_{i,j=1}^{d_\pi}f_{\pi,ij}\pi_{ij}(g),$$
where $d_\pi$ is the dimension of the irreducible representation $\pi$.
Since $f$ is central:
    $$
 f(g)=f(hgh^{-1})=\sum_{\pi\in \widetilde{G}}\sum_{i,j=1}^{d_\pi}f_{\pi,ij}\pi_{ij}(hgh^{-1}),\qquad \forall h\in G.
$$
Hence,   we can obtain by assuming $e_{\pi,i}$ for $i=1,\ldots,d_\pi$ being a basis of the irreducible representation $\pi$:
$$
f(g)=\sum_{\pi\in \widetilde{G}}\frac{1}{|G|}\sum_{h\in G}\sum_{i,j=1}^{d_\pi}f_{\pi,ij}\pi_{ij}(hgh^{-1})=\sum_{\pi\in \widetilde{G}}\sum_{i,j=1}^{d_\pi}f_{\pi,ij}(e_{\pi,i}\widetilde{\pi(g)}|e_{\pi,j}).
 $$
The mapping $\widetilde {\pi(g)}$ is a morphism of the representation $\pi$ and by the Schur theorem is proportional to the identity. It is simple to prove that 
    $$
 f(g)=\sum_{\pi\in \widetilde{G}}\sum_{i=1}^{d_\pi}f_{\pi,ii}\frac{\chi_{\pi}(g)}{d_\pi}.
 $$
Hence, the central function is spanned by the characters of irreducible representations.
