Define the inner product $\langle\varphi, \psi\rangle$ of two class functions from the finite group $G$ into the complex numbers. Prove that characters of the irreducible representations of $G$ form an orthonormal basis for the space of class functions.

Consider the representation $\pi: S_{n} \rightarrow G L_{n}(\mathbb{C})$ of the symmetric group $S_{n}$ by permutation matrices. Show that $\pi$ splits as a direct sum $1 \oplus \rho$ where 1 denotes the trivial representation. Is the $(n-1)$-dimensional representation $\rho$ irreducible?