Exercise 6, § 21

Let X be a finite set on which G acts. We consider the representation $\pi$ of G in $K^X$ such that $\pi (g)e_x=e_{gx}$ for $g\in G,x\in X$ (“permutation representation” associated with X). a) For $g\in G$, prove that $\chi_{\pi}(g)$ is equal to the number of points of X fixed by $g$. b) Prove that the multiplicity of the unit representation in $\pi$ is equal to the number of orbits of G in X. c) Assume from now on that the action of G is transitive; fix a point $x$ of X, and denote its stabilizer by H. Prove that $\pi$ is isomorphic to the representation induced by the unit representation of H and that $\langle \chi_{\pi}, \chi_{\pi}\rangle$ is equal to the number of orbits of H in X. d) Let V be the subspace of $K^X$ consisting of the vectors for which the sum of the coordinates is zero; it is stable under G, and we denote by $\pi_X$ the representation of G in V deduced from $\pi$. Prove that $\pi_X$ is irreducible if and only if the action of G is doubly transitive (I, §5, p. 143, Exercise 14).