Let the finite group $G$ act on finite sets $X$ and $Y$, and denote by $\mathbb{C}[X], \mathbb{C}[Y]$ the associated permutation representations on the spaces of complex functions on $X$ and $Y$. Call their characters $\chi_{X}$ and $\chi_{Y}$.

(i) Show that the inner product $\left\langle\chi_{X} \mid \chi_{Y}\right\rangle$ is the number of orbits for the diagonal action of $G$ on $X \times Y$.

(ii) Assume that $|X|>1$, and let $S \subset \mathbb{C}[X]$ be the subspace of those functions whose values sum to zero. By considering $\left\|\chi_{X}\right\|^{2}$, show that $S$ is irreducible if and only if the $G$-action on $X$ is doubly transitive: this means that for any two pairs $\left(x_{1}, x_{2}\right)$ and $\left(x_{1}^{\prime}, x_{2}^{\prime}\right)$ of points in $X$ with $x_{1} \neq x_{2}$ and $x_{1}^{\prime} \neq x_{2}^{\prime}$, there exists some $g \in G$ with $g x_{1}=x_{1}^{\prime}$ and $g x_{2}=x_{2}^{\prime}$.

(iii) Let now $G=S_{n}$ acting on the set $X=\{1,2, \ldots, n\}$. Call $Y$ the set of 2element subsets of $X$, with the natural action of $S_{n}$. If $n \geqslant 4$, show that $\mathbb{C}[Y]$ decomposes under $S_{n}$ into three irreducible representations, one of which is the trivial representation and another of which is $S$. What happens when $n=3$ ?

[Hint: Consider $\left\langle 1 \mid \chi_{Y}\right\rangle,\left\langle\chi_{X} \mid \chi_{Y}\right\rangle$ and $\left\|\chi_{Y}\right\|^{2}$.]