Consider the symmetric group $S_n$ acting on $A=\{1,..,n\}$, for any nonnegative integer $k\leq n/2$, denote $A_k$ to be the collection of all $k$-element subsets of $A$. Let $\chi_k$ be the character of the permutation representation of the action of G on $A_k$.
a) Show that for $0\leq l \leq k\leq n/2$, $(\chi_k|\chi_l)= l+1$
b) Let $m = n/2$ if $n$ is even, and $m = (n − 1)/2$ if $n$ is odd. Deduce that $S_n$ has distinct irreducible characters $\chi (n) = 1_G$, $\chi (n−1,1), χ(n−2,2), . . . , χ(n−m,m)$ such that for all $k\leq m$, $\chi_k = χ (n) + χ (n−1,1) + χ (n−2,2) + · · · + χ (n−k,k)$.
In particular, the class functions $\chi_k − \chi_{k−1}$ are irreducible characters of $S_n$ for $1 \leq k \leq n/2$ and equal to $χ(n−k,k)$.
For part a), what I have been able to deduce thus far is that $\chi_k(1)=|A_k|={n\choose k}$ and that $(\chi_k|\chi_l)= \sum_{g\in G} |Fix_{A_k}(g)||Fix_{A_l}(g)|$ by looking at the matrices since for each $g\in S_n$, the action of $g$ on $V=<e_P>_{P\in A_k} = \mathbb{C}^{|A_k|}$ is given by $e_{P} \mapsto e_{g \cdot P}$
I think I might need to use Burnside's lemma but I'm stuck at the moment. I'm very grateful for any help or suggestion.