Exercise 25, § 21

Consider the representation $\pi_{\mathbf{n}}$ of $\mathfrak{S}_n$ associated with the action of $\mathfrak{S}_n$ on the set $\mathbf{n}=[1, n]$ (Exercise 6). Let $k$ be a positive integer and $\chi_k$ the character of the representation $\wedge^k\pi_{\mathbf{n}}$.

a) Let $\sigma \in \mathfrak{S}_n$. Prove the formula $\chi_k(\sigma ) =\sum\varepsilon (\sigma |S)$, where the sum is taken over the set of subsets S of $\mathbf{n}$ with $k$ elements such that $\sigma (S) = S$.

b) Prove that $\wedge^k\pi_{\mathbf{n}}$ is the irreducible representation of $\mathfrak{S}_n$ associated with the element $(n-k,1, . . . ,1)$ of $\mathscr{S}_n$ (calculate the character of this representation using the Frobenius formula, observing that the only terms of the sum

Δ($\mathbf{X}$)$f^{\boldsymbol{\alpha }}(\mathbf{X}) =\sum_{\sigma\in\mathfrak{S}_n}\varepsilon (\sigma )X^{\sigma(n)-1}_1. . .X^{\sigma(1)-1}_nf^{\boldsymbol{\alpha }}(\mathbf{X})$

for which the coefficient of $\mathbf{X}^{\mathbf{p}'}$ is nonzero correspond to the permutations $\sigma$ that satisfy $\sigma (i)\leqslant i+ 1$ for every $i$ and $\sigma (i) =i$ for $1\leqslant i < n-k.)$