Solution
= Solution
Let $s_i=(i\ i+1)$. Since $\mu-\mathrm{id}=s_i(\lambda-\mathrm{id})$, reindex the alternating sum defining $\psi^\mu$ by $\pi\mapsto s_i\pi$. This preserves all permutation-character terms and reverses every <sign of a permutation>, because $\operatorname{sgn}(s_i\pi)=-\operatorname{sgn}(\pi)$. Hence $\psi^\mu=-\psi^\lambda$.