Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 160 / 2 / a / i / Solution

Write $a=\lambda -\mathrm{id}$. The two modified entries satisfyThus $\mu -\mathrm{id}$ is obtained from $\lambda -\mathrm{id}$ by the adjacent transposition $\tau =(i,i+1)$. In the alternating definition of $\psi$, reindexing $\pi$ by $\tau \pi$ preserves every induced permutation character and reverses every sign. Therefore the straightening of a symmetric-group character indexed by a composition gives