Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 160 / 2 / a / Solution

Realize ${\xi }^{\lambda }$ as the permutation character on ordered set partitions with row sizes ${\lambda }_1,{\lambda }_2,\dots$. Under $S_m\times S_k$, an orbit is determined by the weak composition $\mu$ whose $i$th part counts elements of $\{m+1,\dots ,n\}$ in row $i$. Its stabilizer is the product of the Young subgroups for $\lambda -\mu$ and $\mu$. The orbit character is therefore the outer tensor product ${\xi }^{\lambda -\mu }\#{\xi }^{\mu }$, and summing the orbits giveswith impossible compositions contributing zero.

Insert this identity into the alternating definition of ${\psi }^{\lambda }$. Group the weak compositions by permutations of their parts and use character straightening; the alternating sum in the first tensor factor is ${\psi }^{\lambda -\mu }$, while the second factors combine once for each partition $\mu ⊢k$. Thus