Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 160 / 1 / c

Conjugation by a permutation merely permutes the transpositions. Their sum $C$ is therefore a conjugacy class sum and belongs to the center of an associative algebra $\mathbb{C}{S}_n$. Since the complex Specht module is an irreducible representation, Schur lemma says that $C$ acts on it as a scalar, say $c(\lambda )$.

Compare the coefficient of $\{t\}$ in $Ce(t)$. For a transposition $\tau$, a summand $\tau \pi \{t\}$ with $\pi \in C(t)$ equals $\{t\}$ exactly when $\tau \pi \in R(t)$, equivalently $\tau \in R(t)C(t)$. Part b(i) then leaves two cases: a row transposition contributes $+1$ through $\pi =1$, while a column transposition contributes $-1$ through $\pi =\tau$. Every other transposition contributes zero.

The coefficient is consequently the number of row transpositions minus the number of column transpositions, which part b(ii) identifies with ${\sum }_{(i,j)\in \lambda }{c}_{i,j}(\lambda )$. The coefficient of $\{t\}$ in $c(\lambda )e(t)$ is $c(\lambda )$, so