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 $c(\lambda)$.

Solution

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

 Solution (source code)