 Solution

For $k=1$, the product is empty and equals the identity, the only $1$-cycle in $S_1$. Suppose the cycle-sum identity for Young–Jucys–Murphy elements holds at $k-1$:Multiplication by $X_k={\sum }_{a<k}(ak)$ gives terms $\sigma (ak)$. With permutation products acting right to left, this inserts $k$ immediately after $a$ in the cycle of $\sigma$: the arrows become $a\mapsto k\mapsto \sigma (a)$, with the other arrows unchanged.

Conversely, in any $k$-cycle there is a unique predecessor $a$ of $k$. Delete $k$ and join that predecessor to the successor of $k$; this uniquely recovers $\sigma$ and the factor $(ak)$. Hence every $k$-cycle occurs exactly once, and no other permutation occurs. Induction provesThus ${\xi }_k$ is a conjugacy-class sum in $\mathbb{C}{S}_k$, although it need not be central in $\mathbb{C}{S}_n$ when $k<n$.