Cycle-sum identity for Young–Jucys–Murphy elements ($X_2\cdots X_n={\sum }_{\pi \text{ an }n\text{-cycle}}\pi$)

In the group algebra of $S_n$, the product of the Young–Jucys–Murphy elements $X_2,\dots ,X_n$ is the sum of all $n$-cycles, each with coefficient one. To prove it, multiply the sum of all $m$-cycles by $X_{m+1}$. Right multiplication by $(jm+1)$ inserts $m+1$ immediately after $j$ in the cycle. Every $(m+1)$-cycle has a unique predecessor of $m+1$, so deletion inverts this insertion bijectively. Induction starts at $X_2=(12)$. The identity turns a product of cell contents into a central character value of a conjugacy-class sum.