= Cycle-sum identity for Young–Jucys–Murphy elements
{title2=$X_2\cdots X_n=\sum_{\pi\text{ an }n\text{-cycle}}\pi$}
In the <group algebra> of $S_n$, the product of the <Young–Jucys–Murphy elements> $X_2,\ldots,X_n$ 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 $(j\ m+1)$ 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 $X_2=(1\ 2)$. The identity turns a product of cell contents into a <central character value of a conjugacy-class sum>.
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