Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 103 / 1 / a / Solution

Work over $\mathbb{C}$, with the standard inclusions $S_r\subseteq S_n$ fixing the letters $r+1,\dots ,n$. In the group algebra, the Young–Jucys–Murphy elements areThus $X_i$ adds all transpositions involving the newest letter. Conjugating by $S_{i-1}$ merely permutes the summands, so $X_i$ commutes with $\mathbb{C}{S}_{i-1}$. In particular, if $j<i$, then $X_j\in \mathbb{C}{S}_{i-1}$ and $X_i{X}_j=X_j{X}_i$. Hence the whole family is commuting.

Put ${\Omega }_r={\sum }_{1\le a<b\le r}(ab)$, the transposition conjugacy-class sum in $S_r$, with ${\Omega }_1=0$. ThenThis identity will connect the Young–Jucys–Murphy elements with the centres of the successive group algebras.