= Deletion projection for a symmetric group
{title2=$\Pi_n$}
Delete $n$ from the cycle notation of a <permutation> in $S_n$ to obtain a permutation in $S_{n-1}$. Extend this set map linearly to $\Pi_n:\mathbb C S_n\to\mathbb C S_{n-1}$. It is an $S_{n-1}$-bimodule map and the identity on $\mathbb C S_{n-1}$, but is not in general an algebra homomorphism. In particular,
$$
\Pi_n(X_n)=(n-1)1,\qquad \Pi_n(X_n^2)=(n-1)1+2\sum_{i<j<n}(i\ j).
$$
For $n\geq5$ it is the unique bimodule-equivariant set map $S_n\to S_{n-1}$; at $n=4$ another such map exists because $C_{S_3}(S_2)=S_2$.
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