= Olshanskii centralizer lemma
{c}
For the standard inclusion $S_{n-1}\subset S_n$, over the <complex numbers>,
$$
C_{\mathbb C S_n}(\mathbb C S_{n-1})=\operatorname{alg}(Z(\mathbb C S_{n-1}),X_n).
$$
Here $X_n$ is a <Young–Jucys–Murphy element>. One way to see generation is to use <multiplicity-free restriction>: a central idempotent of $S_{n-1}$ selects a preceding shape, and the distinct contents of its <addable nodes of a Young diagram> distinguish all possible succeeding shapes. Polynomial interpolation in $X_n$ supplies every diagonal projection in the <centralizer of a subalgebra>.
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