 Olshanskii centralizer lemma

For the standard inclusion $S_{n-1}\subset S_n$, over the complex numbers,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.