Jucys–Murphy description of the center of a symmetric-group algebra ($Z(F{S}_n)=\{f(J_1,\dots ,J_n):f\text{ symmetric}\}$)

The Young–Jucys–Murphy elements commute, and symmetric polynomials in them are central. The full center theorem holds integrally as well as over fields, so it survives modular reduction. On an ordinary Specht module, the scalar of such a polynomial is its value on the cell-content multiset. The integral generation step is stronger than knowing this description only over a characteristic-zero field and is essential in the modular block argument.