 Residue-content criterion for symmetric-group blocks

For shapes of the same size, equal residue-content multiplicities are equivalent to equal cores of a partition at modulus $p$. Removing a rim $p$-hook subtracts one of each residue. With the number of beta numbers divisible by $p$, the packed-runner charges are $q_r=c_r-c_{r+1}$, so the multiplicities recover the core. Symmetric polynomials in Young–Jucys–Murphy elements act by their evaluations on these residues, and generate the modular center. Equality of its characters is precisely the block-equivalence criterion for simple modules.