= Residue-content criterion for symmetric-group blocks
{title2=$c_r(\lambda)=\#\{(i,j)\in\lambda:j-i\equiv r\pmod p\}$}
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>.
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