= Maximum total angular momentum of equivalent fermions
{title2=$J_{\max}=Nj-N(N-1)/2$}
For $N$ identical <fermions> restricted to one angular-momentum multiplet $j$, with all other one-particle labels fixed, the <Pauli exclusion principle> permits distinct <angular momentum> projections $j,j-1,\ldots,j-N+1$. Their sum is $Nj-N(N-1)/2$, provided $N\leq2j+1$. The resulting <Slater determinant> is killed by the total raising <angular momentum ladder operator> because raising any occupied state either exceeds $j$ or duplicates an occupied state. It is therefore a <highest-weight vector> of precisely this total <angular momentum>. The formula is invariant under replacing particles by holes in the full shell.
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