Maximum total angular momentum of equivalent fermions ($J_{\max }=Nj-N(N-1)/2$)

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 $j,j-1,\dots ,j-N+1$. Their sum is $Nj-N(N-1)/2$, provided $N\le 2j+1$. 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.