 Solution

The Pauli exclusion principle allows each one-particle magnetic substate only once, since the fermions share the other quantum numbers. To maximize the total angular momentum projection, occupy $m_j=5/2,3/2,1/2,-1/2$, givingThe corresponding Slater determinant is a highest-weight vector: every attempted raising operation either tries to exceed $j$ or duplicates an occupied substate and vanishes. Therefore it belongs to a multiplet with $\boxed{J_{\max }=4}$, not the distinguishable-particle value $4j=10$. Equivalently, four particles in a six-state shell have two holes whose maximum angular momentum projection is $5/2+3/2=4$. The general maximum total angular momentum of equivalent fermions is $J_{\max }=Nj-N(N-1)/2$ for $N\le 2j+1$.