 Solution

Total symmetry makes a component depend only on its number $n_{\uparrow }$ of up indices. The possible values are $0,1,\dots ,N$, givingA normalized basis is $|n_{\uparrow },N-n_{\uparrow }⟩$, with $S_z$ eigenvalue $\hslash (n_{\uparrow }-N/2)$. It is precisely one spin-$N/2$ multiplet. These are independent complex amplitudes before normalization and removal of a common quantum phase, rather than $N+1$ independent real physical parameters.