Magnetization sectors of a spin chain ($S=N-2k,\dim {\mathcal{H}}_k=\left(\genfrac{}{}{0ex}{}{N}{k}\right)$)

With $S={\sum }_{n=1}^N{Z}_n$, a computational basis vector containing $k$ excitations has eigenvalue $N-2k$. There are $N+1$ distinct eigenspaces, of dimensions $\left(\genfrac{}{}{0ex}{}{N}{k}\right)$. Any Hamiltonian commuting with $S$ preserves each sector and their population weights. In particular, a state supported in one sector cannot be transferred to a different sector by such controls.