Exact Ground State of Lieb-Mattis Hamiltonian as a Superposition of N\'eel states

We show that the exact ground state of the Lieb-Mattis Hamiltonian is an equal-weight superposition of all possible classical N'{e}el states, and provide an exact formulation of this superposition in the $z$-spin basis for both $S=1\mathrm{/}2$ and general $S$ using Schwinger bosons. In general, a superposition of possible rotations on a general initial state is symmetric if and only if the initial state has a nonzero overlap with a singlet state and is otherwise made up of states that vanish due to the symmetrization. Most notably, $\mathrm{\mid }s,m=0⟩$ states will vanish if symmetrized, which explains how a superposition of N'{e}el states projects onto its singlet component.