Raising and lowering operators commuting with the periodic Motzkin Hamiltonian

Prove that there exist nonlocal operators Σ^+ and Σ^- on (C^3)^{⊗N}, built from local spin-1 operators s_i^± by Σ^± = ∑_{r_1,…,r_N∈{−2,−1,0,1,2}, r_1+⋯+r_N=±1} s_1^{r_1}⋯s_N^{r_N} (with s_i^0 = I, s_i^{±1} = s_i^±, s_i^{±2} = (s_i^±)^2) that commute with the periodic Motzkin Hamiltonian H^Periodic and act as ladder operators on the conjectured ground-state basis {v_{S^z}} by Σ^± v_{S^z} = c_±(S^z) v_{S^z±1} for S^z ≠ ±N and Σ^± v_{±N} = 0, equivalently Σ^± = Res_{λ=0} ∏_{i=1}^N (λ^{−2}(s_i^±)^2 + λ^{−1}s_i^± + I + λ s_i^∓ + λ^2 (s_i^∓)^2).

Background

Motivated by the observed (2N+1)-fold ground-state degeneracy labeled by S^z, the authors propose the existence of global ladder operators that connect adjacent S^z sectors within the ground-state manifold and commute with the Hamiltonian.

They provide explicit nonlocal formulas for the proposed operators Σ^± in terms of local spin-1 raising/lowering matrices, together with an equivalent residue representation, and assert their commutation with H^Periodic and ladder action on the v_{S^z}.