Characterization of the annihilator ideals of Gelfand–Tsetlin trapezoid enumeration polynomials

Prove that the family of polynomial ideals \(\mathcal{I}_{h,n}=\{R(\mathbf{X}_n)\in\mathbb{Q}[\mathbf{X}_n]\mid R(\Delta_{\mathbf{k}_n})\GT_h(\mathbf{k}_n)=0\}\) is generated by the polynomials listed in Proposition \ref{list}, equivalently, determine whether those listed polynomials generate the annihilator ideal of \(\GT_h(\mathbf{k}_n)\).

Background

The paper introduces $\mathcal{I}_{h,n}$ as the annihilator ideal of the Gelfand–Tsetlin trapezoid enumeration polynomial under forward-difference operators. The extreme cases $h=0$, $h=n-1$, and $h=n$ are described, and several closure properties and families of elements in these ideals are established. The authors then propose that the listed relations generate the entire ideal, which would provide an inductive description of all annihilator ideals and extend the coinvariant-algebra picture associated with monotone triangles.