The single-relation conjecture for Calogero–Moser spaces and invariant commuting varieties

Let $n\geq 1$ and set

$$k=\frac{n(n+3)}{2}.$$

For the Calogero–Moser space $\mathcal{C}_n$ and the invariant commuting variety $\mathcal{Com}_n$, let $I$ and $J$ denote ideals in the corresponding polynomial algebras, generated via the Poisson bracket $\\{-,-\\}$.

Single-relation conjecture. The coordinate rings satisfy

$$\mathcal{C}[\mathcal{C}_n] \cong \mathbb{C}[a_1,a_2] \otimes \dfrac{\mathbb{C}[a_3,\dots, a_{k}]}{I},$$ $$\mathcal{C}[{\mathcal Com}_n] \cong \mathbb{C}[a_1,a_2]\otimes \dfrac{\mathbb{C}[a_3,\dots, a_{k}]}{J},$$

where $I$ and $J$ are generated via the bracket $\\{-,-\\}$ by a single relation naturally obtained using the Cayley–Hamilton theorem on the corresponding varieties.

For $n\,\dots,4$, the analogous ideals are generated by a single relation, but whether this pattern holds for every $n$ remains open.