On the coordinate rings of Calogero-Moser spaces and the invariant commuting variety of a pair of matrices

TL;DR

This work provides an explicit presentation of the coordinate rings for the fourth Calogero-Moser space $\mathcal{C}_4$ and the invariant commuting variety $\mathcal{C}om_4$ by reducing to 14 trace generators and deriving a single key relation $r_1$ via the Fourier map and Cayley–Hamilton identities. It shows that $\mathbb{C}[\mathcal{C}_4] \cong \mathbb{C}[a_1,a_2]\otimes \mathbb{C}[a_3,...,a_{14}]/I$ and $\mathbb{C}[\mathcal{C}om_4] \cong \mathbb{C}[a_1,a_2]\otimes \mathbb{C}[a_3,...,a_{14}]/J$, with $I$ generated by $r_1$ (under a Poisson bracket) and $J$ by $r_1-8a_3$, establishing a Poisson-algebraic presentation. The authors compute the Grothendieck-ring classes $[\mathcal{C}om_4]=\mathbb{L}^8$ and $[\mathcal{C}_4]=\mathbb{L}^8-\mathbb{L}^7+2\mathbb{L}^6-\mathbb{L}^5$, and use stratification and Gröbner-basis methods to confirm these results, also linking to the Hilbert scheme $\mathrm{Hilb}_n(\mathbb{C}^2)$ via the Hilbert-Chow morphism. Beyond $n=4$, they conjecture a universal description in terms of a single relation for $\mathbb{C}[\mathcal{C}_n]$ and $\mathbb{C}[\mathcal{C}om_n]$, and propose a Betti-number–driven formula for their Grothendieck classes, together with connections to necklace Lie algebras and restriction theorems.

Abstract

This paper presents a comprehensive description of the coordinate rings and Poisson brackets associated with the fourth Calogero-Moser space and invariant commuting pairs of matrices of size four. As an application, we compute their respective classes in the Grothendieck ring of the category of complex varieties and we offer some novel insights about the geometry of the Hilbert scheme of points on the affine plane.

Theorems & Definitions (20)

• Theorem 1.1
• Conjecture 1
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Lemma 3.1
• Proposition 3.2: G
• Lemma 3.3: N
• Remark 3.4