Topology of \(\mathbb{Z}_3\)-equivariant Hilbert schemes (Q1733927)

Summary: Motivated by work of \textit{S. Gusein-Zade} et al. [Mosc. Math. J. 10, No. 3, 593--602 (2010; Zbl 1206.14014)], we study a specific generating series of arm and leg statistics on partitions, which is known to compute the Poincaré polynomials of \(\mathbb{Z}_3\)-equivariant Hilbert schemes of points in the plane, where \(\mathbb{Z}_3\) acts diagonally. This generating series has a conjectural product formula, a proof of which has remained elusive over the last ten years. We introduce a new combinatorial correspondence between partitions of \(n\) and \(\{1,2\}\)-compositions of \(n\), which behaves well with respect to the statistic in question. As an application, we use this correspondence to compute the highest Betti numbers of the \(\mathbb{Z}_3\)-equivariant Hilbert schemes.