Topology of Z_3 equivariant Hilbert schemes

TL;DR

This paper investigates the topology of Z_3-equivariant Hilbert schemes by analyzing a generating series related to partitions, introduces a new combinatorial correspondence, and computes the highest Betti numbers, advancing understanding of their geometric structure.

Contribution

It introduces a novel combinatorial correspondence between partitions and compositions that facilitates the computation of topological invariants of Z_3-equivariant Hilbert schemes.

Findings

Established a new combinatorial method for analyzing partitions.

Computed the highest Betti numbers of Z_3-equivariant Hilbert schemes.

Provided insights into the conjectural product formula for the generating series.

Abstract

Motivated by work of Gusein-Zade, Luengo, and Melle-Hern\'andez, we study a specific generating series of arm and leg statistics on partitions, which is known to compute the Poincar\'e polynomials of Z_3-equivariant Hilbert schemes of points in the plane, where 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 Z_3 equivariant Hilbert schemes.