Ansatz for $(-1)^{n-1}\nabla p_n$

TL;DR

This paper constructs a family of equivariant coherent sheaves on the Hilbert scheme of points in the affine plane, linking their Euler characteristics to symmetric functions and proving a higher cohomology vanishing result.

Contribution

It introduces a new family of equivariant sheaves on the Hilbert scheme with connections to symmetric functions and establishes a higher cohomology vanishing theorem.

Findings

Euler characteristics related to symmetric functions

Higher cohomology vanishing proven

Effective $S_n$ module structure established

Abstract

We construct a special family of equivariant coherent sheaves on the Hilbert scheme on $n$-points in the affine plane. The equivariant Euler characteristic of these sheaves are closely related to the symmetic functions $(-1)^{n-1} \nabla p_n$. We prove a higher cohomology vanishing result of these sheaves. It follows from the Bridgeland-King-Reid correspondence that there is an effective $S_n$ module underlying the aforementioned family of symmetric functions.