Equivariant Chern character operators and Okounkov's conjecture

TL;DR

This paper investigates equivariant Chern character operators on Hilbert schemes of points in the complex plane, connecting them to symmetric functions and partially verifying Okounkov's conjecture in this context.

Contribution

It introduces a novel approach linking equivariant cohomology of Hilbert schemes with symmetric functions via deformed vertex operators, advancing understanding of Okounkov's conjecture.

Findings

Partial verification of Okounkov's conjecture in the Hilbert scheme setting.

Establishment of a connection between equivariant cohomology and symmetric functions.

Application of deformed vertex operators to study Chern character operators.

Abstract

In this paper, we study the Chern character operators on the equivariant cohomology of the Hilbert schemes of points in the complex affine plane $C^2$ with the action of the torus $(C^*)^2$, and partially verify Okounkov's Conjecture [Oko, Conjecture 2] in this setting. Our main idea is to apply the connection between the equivariant cohomology of these Hilbert schemes and the ring of symmetric functions, via the deformed vertex operators of Cheng and Wang [CW], (the integral form of) the Jack symmetric functions and the transformed Macdonald symmetric functions of Garsia and Haiman [GH, Hai].