Hall-Littlewood polynomials and vector bundles on the Hilbert scheme

TL;DR

This paper provides an explicit asymmetric formula for the equivariant Euler characteristic of certain vector bundles on the Hilbert scheme, revealing nonnegative coefficients and connecting to Hall-Littlewood polynomials.

Contribution

It introduces a novel explicit formula for the Euler characteristic of bundles on the Hilbert scheme using Hall-Littlewood polynomials, expanding contour integral formulas.

Findings

The Euler characteristic has nonnegative coefficients in torus variables.

Derived an explicit asymmetric formula for the Euler characteristic.

Connected the formula to Hall-Littlewood vertex operators.

Abstract

Let $E$ be the bundle defined by applying a polynomial representation of $GL_n$ to the tautological bundle on the Hilbert scheme of $n$ points in the complex plane. By a result of Haiman, the Cech cohomology groups $H^i(E)$ vanish for all $i>0$. It follows that the equivariant Euler characteristic with respect to the standard two-dimensional torus action has nonnegative coefficients in the torus variables $z_1,z_2$, because they count the dimensions of the weight spaces of $H^0(E)$. We derive a very explicit asymmetric formula for this Euler characteristic which has this property, by expanding known contour integral formulas for the Euler characteristic stemming from the quiver description in $z_2$, and calculating the coefficients using Jing's Hall-Littlewood vertex operator with parameter $z_1$.