Generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes

TL;DR

This paper proves a remarkable infinite product formula for the generating series of Poincare polynomials of quasihomogeneous Hilbert schemes and relates it to affine Lie algebra characters.

Contribution

It introduces a new infinite product decomposition for these generating series and connects them to affine Lie algebra representations.

Findings

Infinite product decomposition of generating series

Explicit computation of quasihomogeneous components

Connection to affine Lie algebra characters

Abstract

In this paper we prove that the generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes of points in the plane has a beautiful decomposition into an infinite product. We also compute the generating series of the numbers of quasihomogeneous components in a moduli space of sheaves on the projective plane. The answer is given in terms of characters of the affine Lie algebra $\hat{sl}_m$.