Jack polynomials and Hilbert schemes of points on surfaces

TL;DR

This paper provides a geometric realization of Jack polynomials using Hilbert schemes of points on a specific class of surfaces, connecting algebraic combinatorics with geometric topology.

Contribution

It introduces a novel geometric framework linking Jack polynomials to Hilbert schemes of points on surfaces, extending the known Schur polynomial realization.

Findings

Jack polynomials relate to Hilbert schemes of points on a surface

Parameter α corresponds to the self-intersection number of a zero section

Geometric realization generalizes classical symmetric polynomial representations

Abstract

The Jack symmetric polynomials $P_\lambda^{(\alpha)}$ form a class of symmetric polynomials which are indexed by a partition $\lambda$ and depend rationally on a parameter $\alpha$. They reduced to the Schur polynomials when $\alpha=1$, and to other classical families of symmetric polynomials for several specific parameters. It is well-known that Schur polynomials can be realized as certain elements of homology groups of Grassmann manifolds. The purpose of this paper is to give a similar geometric realization for Jack polynomials. However, spaces which we use are totally different. Our spaces are Hilbert schemes of points on a surface X which is the total space of a line bundle L over the projective line. The parameter $\alpha$ in Jack polynomials relates to our surface X by $\alpha = -<C,C>$, where C is the zero section, and <C,C> is the self-intersection number of C.