Jack polynomials and Hilbert schemes of points on surfaces

Abstract

The Jack symmetric polynomials $P_λ^{(α)}$ form a class of symmetric polynomials which are indexed by a partition $λ$ and depend rationally on a parameter $α$. They reduced to the Schur polynomials when $α=1$, 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 $α$ in Jack polynomials relates to our surface X by $α= -<C,C>$, where C is the zero section, and <C,C> is the self-intersection number of C.