Jack polynomials and Hilbert schemes of points on surfaces

Details Jack polynomials and Hilbert schemes of points on surfaces Jack polynomials and Hilbert schemes of points on surfaces

1996-10-31

arxiv.org/abs/alg-geom/9610021v1

Physics

High Energy Physics

High Energy Physics - Theory

AMSLaTeXv1.2 + epic.sty + eepic.sty + youngtab.sty, 20pages

Scientific paper

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 $\alpha = -$, where C is the zero section, and is the self-intersection number of C.

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