Jack polynomials and Hilbert schemes of points on surfaces

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.

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References in corpus (1)

• A recursion and a combinatorial formula for Jack polynomials

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