1996/10/31 by Hiraku Nakajima, Nakajima, Hiraku
Mathematics · Physics and Astronomy · #05E05 #14C05 #Algebraic Geometry (math.AG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA) #alg-geom #hep-th #math.AG #math.QA #msc:05E05 #msc:14C05 #nlin.SI #q-alg #solv-int
paper · pdf · doi:10.48550/arxiv.alg-geom/9610021
AMSLaTeXv1.2 + epic.sty + eepic.sty + youngtab.sty, 20pages
arxiv created 1996/10/31 · arxiv updated 2009/11/30
The Jack symmetric polynomials Pλ(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.