Schur polynomials and Weighted Grassmannians

Abstract

In this paper, we introduce a family of symmetric polynomials by specializing the factorial Schur polynomials. These polynomials represent the weighted Schubert classes of the cohomology of the weighted Grassmannian introduced by Corti-Reid, and we regard these polynomials as analogue of the Schur polynomials. We show that those twisted Schur polynomials are the characters of certain representations. Thus we give an interpretation of the Schubert structure constants of the weighted Grassmannians as the (rational) multiplicities of tensor products of the representations. Furthermore, we derive two types of determinantal formulas for the weighted Schubert classes, in terms of special weighted Schubert classes, and also in terms of Chern classes of tautological orbi-bundles.