Schur polynomials and Weighted Grassmannians

TL;DR

This paper introduces weighted Schur polynomials as symmetric functions representing cohomology classes of weighted Grassmannians, linking them to representation characters and providing determinantal formulas for Schubert classes.

Contribution

It defines weighted Schur polynomials as analogues of Schur polynomials, interprets Schubert structure constants via representation multiplicities, and derives determinantal formulas for weighted Schubert classes.

Findings

Weighted Schur polynomials represent weighted Grassmannian cohomology classes.

Schur polynomials are interpreted as characters of certain representations.

Determinantal formulas for weighted Schubert classes are established.

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.