Gysin maps, duality and Schubert classes

TL;DR

This paper develops a Gysin formula and duality theorem in Schubert calculus, enabling new proofs of classical formulas for vector bundles and advancing the understanding of Schubert classes in Grassmann bundles.

Contribution

It introduces a Gysin formula for Schubert bundles and a strong duality theorem, providing a novel approach to computing fundamental classes and proving the Giambelli formula.

Findings

Established a Gysin formula for Schubert bundles

Proved a strong duality theorem in Schubert calculus

Provided a new proof of the Giambelli formula

Abstract

We establish a Gysin formula for Schubert bundles and a strong version of the duality theorem in Schubert calculus on Grassmann bundles. We then combine them to compute the fundamental classes of Schubert bundles in Grassmann bundles, which yields a new proof of the Giambelli formula for vector bundles.