A Geometric Definition Of Schubert Polynomials and Dual Schubert Polynomials For Classical Lie Groups

TL;DR

This paper introduces a geometric approach to defining Schubert and dual Schubert polynomials for classical Lie groups by exploring the topology of projective Stiefel manifolds and their embeddings.

Contribution

It provides a new geometric framework for Schubert polynomials using topological properties of Stiefel manifolds and their embeddings into infinite-dimensional spaces.

Findings

Computed cohomology rings of projective Stiefel manifolds

Classified cohomology endomorphisms of these manifolds

Defined and analyzed properties of Schubert and dual Schubert polynomials

Abstract

In this paper, we first discuss the topological properties of projective Stiefel manifolds, we compute their cohomology rings and classify their cohomology endomorphisms; Then by embedding the flag manifold of a classical Lie group into its corresponding infinite dimensional projective Stiefel manifold(which is homotopic to the product of infinite dimensional complex projective space $\mathbb{C}P^{\infty}$), we define the Schubert polynomials and dual Schubert polynomials. Finally we discuss the property and the computation of these polynomials.