Giambelli and degeneracy locus formulas for classical G/P spaces

TL;DR

This paper provides explicit formulas for cohomology classes of Schubert varieties in classical G/P spaces, connecting combinatorial Weyl group data with degeneracy locus formulas, advancing the understanding of algebraic geometry of flag varieties.

Contribution

It offers a comprehensive combinatorial framework for Giambelli and degeneracy locus formulas in classical G/P spaces, extending previous partial results.

Findings

Derived explicit cohomology formulas for Schubert varieties

Connected Weyl group combinatorics with degeneracy loci

Enhanced tools for algebraic geometry of flag varieties

Abstract

Let G be a classical complex Lie group, P any parabolic subgroup of G, and X = G/P the corresponding homogeneous space, which parametrizes (isotropic) partial flags of subspaces of a fixed vector space. In the mid 1990s, Fulton, Pragacz, and Ratajski asked for global formulas which express the cohomology classes of the universal Schubert varieties in flag bundles -- when the space X varies in an algebraic family -- in terms of the Chern classes of the vector bundles involved in their definition. This has applications to the theory of degeneracy loci of vector bundles and is closely related to the Giambelli problem for the torus-equivariant cohomology ring of X. In this article, we explain the answer to these questions which was obtained in arXiv:0908.3628, in terms of combinatorial data coming from the Weyl group.