A Giambelli formula for classical $G/P$ spaces

TL;DR

This paper presents an explicit combinatorial Giambelli formula for classical partial flag varieties, expressing Schubert classes as polynomials in special generators, with extensions to equivariant cohomology and degeneracy loci.

Contribution

It introduces a new combinatorial Giambelli formula for classical G/P spaces, extending previous results to equivariant cohomology and degeneracy loci settings.

Findings

Provides explicit formulas for Schubert classes in cohomology.

Extends formulas to torus-equivariant cohomology.

Applies to symplectic and orthogonal degeneracy loci.

Abstract

Let $G$ be a classical complex Lie group, $P$ any parabolic subgroup of $G$, and $G/P$ the corresponding partial flag variety. We prove an explicit combinatorial Giambelli formula which expresses an arbitrary Schubert class in the cohomology ring of $G/P$ as a polynomial in certain special Schubert class generators. Our formula extends to one that applies to the torus-equivariant cohomology ring of $G/P$ and to the setting of symplectic and orthogonal degeneracy loci.