Brion atoms for classical types

TL;DR

This paper introduces a uniform description of Brion atoms for classical groups, extending previous work to type D, and proposes involution Schubert polynomials with related conjectures.

Contribution

It provides a comprehensive treatment of Brion atoms for type D and introduces involution Schubert polynomials for all classical types.

Findings

Uniform description of Brion atoms for classical groups

Extension of results to type D

Introduction of involution Schubert polynomials and related conjectures

Abstract

Let $G$ be a classical group defined over the complex numbers with a Borel subgroup $B$. Choose a holomorphic involution of $G$ and let $K$ be its set of fixed points. The group $K$ acts on the flag variety $G/B$ with finitely many orbits and Brion has derived a general formula for the cohomology classes of the corresponding orbit closures as linear combinations of Schubert classes. This article provide a uniform description of the sets of Weyl group elements (which we refer to as Brion atoms) indexing the terms in this formula. This builds on prior work addressing types A, B, and C. The main novelty of our results is a thorough treatment of type D. As one application, we introduce a notion of involution Schubert polynomials for all classical types and present several conjectures related to these objects.