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Brion atoms for classical types

Eric Marberg

TL;DR

Brion's cohomology formula expresses K-orbit closures on G/B as sums of Schubert classes with coefficients powers of two; this work delivers a uniform, combinatorial description of the indexing Brion atoms across all classical types, with a thorough treatment of type D. It develops a cohesive framework based on Richardson–Springer data, extended Brion atoms, and a shape/matching approach built from clans, Demazure products, and NCSP matchings, enabling explicit decomposition and graded poset structures for the Brion terms. The main theorems provide constructive generators and partial orders that let one enumerate the Brion atoms and compute Brion coefficients efficiently, while connecting to involution Schubert polynomials for all classical types and proposing conjectures. The results unify A–D types under a single combinatorial paradigm, and the type-D treatment yields a robust, algorithmic pathway to generalized Schubert calculus in symmetric varieties.

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.

Brion atoms for classical types

TL;DR

Brion's cohomology formula expresses K-orbit closures on G/B as sums of Schubert classes with coefficients powers of two; this work delivers a uniform, combinatorial description of the indexing Brion atoms across all classical types, with a thorough treatment of type D. It develops a cohesive framework based on Richardson–Springer data, extended Brion atoms, and a shape/matching approach built from clans, Demazure products, and NCSP matchings, enabling explicit decomposition and graded poset structures for the Brion terms. The main theorems provide constructive generators and partial orders that let one enumerate the Brion atoms and compute Brion coefficients efficiently, while connecting to involution Schubert polynomials for all classical types and proposing conjectures. The results unify A–D types under a single combinatorial paradigm, and the type-D treatment yields a robust, algorithmic pathway to generalized Schubert calculus in symmetric varieties.

Abstract

Let be a classical group defined over the complex numbers with a Borel subgroup . Choose a holomorphic involution of and let be its set of fixed points. The group acts on the flag variety 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.

Paper Structure

This paper contains 50 sections, 66 theorems, 287 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Symmetric subroups
  3. Classical groups and their Weyl groups
  4. Schubert varieties and Schubert classes
  5. Brion's cohomology formula
  6. Main results
  7. Orbit indexing sets and the weak order graph
  8. Permutations
  9. Twisted involutions
  10. Clans
  11. Clans for type A
  12. Clans for types B and C
  13. Clans for type D
  14. Orbit parametrizations
  15. Demazure products
  16. ...and 35 more sections

Key Result

Theorem 1.3

Fix a pair $(G,K)$ in classical type. Suppose $\gamma \in \Gamma^G_K$ and $z = \psi^G_K(\gamma) \in I^G_K$. Then Moreover, there is a map $d_z : \mathcal{E}^G_K(z)\to \mathbb{N}$ such that $d_\gamma(w) = d_z(w)$ for all $w \in \mathcal{W}^G_K(\gamma)$.

Figures (1)

  • Figure 1: Paths in weak order for types BI (left) and CI (right) lifting the sequence $P$ in Example \ref{['lift-ex']}. The path in type BI has 3 doubled arrows while the path in type CI only has one.

Theorems & Definitions (187)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • ...and 177 more