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On algebro-geometric quotients of torus-invariant subvarieties of the flag variety

Luis Y. Meza-Pérez, Pedro L. del Ángel R., Carlos Pompeyo-Gutiérrez, Miguel Angel Dela-Rosa

TL;DR

This work develops a combinatorial framework using matroids to decompose complex flag varieties into affine thin Schubert cells, enabling a structured analysis of torus-invariant subvarieties.It proves the existence of universal geometric and categorical quotients for torus actions on these cells and their intersections, and establishes a surjective bridge from these quotients to the flag variety's $T$-invariant points, connecting to invariant homology.The authors provide a complete classification and counting formulas for thin Schubert cells in the flag variety $F_{1<n-1}$, including explicit incidence criteria and dimension counts, and offer computational tools to realize the counts.Overall, the paper extends prior Grassmannian results to flag varieties, yielding a robust quotient/homology framework that can inform further study of invariant geometry and Schubert-theoretic structures.

Abstract

In this paper, we study the subvarieties of a complex flag variety that are invariant under the action of a maximal torus. Using combinatorial techniques derived from matroid theory, we introduce a decomposition of this variety into affine, locally closed subsets, which we refer to as thin Schubert cells, each indexed by an element of a Cartesian product of matroids. We also show that the set of orbits for each of these thin Schubert cells under the action of the torus is, in fact, an orbit space, which gives rise to topologically trivial fiber bundles. As a consequence of this, we prove the existence of algebro-geometric quotients in the sense of Mumford's Geometric Invariant Theory. The main result of this work is the existence of a surjective map from the set of geometric quotients of thin Schubert cells to the invariant scheme-theoretic points of a complex flag variety, which allows us to decompose it in terms of such quotients and also define a map from the set of these geometric quotients to the invariant homology of the variety. Finally, we give a complete description of the thin Schubert cells for the special case of the flag variety $\mathds{F}_{1<n-1}(\mathds{C}^n)$ and derive explicit counting formulas.

On algebro-geometric quotients of torus-invariant subvarieties of the flag variety

TL;DR

This work develops a combinatorial framework using matroids to decompose complex flag varieties into affine thin Schubert cells, enabling a structured analysis of torus-invariant subvarieties.It proves the existence of universal geometric and categorical quotients for torus actions on these cells and their intersections, and establishes a surjective bridge from these quotients to the flag variety's $T$-invariant points, connecting to invariant homology.The authors provide a complete classification and counting formulas for thin Schubert cells in the flag variety $F_{1<n-1}$, including explicit incidence criteria and dimension counts, and offer computational tools to realize the counts.Overall, the paper extends prior Grassmannian results to flag varieties, yielding a robust quotient/homology framework that can inform further study of invariant geometry and Schubert-theoretic structures.

Abstract

In this paper, we study the subvarieties of a complex flag variety that are invariant under the action of a maximal torus. Using combinatorial techniques derived from matroid theory, we introduce a decomposition of this variety into affine, locally closed subsets, which we refer to as thin Schubert cells, each indexed by an element of a Cartesian product of matroids. We also show that the set of orbits for each of these thin Schubert cells under the action of the torus is, in fact, an orbit space, which gives rise to topologically trivial fiber bundles. As a consequence of this, we prove the existence of algebro-geometric quotients in the sense of Mumford's Geometric Invariant Theory. The main result of this work is the existence of a surjective map from the set of geometric quotients of thin Schubert cells to the invariant scheme-theoretic points of a complex flag variety, which allows us to decompose it in terms of such quotients and also define a map from the set of these geometric quotients to the invariant homology of the variety. Finally, we give a complete description of the thin Schubert cells for the special case of the flag variety and derive explicit counting formulas.

Paper Structure

This paper contains 11 sections, 21 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matroids and the Grassmannian variety $G(d,n)$
  4. Thin Schubert cells and matroidal decompositions on flag varieties
  5. Algebro-geometric quotients and the homology of $T$-invariant subvarieties
  6. Homology of $T$-equivariant subvarieties
  7. Thin Schubert cells in the flag variety $\mathds{F}_{1<n-1}(\mathds{C}^n)$
  8. Conclusions
  9. Programming codes
  10. Maple (v. 2024.2)
  11. Mathematica (v. 14.2)

Key Result

Proposition 2.2

The Plücker embedding $\psi$ is $\mathfrak{S}_n$-equivariant, i.e., the following diagram \begin{tikzcd} \mathfrak{S}_n\times G(d,n)\arrow{r}{\sbullet}\arrow{d}[swap]{\operatorname{id}\times\,\psi} &G(d,n) \arrow[hook]{d}{\psi}\\ \mathfrak{S}_n\times\mathds{P}_{\math

Theorems & Definitions (48)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Remark 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • ...and 38 more