Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tropical matroid Schubert varieties and the graded Möbius algebra

Seungkyu Lee

Abstract

We introduce tropical matroid Schubert varieties, a tropical analogue of arrangement Schubert varieties associated with realisable matroids. We prove that the tropical cohomology ring of the tropical matroid Schubert variety associated to any matroid $M$ is isomorphic to the graded Möbius algebra $\operatorname{B}^\bullet(M)$. This yields a geometric model for $\operatorname{B}^\bullet(M)$, extending the geometric setting of arrangement Schubert varieties to arbitrary matroids, including non-realisable ones.

Tropical matroid Schubert varieties and the graded Möbius algebra

Abstract

We introduce tropical matroid Schubert varieties, a tropical analogue of arrangement Schubert varieties associated with realisable matroids. We prove that the tropical cohomology ring of the tropical matroid Schubert variety associated to any matroid is isomorphic to the graded Möbius algebra . This yields a geometric model for , extending the geometric setting of arrangement Schubert varieties to arbitrary matroids, including non-realisable ones.

Paper Structure

This paper contains 17 sections, 28 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Idea of the proofs
  4. Preliminaries
  5. Matroids and (augmented) Bergman fans
  6. Tropical varieties
  7. Combinatorial identity for Whitney numbers and independence complexes
  8. Geometry and stratification of $Y_M$
  9. Induced polyhedral structure
  10. Polyhedral structure of $Y_M$
  11. Cohomology of $Y_M$ via the rank filtration
  12. Pullback to the canonical compactification of $\Sigma_M^+$
  13. Comparison with arrangement Schubert varieties
  14. Examples
  15. Boolean matroid of rank $2$
  16. ...and 2 more sections

Key Result

Theorem 1.1

The tropical cohomology of $Y_M$ is concentrated in bidegrees $(p,p)$, that is, for $p \neq q$, $\operatorname{H}^{p,q}(Y_M) = 0$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (81)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: \ref{['rankstrat']} and \ref{['stratYM']}
  • Proposition 1
  • Definition 1
  • Lemma 1: Brylawski
  • Definition 2: Huh2017
  • Definition 3: MR4477425
  • Definition 4: MR4477425
  • Remark 1
  • ...and 71 more