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The external activity complex of a pair of matroids

Andrew Berget, Alex Fink

TL;DR

The paper develops a comprehensive framework linking geometry of Schubert-like varieties for pairs of linear spaces to combinatorics of matroid pairs via the diagonal Dilworth truncation. By performing Gröbner degenerations to initial ideals governed by external activity data, it defines the external activity complex $\Delta_w(M_1,M_2)$, proves its Cohen–Macaulayness, and expresses its finely graded $K$-polynomial in terms of exterior powers of tautological quotient classes. This yields a bivaluative invariant framework and a nonnegative formula for the matroid invariant $\omega(M)$, which in turn proves Speyer’s tropical $f$-vector conjecture. The results unify tropical intersection theory, equivariant $K$-theory, and matroid Hodge-theoretic ideas, providing new positivity results and deep connections between algebraic geometry and matroid combinatorics with implications for valuations, Schubert calculus, and tropical geometry.

Abstract

We introduce the Schubert variety of a pair of linear subspaces in $\mathbf{C}^n$ and the external activity complex of a pair of not necessarily realizable matroids. Both of these generalize constructions of Ardila et al., which occur when one of the linear spaces is one-dimensional. We prove that our external activity complex is Cohen-Macaulay and deduce a formula for its $K$-polynomial in terms of exterior powers of the dual tautological quotient classes of matroids. As a consequence, we deduce a non-negative formula for the matroid invariant $ω(M)$ of Fink, Shaw, and Speyer in terms of certain homology groups of links within an external activity complex, proving the 2005 tropical $f$-vector conjecture of Speyer.

The external activity complex of a pair of matroids

TL;DR

The paper develops a comprehensive framework linking geometry of Schubert-like varieties for pairs of linear spaces to combinatorics of matroid pairs via the diagonal Dilworth truncation. By performing Gröbner degenerations to initial ideals governed by external activity data, it defines the external activity complex , proves its Cohen–Macaulayness, and expresses its finely graded -polynomial in terms of exterior powers of tautological quotient classes. This yields a bivaluative invariant framework and a nonnegative formula for the matroid invariant , which in turn proves Speyer’s tropical -vector conjecture. The results unify tropical intersection theory, equivariant -theory, and matroid Hodge-theoretic ideas, providing new positivity results and deep connections between algebraic geometry and matroid combinatorics with implications for valuations, Schubert calculus, and tropical geometry.

Abstract

We introduce the Schubert variety of a pair of linear subspaces in and the external activity complex of a pair of not necessarily realizable matroids. Both of these generalize constructions of Ardila et al., which occur when one of the linear spaces is one-dimensional. We prove that our external activity complex is Cohen-Macaulay and deduce a formula for its -polynomial in terms of exterior powers of the dual tautological quotient classes of matroids. As a consequence, we deduce a non-negative formula for the matroid invariant of Fink, Shaw, and Speyer in terms of certain homology groups of links within an external activity complex, proving the 2005 tropical -vector conjecture of Speyer.

Paper Structure

This paper contains 46 sections, 81 theorems, 144 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Collapsing of vector bundles
  3. Gröbner degenerations
  4. Pairs of matroids and our main results
  5. The tropical $f$-vector conjecture
  6. Relation to previous work
  7. Organization
  8. Combinatorial background
  9. Simplical complexes
  10. Matroids
  11. Basis activities
  12. Polytopes and valuations
  13. The permutohedron and its fan
  14. Valuations
  15. Intersection theory
  16. ...and 31 more sections

Key Result

Theorem A

Let $L_1,L_2$ be a pair of linear subspaces of $\mathbf{C}^n$ with respective matroids $M_1$ and $M_2$, whose affine Schubert variety is $\hat{Y}_{L_1,L_2} \subset \mathbf{C}^n \times \mathbf{C}^n$. Then

Figures (3)

  • Figure 1: The graphs $G(p)$ of \ref{['ex:6.1']}.
  • Figure 2: Intersecting Chern classes of $\mathcal{Q}_{M_1}$ and $\mathcal{Q}_{M_2}$.
  • Figure 3: The compactified tropical cell complex encoding $\Delta_w(U_{2,3},U_{1,3})$.

Theorems & Definitions (180)

  • Conjecture 1.1: The tropical $f$-vector conjecture speyerThesis
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem : Fink--Shaw--Speyer FSS
  • Theorem E
  • proof
  • Theorem 2.1
  • Theorem 2.2: Hochster's formula, dual version
  • ...and 170 more