The external activity complex of a pair of matroids

TL;DR

This paper introduces the external activity complex for pairs of matroids, generalizing previous work, and proves it is Cohen-Macaulay, providing new formulas for matroid invariants and confirming a longstanding conjecture.

Contribution

It extends the concept of the external activity complex to pairs of matroids, proves its Cohen-Macaulay property, and derives formulas for matroid invariants, resolving a tropical $f$-vector conjecture.

Findings

The external activity complex is Cohen-Macaulay.

A formula for the $K$-polynomial in terms of exterior powers.

A non-negative formula for the matroid invariant $(M)$.

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 $\omega(M)$ 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.