Lagrangian combinatorics of matroids

TL;DR

This paper advances the Lagrangian geometric approach to matroids by linking the combinatorics of biflats and biflags to basis activities, providing explicit formulas for mixed intersections and strengthening the understanding of the h-vector.

Contribution

It develops the Lagrangian combinatorics of matroids, connecting biflats and biflags to basis activities, and provides explicit combinatorial formulas for mixed intersections.

Findings

Proved the h-vector entries are degrees of mixed intersections of convex functions.

Established a combinatorial formula for mixed intersections using biflags and nbc-bases.

Strengthened the connection between Lagrangian geometry and matroid combinatorics.

Abstract

The Lagrangian geometry of matroids was introduced in [ADH20] through the construction of the conormal fan of a matroid M. We used the conormal fan to give a Lagrangian-geometric interpretation of the h-vector of the broken circuit complex of M: its entries are the degrees of the mixed intersections of certain convex piecewise linear functions $\gamma$ and $\delta$ on the conormal fan of M. By showing that the conormal fan satisfies the Hodge-Riemann relations, we proved Brylawski's conjecture that this h-vector is a log-concave sequence. This sequel explores the Lagrangian combinatorics of matroids, further developing the combinatorics of biflats and biflags of a matroid, and relating them to the theory of basis activities developed by Tutte, Crapo, and Las Vergnas. Our main result is a combinatorial strengthening of the $h$-vector computation: we write the k-th mixed intersection of $\gamma$ and $\delta$ explicitly as a sum of biflags corresponding to the nbc-bases of internal activity k+1.