Face enumeration for split matroid polytopes

TL;DR

This paper studies the face numbers of split matroid polytopes, providing formulas to compute their f-vectors based on combinatorial data, thus avoiding complex convex hull computations.

Contribution

It introduces explicit formulas for face numbers of split matroid polytopes, linking them to cyclic flats and modular pairs, and explores special cases like sparse paving and rank two matroids.

Findings

Face numbers depend on cyclic flats and modular pairs.

Formulas enable f-vector computation without convex hulls.

Analysis of sparse paving and rank two matroids.

Abstract

This paper initiates the explicit study of face numbers of matroid polytopes and their computation. We prove that, for the large class of split matroid polytopes, their face numbers depend solely on the number of cyclic flats of each rank and size, together with information on the modular pairs of cyclic flats. We provide a formula which allows us to calculate $f$-vectors without the need of taking convex hulls or computing face lattices. We discuss the particular cases of sparse paving matroids and rank two matroids, which are of independent interest due to their appearances in other combinatorial and geometric settings.