Face numbers of Engstr\"om representations of matroids

TL;DR

This paper calculates the face numbers of Engström representations of matroids based on their indexing complexes and provides bounds on the total number of faces, revealing polynomial growth related to the matroid's size.

Contribution

It derives explicit formulas for face numbers of Engström representations and establishes upper bounds on their complexity based on the properties of the indexing complexes.

Findings

Face numbers expressed in terms of indexing complexes

Upper bounds on total face counts for Engström representations

Polynomial bounds related to matroid size and rank

Abstract

A classic problem in matroid theory is to find subspace arrangements, specifically hyperplane and pseudosphere arrangements, whose intersection posets are isomorphic to a prescribed geometric lattice. Engstr\"om recently showed how to construct an infinite family of such subspace arrangements, indexed by the set of finite regular CW complexes. In this note, we compute the face numbers of these representations (in terms of the face numbers of the indexing complexes) and give upper bounds on the total number of faces in these objects. In particular, we show that, for a fixed rank, the total number of faces in the Engstr\"om representation corresponding to a codimension one homotopy sphere arrangement is bounded above by a polynomial in the number of elements of the matroid with degree one less than its rank.