Flawlessness of $h$-vectors of broken circuit complexes

TL;DR

This paper proves that for matroids representable over a field of characteristic zero, the $h$-vector of their broken circuit complex satisfies certain symmetry and monotonicity inequalities, confirming a major open conjecture.

Contribution

It establishes the validity of the $h$-vector inequalities for a broad class of matroids, specifically those representable over characteristic zero fields.

Findings

Confirmed the inequalities for $h$-vectors of broken circuit complexes in characteristic zero.

Provided a positive answer to a major open question in matroid theory.

Enhanced understanding of the combinatorial properties of matroids.

Abstract

One of the major open questions in matroid theory asks whether the $h$-vector $(h_0,h_1,\ldots,h_s)$ of the broken circuit complex of a matroid $M$ satisfies the following inequalities: $$ h_0\leq h_1\leq \cdots\leq h_{\lfloor s/2\rfloor} \quad \text{and}\qua h_i\le h_{s-i}\ \text{ for }\ 0\leq i \leq \lfloor s/2\rfloor. $$ This paper affirmatively answers the question for matroids that are representable over a field of characteristic zero.