  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.
