When Alexander dual of a simplicial complex is a matroid?

Question

Let $C$ be a simplicial complex on a finite set $V$: that means $C$ is a collection of subsets of $V$ such that if $\sigma\in C$ and $\tau\subseteq \sigma$, then $\tau\in C$.

The Alexander dual $D(C)$ of a simplicial complex $C$ is defined as for $\sigma\subseteq V$, $$\sigma\in D(C) \text{ if and only if } V\setminus \sigma \not\in C.$$

I am wondering is there any criteria of $C$ so that $D(C)$ is a matroid? Or at least, are there interesting examples of some simplicial complex $C$ such that $D(C)$ is a matroid?

Answer 1

There are many simplicial complexes one may associate with a matroid $\mathcal{M}$. Perhaps the most common one is the independence complex consisting of the independent sets of $\mathcal{M}$. Their Alexander duals are easy to recognize via the hyperplane axioms for matroids.

Fact: The Alexander dual of a simplicial complex $\Delta$ is the independence complex of a matroid $\mathcal{M}$ if and only if the facets of $\Delta$ are the hyperplanes of the dual matroid $\mathcal{M}^*$.

Proof. Let $\Delta$ be a simplicial complex whose facets are cohyperplanes of a matroid $\mathcal{M}$ on the ground set $E$. Then $E \setminus \sigma \notin \Delta$ for some some subset $\sigma \subseteq E$ if and only if $E \setminus \sigma$ contains a cobasis of $\mathcal{M}$. But this is equivalent to $\sigma$ being contained in a basis of $\mathcal{M}$, so that $\sigma$ is in the independence complex of $\mathcal{M}$.