# Fundamental circuit characterization of matroid independence complexes

I have the following characterization of independence complexes of matroids, which I think is standard but I can't find a reference. Here it goes:

A pure simplicial complex $$\Delta$$ is the independence complex of a matroid if and only for every facet $$F$$ of $$\Delta$$ and every vertex $$v$$ that is not in $$F$$, there is a unique minimal non face $$G$$ of $$\Delta$$ that is contained in $$F\cup\{v\}$$.

In matroid theory language this is caled the fundamental circuit of $$F$$ and $$v$$ (usually written $$Circ(B, b)$$ in that language).

The proof is quite simple, but I am wondering if it is known.

Edit: Let me add an analogous formulation that may be of interest, and perhaps of some help. It is well known (found for example in Stanley's green book), that a simplicial complex is a matroid independence complex if and only every induced subcomplex is pure.

What the characterization above tells us is that we only have to check this purity for of size two larger than the dimension that contain a facet. In other words, there is much more 'economical' way to check for matroidness in terms of purity.

New contributor
José Alejandro Samper is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.