Another characterization of matroids Has anyone seen the following characterization of matroids?
Let $\Delta$ be a simplicial complex on finite ground set $E$.  Then $\Delta$ is a matroid complex if and only if, for every $X\subseteq E$ and any two facets (maximal faces) $\sigma,\tau$ of the induced subcomplex $\Delta|_X$, the links of $\sigma$ and $\tau$ in $\Delta$ are equal.
This seems to be nontrivial, but not too hard to prove (perhaps a [2] or [2+] on Stanley's scale).  The $\implies$ direction is conceptually clearer: the two links are both the matroid contractions of $A$, and the proof is by basis exchange and induction.  For $\impliedby$, one can prove the contrapositive, using the criterion that $\Delta$ is a matroid complex iff every induced subcomplex is pure.
I have not seen this in the literature.  Has anyone come across it before?
 A: Yes though this is commonly stated using clutters and their associated minors, (identify your ASC with its clutter of facets) basically you're describing a forbidden minor characterisation of a matroid basis clutter (the forbidden clutter minor is $M=(\{1,2,3\},\{\{1\},\{2,3\}\})$ explaining why you can rephrase the absence of said minor in terms of the links of facets in all restrictions coinciding). Seymour for example gave a forbidden clutter minor characterisation of not just the basis clutters of matroids but of binary matroids as well as many other interesting clutters:
https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-12.3.356
Its a cool identity, though yeah its at least 40-50 years old, for example here are some course notes I found online that give your result under the header "proposition 2.1" on page 3 here:
https://sites.lafayette.edu/traldil/files/2010/06/matroids.pdf
I doubt you'll be able to find when it was originally noticed and definitely not using that notation since I think Tutte was still calling undirected cycles in graphs "polygons" during the 1960s which suggests to me that the terminology for simplicial complexes was not even standardized in usage. Though I am surprised no one has answered your question in these past two months, I am very new to matroid theory myself, though I've been trying to read an article on MFMC matroids and many of the lemmas were from an old paper on clutters which is why I recognised your identity.
