Lattice description of matroid duality

Question

Apologies for this very basic question in matroid theory, but I could not find anything about it online after a bit of searching.

There is a well-known bijective correspondence ("cryptomorphism") between matroids and geometric lattices. Duality for matroids is easy to describe: the bases of the dual matroid $M^*$ are the complements of the bases of $M$.

Question: Is there a "nice" lattice-theoretic description of what matroid duality means for geometric lattices?

Notice that matroid duality does not correspond to order duality (as the Wikipedia page on geometric lattices notes). Indeed, the order dual of a geometric lattice will not in general be a geometric lattice.

I've found some discussion of "adjoints" for geometric lattices, a concept which seems not directly related to my question here, but maybe I'm missing something. Of course it's also possible that the reason I can't find any source giving a simple lattice description of matroid duality is because there isn't one.

Answer 1

I believe that there is no really nice characterization of the type you are looking for. There is some discussion of the matter in Section 3 of Chapter 3 of Welsh's book, Matroid Theory. He points out that geometric lattices are not quite equivalent to matroids; they're essentially equivalent to simple matroids. And one snag is that the dual of a simple matroid need not be simple.

In the exercises, Welsh poses the following open question:

Given two geometric lattices $\mathscr{L}_1,\mathscr{L}_2$, is there any way of recognizing the fact that their associated simple matroids $M(\mathscr{L}_1), M(\mathscr{L}_2)$ are dual apart from the obvious rather tedious exhaustive method of testing whether $M(\mathscr{L}_1) \simeq M(\mathscr{L}_2)^*$?

The abstract of Hua Mao's paper A Necessary and Sufficient Condition for Two Associated Simple Matroids to be Dual (Algebra Colloquium 15 (2008), 511–516) says:

Let $\mathscr{L}_1$ and $\mathscr{L}_2$ be finite geometric lattices, and $M(\mathscr{L}_1)$ and $M(\mathscr{L}_2)$ be their associated matroids. This paper gives a necessary and sufficient condition for $M(\mathscr{L}_1)$ and $M(\mathscr{L}_2)^*$ (the dual matroid of $M(\mathscr{L}_2)$) to be isomorphic, which is actually an affirmative answer to a question of Welsh.

I suspect that the necessary and sufficient condition is still not the "nice" characterization you're hoping for, but I haven't read the paper.