What's known about the matroid induced by the Plücker coordinates of the representation of a matroid?

Question

Let $M$ be a linear matroid with ground set $E$ and independent subsets $\mathcal I$, represented by $\rho: E \rightarrow V$. This induces a map $$ \hat\rho: \mathcal I \rightarrow \mathbf P(\Lambda V), \{e_1,\ldots,e_k\} \mapsto [\rho(e_1) \wedge \cdots \wedge \rho(e_k)]. $$ The subset $\operatorname{Im}(\hat\rho) \subset \mathbf P(\Lambda V)$ in turn defines a linear matroid $\hat M$ over $\Lambda V$.

Is there anything known about this matroid $\hat M$, e.g. how it depends on the choice of representation, and what its bases look like in relation to the bases of $M$? I'm interested in particular in the case where $M$ is uniform (for instance, I think the representation shouldn't matter in that case) or arises combinatorially, and would like to obtain combinatorial descriptions of $\hat M$. But any pointers would be greatly appreciated.

Answer 1

The matroid you're considering is the direct sum of the matroids where you take the $k$th wedge power, so it suffices to study those for each $k$.

The question of whether, for an abstract matroid $M$, there is a matroid $\hat{M}$ with the properties that you expect has been studied in the paper "Glueing matroids together: a study of Dilworth truncations and matroid analogues of exterior and symmetric powers" by Mason. I'm not sure if there is a counterexample there (as the paper is hard to access), but one expects the answer to be negative, as there is a counterexample to the analogous question for tensor products in the paper "On products of matroids" by Las Vergnas.

The matroids for $\wedge^0, \wedge^1$ are easy to understand. Brakensiek, Dhar, Gao, Gopi and I have shown that if the field has characteristic $0$ and $\rho$ is generic, then the dual of the matroid for $\wedge^2$ is Kalai's hyperconnectivity matroid. This matroid has been studied a fair amount, but in general there is nothing you could call a combinatorial description of it.

Here you really need $\rho$ to be generic, representing the uniform matroid is not enough: there are realizations of the uniform matroid in characteristic $0$ for which $\wedge^2$ is not dual to the hyperconnectivity matroid. If $\rho$ is generic and the field has positive characteristic, then I don't know if you still get the dual to the hyperconnectivity matroid.