algebraicMatroid -- The algebraic matroid of the tropical cone that induces the matroid

Let $V \subseteq \CC^n$ be an affine variety. The algebraic matroid of $V$ is a matroid whose independent sets $S \subseteq [n]$ are the subsets such that the projection from $V$ to the coordinates indexed by $S$ is a dominant morphism. Similarly, if $C \subseteq \RR^n$ is a polyhedral cone, then the algebraic matroid of $C$ is the matroid whose independent sets $S \subseteq [n]$ are the subsets such that image of the projection of $C$ onto the coordinates indexed by $S$ is full-dimensional.

In the case of the affine cone of Grassmannian under the Pluecker embedding, there are a few different ways to compute its algebraic matroid. One way is to use its tropicalization. The algebraic matroid of the Grassmannian is equal to the matroid whose bases are the union of all bases of the algebraic matroid for all maximal cones of Trop Gr($k$, $n$).

For each coherent matching field, we compute its cone in the tropicalization of the Grassmannian. We compute the algebraic matroid of this cone. To view the bases of this matroid in terms of the $k$-subsets of $[n]$, use the function algebraicMatroidBases. Similarly, to view its circuits use algebraicMatroidCircuits