Let $k$ be a field, $V$ is a finite dimensional vector space. If $A: V \to V$ is a linear map, it induces a linear map $\wedge^k(A): \wedge^k(V) \to \wedge^k(V)$. Suppose $B: \wedge^k(V) \to \wedge^k(V)$ is any linear map. What are conditions on $B$ that are equivalent to the existence of $A$ such that $B=\wedge^k(A)$?
Notice that the map $A \to \wedge^k(A)$ is polynomial, so we have a polynomial map of affine spaces $\phi: \mathbb{A}^{n^2} \to \mathbb{A}^N$, where $N ={n \choose k}^2$. From the point of view of algebraic geometry I'm asking to find a set of generators of the ideal $I_X$ of the subvariety $X=\phi(\mathbb{A}^{n^2}) \subset \mathbb{A}^N$.