Let $V$ a real vector space of dimension $d$. Let $1<k < d-1$. Consider the map induced by the exterior algebra functor: 

$$ \psi:\text{Hom}(V,V) \to \text{Hom}(\bigwedge^kV,\bigwedge^kV)  \, \, \, \, , \, \, \,\psi(A)=\bigwedge^k A$$

>Is the image of $\psi$ closed in the standard topology on the $\text{Hom}$-space?

I know that $\psi(\text{GL}(V)) $ is closed in $ \text{GL}(\bigwedge^{k}V)$
but this does not seem to help. (This follows from the fact $\psi|_{\text{GL}(V)}$ is a morphism of algebraic groups. There is also a simple direct argument [here][2]).
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I think this question is equivalent to the following question:

>Do there exist polynomial relations which **characterize** the $k$-minors of a square $d \times d$ matrix  over $\mathbb{R}$? i.e. is $\psi(\text{Hom}(V,V))$ the zero set of some polynomials defined on $\text{Hom}(\bigwedge^kV,\bigwedge^kV)$?

(By "**characterizing**" I mean *necessary and sufficient* conditions for a given sequence of numbers to be the minors of some matrix. See an explicit statement below).



It seems that finding *explicit* relations is an open problem, e.g. see [this paper][1]. However, I am not asking about an explicit construction, only on mere existence. 






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Here is a more concrete phrasing:

Let $b_{(i_1,\dots,i_k),(j_1,\dots,j_k)} \in \mathbb{R}$, where  $(i_1,\dots,i_k),(j_1,\dots,j_k)$ are increasing multi-indices of order $k$ ( $1 \le i_1 <i_2 < \dots<i_k\le d$). 

>Are there necessary and sufficient conditions for the existence of a $d \times d$ matrix $A$, such that
$ b_{(i_1,\dots,i_k),(j_1,\dots,j_k)}$ is the $k$-minor of $A$, corresponding to rows  $(i_1,\dots,i_k)$, and to columns $(j_1,\dots,j_k)$?


In "conditions" I mean here a set of continuous functions $\{ f_{\lambda}  \}$ in the $\binom{d}{k}^2$ variables $b_{(i_1,\dots,i_k),(j_1,\dots,j_k)}$ such that the $b_{(i_1,\dots,i_k),(j_1,\dots,j_k)}$ are minors if and only if $ f_{\lambda}  (b_{(i_1,\dots,i_k),(j_1,\dots,j_k)})=0$ for every $ f_{\lambda}  $.

[1]:https://www.sciencedirect.com/science/article/pii/S0001870813001606
[2]:https://math.stackexchange.com/a/2777068/104576