# How independent are the minors of a matrix?

Let $$M$$ be a $$p\times q$$ matrix (say, $$p\leq q$$) with formal coefficients $$m_{i,j}$$. It has $$pq$$ entries, and $${q\choose p}$$ minors of maximal size $$p\times p$$, which gives us a map $$\varphi_{p,q}: \Bbb Z^{pq}\to \Bbb Z^{q\choose p}$$

I'm interested in the surjectivity of this map. It is easy to see that $$\varphi_{3,4}$$, for instance, is surjective, meaning that we can adjust the coefficients of a $$3\times 4$$ matrix to give each of the $$3\times 3$$ minors a prescribed value.

In most cases, $${q\choose p}$$ is much bigger than $$pq$$, so it feels unlikely that $$\varphi$$ will be surjective. But what if we keep only a relatively small amount of minors?

Question 1: For what values of $$p,q,r\in\Bbb N$$ is there a map $$f:\Bbb Z^{q\choose p}\to \Bbb Z^r$$ defined by keeping only some $$r$$ coordinates, such that $$f\circ \varphi_{p,q}$$ is surjective?

Question 2 (less general but the one I'm really interested in) : Take $$p=3, q=6$$ and forget about the three minors corresponding to columns $$(1, 4, 5), (2, 4, 6), (3, 5, 6)$$. Is the resulting map $$\Bbb Z^{18}\to \Bbb Z^{17}$$ surjective?

• In your very special case does it help that the 20 minors satisfy a quadratic equation (rather a la Klein quadric)? – ancientmathematician Jun 16 at 13:42
• @ancientmathematician I think I see what you mean - stacking two copies of M one on top of the other. It does prove that $\varphi_{3,6}$ is not surjective, and it's certainly a relation to have in mind. I can't see yet how it would imply the non-surjectivity of the 17 minors map. Thanks for the idea. – Arnaud Mortier Jun 16 at 13:56
• My geometric insight is poor, but I can't see how one could project even all of this $(10,-10)$ quadric onto the whole of a 17-space. – ancientmathematician Jun 16 at 14:47
• @ancientmathematician Actually there are 35 independent such quadratic equations, and yes this does help a lot - see answer below. Thanks again. – Arnaud Mortier Jun 19 at 12:56

Thanks to a comment by @ancientmathematician, I was able to focus my research and to answer Question 2: the $$\Bbb Z^{18}\to \Bbb Z^{17}$$ map from the question is not surjective. However, the way to the answer yields new interesting questions. First, here is the proof.
Theorem: an ordered collection of $$q\choose p$$ integers is the collection of (lexicographically ordered) maximal minors of some $$p\times q$$ matrix if and only if these numbers satisfy the so-called Plücker equations.
Plücker equations for $$(p,q)=(3,6)$$ can be displayed by typing Grassmannian(2,5) into Macaulay2 (the $$2$$ and $$5$$ come from projective reasons). Here is one of these equations: $$p_{2,3,4} p_{1,3,6} -p_{1,3,4}p_{2,3,6} +p_{1,2,3}p_{3,4,6}$$ This one, and five others, involve only determinants that I wanted to keep (none of $$p_{1,4,5}, p_{2,4,6}, p_{3,5,6}$$ is involved). Hence the result: the $$\Bbb Z^{18}\to \Bbb Z^{17}$$ map from the question is not surjective.