**TL;DR version.**


Given a commutative ring $\mathbf{k}$ and $n+m$ "generic" vectors $\mathbf{x}\_1, \mathbf{x}\_2, \ldots, \mathbf{x}\_n, \mathbf{y}\_1, \mathbf{y}\_2, \ldots, \mathbf{y}\_m$ in $\mathbf{k}^k$ (really over a polynomial ring, not over $\mathbf{k}$), how can we describe the polynomial relations that hold between the $nm$ scalars $\left(\mathbf{x}\_i\right)^T \mathbf{y}\_j$ ? Is the ideal of these relations spanned by the $\left(k+1\right)\times\left(k+1\right)$-determinants that one would expect?




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**Long version.**


Here is a classical result.


Let $\mathbf{k}$ be a commutative ring with $1$. Let $n$ and $m$ be two nonnegative integers. Let $A$ be the polynomial ring $\mathbf{k}\left[z\_{i,j} \mid 1\leq i\leq n, \ 1 \leq j \leq m\right]$, and let $B$ be the polynomial ring $\mathbf{k}\left[x\_1,x\_2,\ldots,x\_n, y\_1,y\_2,\ldots,y\_m\right]$. We define a $k$-algebra homomorphism $\phi : A \to B$ by sending each $z\_{i, j}$ to $x\_i y\_j$. Then, $\operatorname{Ker}\phi$ is the ideal of $A$ generated by all elements of the form $z\_{a, b} z\_{c, d} - z\_{a, d} z\_{c, b}$ with $1 \leq a \leq n$, $1 \leq b \leq m$, $1 \leq c \leq n$ and $1 \leq d \leq m$.


A nice proof that Bjorn Poonen suggested to me in 2011 can be found [on m.se](https://math.stackexchange.com/questions/353846/hartshorne-problem-1-2-14-on-segre-embedding) (see also [the original AoPS thread](http://www.artofproblemsolving.com/community/c7h136853p2450856)).




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Now, let me attempt to generalize this, inspired by a [suggestion from Alexander Chervov on a rather different problem in MO question #102874](https://mathoverflow.net/questions/102874).


Let $\mathbf{k}$ be a commutative ring with $1$. Let $k$, $n$ and $m$ be three nonnegative integers. Let $A$ be the polynomial ring $\mathbf{k}\left[z\_{i,j} \mid 1\leq i\leq n, \ 1 \leq j \leq m\right]$ (this is the same $A$ as before), and let $B\_k$ be the polynomial ring $\mathbf{k}\left[\left(x\_{i, \ell} \mid 1 \leq i \leq n,\ 1 \leq \ell \leq k \right) \cup \left(y\_{j, \ell} \mid 1 \leq j \leq m, \ 1 \leq \ell \leq k\right)\right]$ (this is a polynomial ring in $nk+mk$ variables). For every $1 \leq i \leq n$, we let $\mathbf{x}\_i$ be the vector $\left(x\_{i,1}, x\_{i,2}, \ldots, x\_{i,k}\right)^T \in \left(B\_k\right)^k$. For every $1 \leq j \leq m$, we let $\mathbf{y}\_j$ be the vector $\left(y\_{j,1}, y\_{j,2}, \ldots, y\_{j,k}\right)^T \in \left(B\_k\right)^k$. We define a $k$-algebra homomorphism $\phi\_k : A \to B\_k$ by sending each $z\_{i, j}$ to $\left(\mathbf{x}\_i\right)^T \mathbf{y}\_j$. The question is what $\operatorname{Ker}\left(\phi\_k\right)$ is.


I have a guess: Let $W\_k$ be the ideal of $A$ formed by all determinants of the form


$\det\left( \left( \left( \mathbf{x}\_{i\_a} \right)^T \mathbf{y}\_{j\_b} \right) \_{1 \leq a \leq k+1, \ 1 \leq b \leq k+1} \right)$


ranging over all pairs $\left(\left(i\_1,i\_2,\ldots,i\_{k+1}\right), \left(j\_1,j\_2,\ldots,j\_{k+1}\right)\right) \in \left\{ 1, 2, \ldots, n \right\}^k \times \left\{ 1, 2, \ldots, m \right\}^k$. (Of course, the ideal does not change if we restrict these pairs by requiring $i\_1 < i\_2 < \cdots < i\_{k+1}$ and $j\_1 < j\_2 < \cdots < j\_{k+1}$.)


I suspect that $\operatorname{Ker}\left(\phi\_k\right) = W\_k$. At least $W\_k \subseteq \operatorname{Ker}\left(\phi\_k\right)$ is easy to prove, so the guess looks reasonable.


This seems to be related to semistandard Young tableaux -- after all, there is a bitableau basis of $A$, of which I know nothing beyound the definition; and there is a straightening rule that reduces a monomial in $A$ modulo the ideal $W\_k$, giving a sum of monomials which contain no products $z\_{i\_1, j\_1} z\_{i\_2, j\_2} \cdots z\_{i\_k, j\_k}$ with $i\_1 < i\_2 < \cdots < i\_k$ and $j\_1 < j\_2 < \cdots < j\_k$ (a condition that seems to make more sense in terms of the RSK tableau corresponding to the monomials). But I am not sure if this straightening rule is confluent and, generally, whether this is a good way to move forward. The whole thing could also be the second fundamental theorem for some algebraic group, whence the invariant-theory tag.



A vector version of the Segre embedding: what is the kernel of the ring map?
This is the second fundamental theorem of classical invariant theory for $GL$ acting on vectors and covectors. This is Theorem 8.1 from Ch. 13 of the book [Lie Groups: An Approach Through Invariants and Representations](https://books.google.com/books?id=Sl8OAGYRz_AC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false) by C. Procesi (I still don't have it in front of me but I trust Darij on this).


Alternatively, it is Theorem 5.2.15 (SFT for $GL(n)$) in the book [Representations and Invariants of the Classical Groups](https://books.google.com/books?id=KBFoV2jpnPUC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false) by Goodman and Wallach. I have that one on front of me, the old 1998 edition.


The problem is about finding equations for the
variety of $n\times m$ matrices $Z$ which factor as $XY$ where $X$ is $n\times k$ and $Y$ is $k\times m$.
The condition is that the rank is at most $k$. By elementary linear algebra the $(k+1)\times(k+1)$ minor determinants are set-theoretic equations, however Darij's question is about showing these are also ideal-theoretic equations and that needs a bit more work.


