A vector version of the Segre embedding: what is the kernel of the ring map?

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?

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 (see also the original AoPS thread).

Now, let me attempt to generalize this, inspired by a suggestion from Alexander Chervov on a rather different problem in MO question #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.

• I think you want the second fundamental theorem of classical invariant theory for vectors and covectors, the group being $GL_n$. I don't have it in front of me now but I think Weyl's book on classical groups has that. You might also try the recent textbooks by Procesi or Nolan and Wallach. Commented Apr 4, 2015 at 22:43
• @AbdelmalekAbdesselam: Good reference! Procesi's "Lie Groups", on page 536, says that "the double standard tableaux in these elements $\overline{x}_{ij}$ with at most $m$ columns are a basis of the ring $A_m$", which at least sounds close to what I am looking for. I'll need to make sure that it means what I think it means... Commented Apr 4, 2015 at 22:56
• Ah, even better, what I want is the Theorem in §8.1 of his Chapter 13. Care to post this as an answer? Commented Apr 4, 2015 at 22:57
• That's fine. If you got the answer you were looking for, this is all that matters. Commented Apr 4, 2015 at 23:34
• Surely I would not mind something self-contained and elementary, but what you said is perfectly a good answer. Commented Apr 4, 2015 at 23:42

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 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 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.

• The appropriate reference in my edition of Goodman and Wallach (2009) seems to be Theorem 12.2.12. It actually does the ideal-theoretic equations, not just the zero locus. However, it being in a section on multiplicity-free spaces seems to suggest that the proof uses characteristic $0$ in a nontrivial way. Then again, this is a book that proves PBW using Ado's theorem, so I am not surprised by some roundaboutness :) Commented Apr 5, 2015 at 20:24
• Good to point that out. The two editions differ a lot as the authors added a lot of material in the new one. Commented Apr 5, 2015 at 20:31