I am sorry for spamming MO with questions I have not thought about for more than 3 hours, but currently I am quite busy with preparing a talk on representations of $S_n$, and I don't want these to get lost. I hope this one is not quite as vague as the last one.

This here is an attempt to generalize Exercise I.21 in Kraft-Procesi, *Classical Invariant Theory*.

Let $K$ be a field - say, infinite, since we are going to do classical invariant theory. Let $K\left[\mathrm{SL}_n K\right]$ denote the $K$-algebra of polynomial functions on $\mathrm{SL}_n K$, which I define either as

$\left\lbrace f\mid_{\mathrm{SL}_n K} \ \mid \ f\in K\left[\mathrm{M}_n K\right]\right\rbrace$

or as $K\left[\mathrm{M}_n K\right]\diagup \left(\det-1\right)$ (proving the equivalence of these two definitions is not the matter, it's rather easy - even easier than Kraft and Procesi try to make one believe).

Now, the group $\mathrm{U}_n K$ of unipotent upper triangular matrices acts on $\mathrm{SL}_n K$ from the left. What is the invariant ring? It is easily seen that

$\det\left(\text{the submatrix formed by the intersection of the rows }i,i+1,...,n\text{ with the columns }j,i+1,i+2,...,n\right)$

is an invariant for any $i\geq j$. These generate the fraction field of the invariants, but do they also generate the ring of the invariants itself?

(The above-mentioned exercise is the above for $n=2$.)

Arguments using Victorian age methods (as opposed to Zariski-topological or other algebro-geometrical) would be particularly preferred.

**EDIT:** As Allen Knutson has pointed out, my question has a negative answer. However, the (larger) collection of determinants of the form

$\det\left(\text{the submatrix formed by the intersection of the rows }i,i+1,...,n\text{ with the columns }j_1, j_2, ..., j_{n-i+1}\right)$

for $1 < i \leq n$ and $1 \leq j_1 < j_2 < ... < j_{n-i+1} \leq n$ does generate the ring of invariants. When $K$ has characteristic $0$, this can be proven using the standard theory of highest-weight modules and multiplicity-free algebras explained in Kraft-Procesi (see my errata, "Page 9, Exercise 21" for a proof). I am still wondering whether it holds for arbitrary $K$ and has a more elementary or combinatorial proof.

unipotentupper triangular matrices, whereas the flag variety of $SL_n$ should be something like $SL_n$ modulo left multiplication byallupper triangular matrices, am I right? Though probably the diagonal factors shouldn't matter too much. $\endgroup$ – darij grinberg Jul 19 '10 at 8:25