@SamHopkins well, "the coordinate ring" is something that is not as well defined as the cohomology - you are presumably talking about the coordinate ring of the Plücker embedding - so your question is why the Plücker embedding exists, or not?

@Nati you might also want to look at the paper arxiv.org/abs/1905.04499 which appeared after I wrote my answer; you might find some connections via that work

@GilesGardam yes exactly. And it is equally restrictive for varieties of Lie algebras, for instance. So I think that a lot of people expected a list of possible varieties of nonassociative k-linear algebras with this property to be finite or at least discrete - and we even found continuous families...

I do not think Séminaire Bourbaki is systematically translated into English (long ago there used to be efforts of translating into Russian but the last translated year was 1992, I believe), so it would be just pure chance if such a translation exists. I concur with Aleksandar and Ben : an automatic translation (probably DeepL will be a bit more efficient) is something you can easily do by yourself, and it will serve most imaginable purposes.

@Carl-FredrikNybergBrodda or perhaps one may simply argue that specialisation created a new kind of breakthroughs, the specialised ones. Also, I can easily give speculative examples of what kind of breakthroughs in enumerative combinatorics can seriously impact studying monoidal categories (think Drinfeld associators), so you should be careful about sweeping statements like that ;-)

It is a very bizarre way of describe reality. Research positions usually come with teaching duties. I highly doubt that teaching positions with research duties often lead to breakthroughs.

@darijgrinberg ah ok, that makes sense. I tend to think of the formula "Schur polynomial = alternant divided by Vandermonde determinant" as almost the definition of the Schur polynomial (and a particular case of Weyl character formula), and the formula as the weighted sum of tableaux as some surprising addendum, hence the question!

Out of curiosity, what exactly surprises you in "Schur polynomial = alternant divided by Vandermonde determinant" ?

@LSpice it is indeed!

@Kensmosis no, intersection theory is much deeper than things used in discussions here. A good headline to begin with is dimension theory. In particular, "complete intersection" is an intersection where adding each of the equations reduces the dimension by one, and to understand it well, it is of utmost importance to have full clarity on different ways of how one can think of dimension in algebraic geometry.

As some shameless self-promotion, if one considers varieties of algebras over a field of characteristic zero with one binary operation, a recent result of myself and Umirbaev is that there are infinitely many such varieties that are Nielsen-Schreier (before that, only six such varieties were known - so this was very surprising for us when we proved it).

@Kensmosis For pedagogical purposes, it is also useful to consider the example $x_ix_j=0$ with $(i,j)\ne (1,1)$. There are $n$ unknowns and $\binom{n+1}{2}-1$ linearly independent equations, and still there is a nontrivial solution $(1,0,\ldots,0)$.

@Kensmosis I added two more references that you will perhaps find useful

A trivial observation one can add is that the quadratic relation for the LCS is one of the three relations of the algebra that appears in the paper of Connes and Dubois-Violette as "the quadratic anti self-duality algebra", which is a quotient of the YM algebra. So at least the two algebras discussed by the OP have the same nontrivial quotient that had already been studied.