# "Classical" proof that maximal minors form a Grobner basis under diagonal term order

Let $$R= k[x_{11} , x_{12} \dotsc , x_{nm}]$$ denote the coordinate ring of a generic $$n \times m$$ matrix, $$M$$. It is well known that under the standard diagonal term order, the ideal of maximal minors of $$M$$ forms a Grobner basis (I am aware that it is a universal Grobner basis, but I don't need that result).

I am looking for a classical or elementary proof of this result, relying almost entirely on Buchberger's algorithm. I assume that this is written somewhere, but cannot find this result in any of the standard references. Thanks!

## 1 Answer

Here are a few articles that might be relevant:

Narasimhan, The irreducibility of ladder determinantal varieties, from 1986. It takes about 20 pages to prove the result (that the minors are a standard basis for the ideal they generate). I confess, I don't want to try to read it; perhaps it can be streamlined.

I'm not sure about this one: Caniglia, Guccione, Guccione, Ideals of generic minors, from 1990. I don't have access to the article, and the review doesn't describe the method of proof. So I don't know whether they stick to "elementary" methods.

Herzog and Trung and Sturmfels use Knuth-Robinson-Schensted correspondence and straightening laws.