Some context: The $(k,n)$-Grassmannian is the set of $k$-dimensional subspaces of an $n$-dimensional vector space $V$. It can be realized as a projective variety via the Plücker embedding, and the set of points in $\mathbb{P}(\wedge^k V)$ corresponding to points in the Grassmannian can be specified as the common vanishing of the Plücker relations.
Somewhat in the spirit of this question "What's the "best" proof of quadratic reciprocity?," I am curious about different proofs that the Plücker relations generate the ideal of the Grassmannian. There is one involving some amount of calculus with Young tableaux (you can find the proof in Fulton's book with the same name), which seems to extend to other flag varieties. However, in Shafarevich's introductory book Basic Algebraic Geometry I there is an exercise involving just some basic facts to do with local parameters. How many other proofs of this fact are commonly known?
