For $n\ge 2$ consider the set of Plücker variables $X_{i_1,\dots,i_k}$ with $1\le k\le n-1$ and $1\le i_1<\dots<i_k\le n$ and the ring $R$ of polynomials in these variables (with complex coefficients). We have the ideal $I\subset R$ of Plücker relations. $R$ can be viewed as the homogeneous coordinate ring of $\mathbb P(\wedge^1V)\times\dots\times\mathbb P(\wedge^{n-1}V)$ where $V=\mathbb C^n$. The zero locus of $I$ is then the variety $F$ of complete flags in $V$.
Now, $R$ is graded by the dominant integral weights, i.e. the monoid $\bigoplus_{i=1}^{n-1}\mathbb Z_{\ge 0}\omega_i$, with the grading of $X_{i_1,\dots,i_k}$ being $\omega_k$. The ideal $I$ is seen to be homogeneous. It is well known that $I$ is generated by its quadratic part, i.e. all of its components of gradings $\omega_i+\omega_j$ (where possibly $i=j$). Let us consider the smaller ideal $J\subset I$ generated by all components of $I$ of gradings $\omega_i+\omega_{i+1}$ and $2\omega_i$. The very definition of a flag of subspaces implies that the relations in $J$ "suffice": the zero locus of $J$ is also $F$ meaning that $I$ is the radical of $J$. (Because the definition is "local with respect to the type A Dynkin diagram".)
However, one can show that a similar but less obvious fact holds. Let $S$ be the ring of Laurent polynomials in the Plücker variables. Then, the ideals in $S$ generated by $I\subset R\subset S$ and by $J\subset R\subset S$ coincide, in other words, every element of $I$ may be multiplied by a monomial to obtain an element of $J$. This fact strikes me as fascinating and fairly strong, however, the argument I have in mind is rather ad hoc and I can't say I understand it well. My questions are as follows.
- Is this fact discussed explicitly somewhere in the literature?
- Does it have some nice geometric or representation-theoretic interpretation? (There's the direct scheme-theoretic translation in terms of stalks of the structure sheaf being reduced. If that's nice, then I don't understand why.)
