Consider the Plücker embedding of the variety of complete flags in $\mathbb C^n$: $$F_n\subset\mathbb P(\bigwedge\nolimits^1\mathbb C^n)\times\dots\times\mathbb P(\bigwedge\nolimits^{n-1}\mathbb C^n).$$ It is given by the Plücker ideal $I\subset \mathbb C[\{X_{i_1,\dots,i_k}\}]$ where $1\le k\le n-1$ and $1\le i_1<\dots<i_k\le n$.
Now consider a permutation $w\in\mathcal S_n$ and the corresponding Schubert variety $Y_w\subset F_n$. A set of generators of the defining ideal $I_w\supset I$ of $Y_w$ is well known, one adds a certain set of Plücker variables to $I$. For quite some time I've assumed that $I_w$ can also be described as the kernel of certain map, similarly to $I$. Namely, one considers the $n\times n$ matrix $Z_w$ with $(Z_w)_{i,j}$ equal to the variable $z_{i,j}$ if $j\ge w(i)$ and to 0 otherwise. The map $$\varphi_w:\mathbb C[\{X_{i_1,\dots,i_k}\}]\to\mathbb C[z_{i,j}]_{1\le i,j\le n}$$ then takes $X_{i_1,\dots,i_k}$ to the minor of $Z_w$ spanned by rows $1,\dots,k$ and columns $i_1,\dots,i_k$.
Now, after reconsidering, I still believe that $I_w$ is the kernel of $\varphi_w$, this seems to follow readily from the Bruhat decomposition. However, I'm a bit taken aback by not being able to find a single reference for or even an explicit mentioning of this. Can someone provide me with such a reference (or disprove the claim)?
P.S. A reference for the analogous statement in the Grassmannian case would work just as well.
