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I'd say that the relevant fact here is as follows. For two Borels $B_1$ and $B_2$ with a common maximal torus $T$ let $x_1$ be the unique $T$-fixed point in the open $B_1$-orbit. Then the $B_2$-orbit …

This is essentially an extension of my comment, just to answer the actual "is this the only case?" question. It is not, $Z$ will be affine whenever $S$ is an antichain in the Bruhat order. Indeed, thi …

Since, surprisingly, there are still no answers or even comments, let me note that the answer to the last question is well known to be "yes": the Schubert cell containing a flag $(E_1,\dots,E_m)$ is i …

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) …

Victor Petrov essentially answered your question showing that this projective embedding is, in general, not minimal. I'll just try to explain why this other embedding is, in fact, minimal by dimension …

Consider the Lie group $G=SL_n(\mathbb C)$ with Borel subgroup $B$ and maximal torus $T\subset B$. I'm interested in the (Zariski) closures of $T$-orbits in the flag variety $F=G/B$. Now, as far as I …

To keep the notations simple I'll restrict my attention to the complete flag variety although the question should be equally valid for partial flag varieties. Consider $G=SL_n(\mathbb C)$ with Borel $ …

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). …