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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 can tell, for a generic point in $F$ this closure is the toric variety associated with the permutahedron. Further, let us choose an element $w$ of the Weyl group and let $X_w\subset F$ be the corresponding Schubert variety. Then for a generic point in $X_w$ the closure seems to be the toric variety associated with the convex hull of vertices of the permutahedron corresponding to $w'\ge w$ with respect to the Bruhat order, i.e. the convex hull of the weight diagram of the corresponding Demazure module in an irrep with a regular highest weight.

Questions.
1) Is this last description accurate?
2) My main question. Are, in fact, all orbits of this form? More precisely, let $S$ be the set of all points in $F$ that are generic in some Schubert variety in the above sense. Is it then true that any point outside of $S$ can be mapped to a point in $S$ by the action of the Weyl group?

References to literature are much appreciated.

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    $\begingroup$ I believe your polytope is a case of what Tsukerman and Williams call a “Bruhat interval polytope” in (arxiv.org/abs/1406.5202). This gives a reference to literature but does not answer the question so I will leave it as a comment. $\endgroup$
    – John Machacek
    Commented Jun 20, 2018 at 12:38
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    $\begingroup$ @JohnMachacek, thank you for your comment. The Gelfand-Serganova stratification mentioned in this paper is precisely what I am interested in, the strata classify the various orbit closures. Now, 2) would imply that each Gelfand-Serganova stratum is, in fact, a Schubert variety with respect to some Borel $B'\supset T$. This, of course, is not the case, a counterexample is easily constructed for $n=4$. So 2) is wrong (even for Grassmannians). $\endgroup$
    – Igor Makhlin
    Commented Jun 21, 2018 at 2:03
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    $\begingroup$ I now also watched this very recent and closely related talk Mikiya Masuda gave at the Buchstaber conference: mathnet.ru/eng/present20345 It actually seems to contain some explicit description of the generic orbit in $X_w$, although in different terms than the the one above. $\endgroup$
    – Igor Makhlin
    Commented Jun 21, 2018 at 2:08
  • $\begingroup$ Glad the reference was helpful. The lecture linked looks very interesting. I plan to watch it once I get a chance. $\endgroup$
    – John Machacek
    Commented Jun 21, 2018 at 15:09
  • $\begingroup$ So for anyone still interested, it's not hard to deduce an affirmative answer to 1) from the Gelfand-Serganova paper. Essentially, this is Proposition 1 in Section 5 of the Russian version mathnet.ru/links/3b869d2b21fe77027d34439ebdb3af53/rm2427.pdf (I was not able to find the English translation in open access). This proposition characterizes the image of the moment map of a GS stratum, which is the polytope associated with the corresponding orbit closure toric variety as mentioned in section 3. $\endgroup$
    – Igor Makhlin
    Commented Jul 3, 2018 at 18:50

1 Answer 1

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A point in the Grassmannian $ x \in G(k, n) $ defines a matroid $ M = M(x)$. Associated to this matroid is a matroid polytope $P(M)$. The torus orbit closure through $ x $ is the toric variety associated to $ P(M) $.

Similarly, if we take a point $ x $ in the flag variety, then the can associate a flag matroid $M(x) $ and a flag matroid polytope $ P(M)$. Again the torus orbit closure through $ x $ is the toric variety associated to $ P(M) $.

Thus the classification of torus orbit closures in the flag variety is given by realizable flag matroids. So the answer your second question is no.

The theory of flag matroids (in arbitrary type) and their polytopes was developed in the excellent classic paper by GGMS:

I. M. Gelfand, R. M. Goresky, R. D. MacPherson and V. V. Serganova, Combinatorial Geometries, Convex Polyhedra, and Schubert cells, Adv. Math., 63 (1987), 301–316.

Alex Yong's lecture notes are also quite helpful:

https://faculty.math.illinois.edu/~ayong/Math595TheGrassmannian/Grlecture2matroids.pdf

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  • $\begingroup$ Thank you, this is very useful and I should certainly take a closer look at the paper and lecture notes. But before I do, do you happen to know how specific they get there? Do they end up with a concise explicit descriptions of the polytope for each stratum as an H-polytope or a V-polytope? $\endgroup$
    – Igor Makhlin
    Commented Jun 25, 2018 at 0:15
  • $\begingroup$ From the matroid, I believe that you can describe the hyperplanes definite the polytope. $\endgroup$
    – Joel Kamnitzer
    Commented Jun 26, 2018 at 1:27
  • $\begingroup$ I see. But you do not remember if that gives the descriprion I ask about in 1)? $\endgroup$
    – Igor Makhlin
    Commented Jun 26, 2018 at 1:30
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    $\begingroup$ Let me also mention the book "Coxeter matroids" by Borovik, Gelfand, and White. Note that the definition they consider is analogous to (and in Grassmannians, reproduces) that of matroid, not realizable matroid, which is more in line with what you're asking about. $\endgroup$
    – Allen Knutson
    Commented Aug 31, 2018 at 2:12

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