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Let $\Gamma$ be a Coxeter group on some generating set $S$, with reflection representation $V$. Then $\Gamma$ has two standard partial orders, the weak and strong Bruhat orders.

Moreover, if $\lambda \in V$ is chosen generically (any free orbit will do), then the covering relations in weak order are given exactly by the edge graph of the convex hull of $\Gamma\cdot \lambda$.

Let $[u,v]$ be an interval in strong Bruhat order. Has the edge graph of the polytope $hull([u,v]\cdot \lambda)$ been studied?

For example, the polytope $hull([123,321] \cdot (1,2,3))$ is a hexagon, and the strong Bruhat cover $231 > 132$ defines an edge through the middle of this hexagon, so not a weak cover. Whereas the polytope $hull([132,321] \cdot (1,2,3))$ is a trapezoid, one edge of which connects $231$ and $132$.

EDIT: perhaps I should admit the geometry here. If $\Gamma$ is a Weyl group of a Lie group $G$ -- and I am happy to make this assumption, albeit I want $G$ Kac-Moody -- and $V$ the corresponding weight lattice, and $\lambda$ a dominant weight, then $hull(W\cdot \lambda)$ is the moment polytope for $G/B$ bearing the Borel-Weil line bundle ${\mathcal L}_\lambda$. Within $G/B$ we have the Richardson variety $\overline{BuB}/B \cap \overline{B_- vB}/B$, and $hull([u,v]\cdot \lambda)$ is the moment polytope of that.

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Is it at least true that all edges are strong covers? Small examples suggest so. – David Speyer Feb 10 '13 at 14:41
I think it's true... The possible edges for the moment polytope of any $T$-invariant $X \subseteq G/B$ come from the $T$-fixed ${\mathbb P}^1$s, and each connects $w,wr_\beta$ for some $w \in W$ and root $\beta$ (not necessarily simple!). Now consider the subpolytope $hull([w,wr_\beta]\cdot \lambda) \subseteq hull([u,v]\cdot \lambda)$. If $w \bullet\! -- \!\bullet wr_\beta$ were an edge of the larger, it would be an edge of the smaller, and I think it's not, but I'm stuck on why not at the moment. – Allen Knutson Feb 10 '13 at 17:54
@Allen: It's helpful to clarify whether your question extends to general Coxeter groups or only finite Weyl groups (with $W$ being Bourbaki's choice in both cases). Aside from that, Borel argued persuasively but belatedly that "Chevalley order" is more accurate historically than "Bruhat order" in the classical theory. By now that's a lost cause. The weak order probably doesn't belong to either person, though I'm unclear about the precise origins. (Maybe add a tag Weyl-group or coxeter-group?) – Jim Humphreys Feb 10 '13 at 23:57
Definitely I am not satisfied to look only at finite Weyl groups. Crystalographic yes, but finite no. – Allen Knutson Feb 11 '13 at 0:46

For type $A$, see Appendix A of Kodama-Williams. They prove that all the edges $Hull([u,v]) \cdot \lambda$ correspond to strong Burhat covers. They also show that $Hull([u,v]) \cdot \lambda$ is the Minkowski sum of the polytopes $c_i Hull([u,v]) \cdot \alpha_i$, where $\lambda = \sum c_i \alpha_i$ is the decomposition of $\lambda$ into simple roots.

Skimming the proofs, I think the second sentence should still be true in other types. The first sentence, however, uses the fact that weak and strong order on $S_n/(S_k \times S_{n-k})$ coincide, so I think this should not generalize.

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Could it be possible to generalize this result for other |1|-graded parabolics? – Vít Tuček Aug 26 '13 at 17:47

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