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.

`$W$`

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 23:57notsatisfied to look only at finite Weyl groups. Crystalographic yes, but finite no. – Allen Knutson Feb 11 '13 at 0:46