Are orbit polytopes of rotation subgroup of Coxeter group combinatorially equivalent?

Question

Suppose that $G\subset O(d)$ is a finite reflection (finite Coxeter) group. For any $v\in \mathbb{R}^d$ which is not fixed by any non-trivial $g\in G$, one can consider the orbit polytope (Coxeter) permutahedra \begin{equation} P(G;v)=Conv (G\cdot v) \end{equation} given by the orbit.

Now consider $G^+\subset SO(d)$, the index-two rotation subgroup of $G$. Again one can consider the orbit polytope \begin{equation} P(G^+;v)=Conv(G^+\cdot v) \end{equation} for $v$ as above (i.e., not fixed by any non-trivial $g$ from the original group). Is it necessarily the case that $P(G^+;v)$ is just obtained by $P(G;v)$ by "alternation"?

If $v_1,v_2\in \mathbb{R}^d$ are not fixed by any $g\in G$, it can be shown that $P(G;v_1)$ and $P(G;v_2)$ are combinatorially equivalent. Must the same be true for $P(G^+;v_1)$ and $P(G^+;v_2)$ as well?

This would definitely seem to be the case for a number of examples (e.g., for $G=A_2\times A_2\times A_2$, for which $P(G;v)$ is a box, $P(G^+;v)$ is a tetrahedra).

Answer 1

Let me try to give a rigorous definition to "by alternation" such that the answer to your first question is "yes".

Given a polytope $P$ whose vertex-edge graph is bipartite, one might say that there are two polytope $Q$ is obtained from $P$ "by alternation" if there is a bipartition of the graph such that $Q$ is the convex hull of one block of the bipartition.

Under this definition, yes, your graph $P(G^+,v)$ is obtained from $P(G,v)$ by alternation. The requirement that $v$ is not fixed by any (nontrivial) element of $G$ is equivalent to the requirement that $v$ is not fixed by any reflection, or equivalently, not contained in any reflecting hyperplane. (This is standard...See for example Section 1.12 of Humphreys "Reflection Groups and Coxeter groups".) The reflecting hyperplanes cut the ambient space into simplicial cones, so we're just choosing $v$ in the interior of one of the cones. Then the orbit of $v$ contains exactly one point in each of the cones, and this gives a bijection between the orbit and the elements of the group.

The cones define a fan structure on the ambient space (i.e. any two cones intersect each other in faces), and this fan is the normal fan of $P(G,v)$. If two maximal cones in this fan are adjacent (i.e. share a codimension-1 face), then they are related by a reflection in $G$, so exactly one of the corresponding group elements is in $G^+$. Thus the vertices of $P(G^+,v)$ are one block of a bipartition of the vertices of $P(G,v)$.