I only consider convex polytopes, i.e. convex hulls of finitely many points. The (edge-)graph of a polytope $P\subseteq\Bbb R^d$ is the graph consisting of the polytope's vertices, two are adjacent if they lie on a common edge in $P$.

The orthogonal symmetry group of $P$ induces a permutation symmetry group on its graph. We say that a polytope is vertex-, edge-, arc- and/or half-transitive if the symmetry group induced on its graph acts in the respective way.

Question: Are there half-transitive polytopes, i.e. polytopes which are vertex- and edge-transitive, but not arc-transitive?

I looked a bit into chiral polytopes (i.e. two flag orbits), but I read somewhere that these only exist for abstract polytopes, and not for convex ones. However, I think that chirality is a much stronger requirement for a polytope than half-transitivity (as the latter only speaks about vertices and edges instead of flags).

  • $\begingroup$ I don't see why the graph needs to be half-transitive. The symmetry group of the polytope might not induce the full group of the graph. Graphs with a half-transitive subgroup are much more common. (Cycles, for example.) $\endgroup$ – verret Jan 4 at 18:23
  • $\begingroup$ @verret You are absolutely right. I edited that part out. $\endgroup$ – M. Winter Jan 4 at 18:38

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