Prescribing Symmetries and Automorphisms for Polytopes

Abstract

We study finite groups that occur as combinatorial automorphism groups or geometric symmetry groups of convex polytopes. When Γ is a subgroup of the combinatorial automorphism group of a convex d-polytope, d ≥ 3, then there exists a convex d-polytope related to the original polytope with combinatorial automorphism group exactly Γ. When Γ is a subgroup of the geometric symmetry group of a convex d-polytope, d ≥ 3, then there exists a convex d-polytope related to the original polytope with both geometric symmetry group and combinatorial automorphism group exactly Γ. These symmetry-breaking results then are applied to show that for every abelian group Γ of even order and every involution σ of Γ, there is a centrally symmetric convex polytope with geometric symmetry group Γ such that σ corresponds to the central symmetry.

Non-commutative groups as prescribed polytopal symmetries

“…Not only is the group of combinatorial automorphisms of P equal to Γ, but the group of automorphisms of skel 1 (P) (as a graph) is equal to Γ. The reason for this is that the only tool that [SSW19] uses to determine that extra symmetries have been broken is an analysis of the degrees of the vertices in skel 1 (P). Since we are using Theorem 1.1 as a black box, the majority of our results for non-abelian groups work on the level of automorphisms of 1-skeletons of our polytopes.…”

“…In Doignon's paper, it was shown that every finite group is the automorphism group of a 0/1-polytope. We complete the results of [SSW19] to non-abelian groups with a central involution. Our main results answer the open questions [SSW19, Open Questions 1, 2 and 3] affirmatively.…”

“…Theorem 1.1 ([SSW19]) Let d ≥ 3, let Q be a convex d-polytope with (combinatorial) automorphism group Γ(Q), and let Γ be a subgroup of Γ(Q). Then there exists a finite convex d-polytope P with the following properties:(a) Γ(P) = Γ.…”