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arXiv · 1902.05439

Prescribing Symmetries and Automorphisms for Polytopes

Egon Schulte·Pablo Soberón·Gordon Ian Williams
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Abstract

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

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Egon Schulte, Pablo Soberón, Gordon Ian Williams. 2019-02-14. Prescribing Symmetries and Automorphisms for Polytopes. https://arxiv.org/abs/1902.05439

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