A property of the Birkhoff polytope

Baumeister, Barbara; Ladisch, Frieder

doi:10.5802/alco.6

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2019

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“…The question of finding polytopes with prescribed automorphism group has also been asked as motivated by representation theory, see [Lad16,BL18,FL18]. These articles study orbit polytopes, that is, convex hulls of single point orbits under finite groups acting affinely on a real vector space.…”

Section: Introductionmentioning

confidence: 99%

Prescribing Symmetries and Automorphisms for Polytopes

Schulte¹,

Soberón²,

Williams³

2019

Preprint

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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.

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Section: Introductionmentioning

confidence: 99%