Any Finite Group is the Group of Some Binary, Convex Polytope

Doignon, Jean-Paul

doi:10.1007/s00454-017-9945-0

Cited by 7 publications

(7 citation statements)

References 13 publications

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“…For automorphism groups this question was answered positively by Schulte and Williams [SW15], and later a simpler proof was found by Doignon [Doi18]. In this paper we are studying variations of this question with additional restrictions imposed on the polytopes in question.…”

Section: Introductionmentioning

confidence: 90%

“…For a finite group Γ, we define the (combinatorial) convex polytope dimension of Γ, denoted cpd(Γ), as the smallest dimension d for which there exists a convex d-polytope P whose combinatorial automorphism group is Γ, that is, Γ(P ) = Γ. Note that the results of [SW15,Doi18] are saying that for every finite group Γ, we have cpd(Γ) < ∞.…”

Section: Prescribing Involutions As Central Symmetriesmentioning

confidence: 99%

“…Similarly, the geometric convex polytope dimension of Γ, denoted gcpd(Γ), is defined to be the smallest dimension d for which there is a convex d-polytope P whose geometric symmetry group is Γ, that is, G(P ) = Γ. The results of [Doi18] also imply gcpd(Γ) < ∞.…”

Section: Prescribing Involutions As Central Symmetriesmentioning

confidence: 99%

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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: 90%

Section: Prescribing Involutions As Central Symmetriesmentioning

confidence: 99%

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Prescribing Symmetries and Automorphisms for Polytopes

Schulte¹,

Soberón²,

Williams³

2019

Preprint

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“…(1) Every finite group is the automorphism group of a finite graph [14], even better, a finite 3-regular graph [15,Theorems 2.4 and 4.1], more generally, a finite graph with given connectivity or chromatic number, regular of given degree, and a number of other such constraints [30,Theorem 1.2], some convex polytope, with the group acting either purely combinatorially [33, Theorems 1 and 2] or isometrically [11,Theorem 1.1].…”

Section: Introductionmentioning

confidence: 99%

Prescribed Riemannian Symmetries

Chirvăsitu

2021

SIGMA

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Given a smooth free action of a compact connected Lie group G on a smooth compact manifold M , we show that the space of G-invariant Riemannian metrics on M whose automorphism group is precisely G is open dense in the space of all G-invariant metrics, provided the dimension of M is "sufficiently large" compared to that of G. As a consequence, it follows that every compact connected Lie group can be realized as the automorphism group of some compact connected Riemannian manifold; this recovers prior work by Bedford-Dadok and Saerens-Zame under less stringent dimension conditions. Along the way we also show, under less restrictive conditions on both dimensions and actions, that the space of G-invariant metrics whose automorphism groups preserve the G-orbits is dense G δ in the space of all G-invariant metrics.

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“…More recently, Schulte and Williams showed that every finite group can be realized as the combinatorial symmetry group of some polytope [25]. (A simpler proof has been given by Doignon [6]. )…”

Section: Introductionmentioning

confidence: 99%

Classification of affine symmetry groups of orbit polytopes

Friese

Ladisch

2017

J Algebr Comb

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A b s t r ac t . Let G be a finite group acting linearly on a vector space V . We consider the linear symmetry groups GL(Gv) of orbits Gv ⊆ V , where the linear symmetry group GL(S) of a subset S ⊆ V is defined as the set of all linear maps of the linear span of S which permute S. We assume that V is the linear span of at least one orbit Gv. We define a set of generic points in V , which is Zariskiopen in V , and show that the groups GL(Gv) for v generic are all isomorphic, and isomorphic to a subgroup of every symmetry group GL(Gw) such that V is the linear span of Gw. If the underlying characteristic is zero, "isomorphic" can be replaced by "conjugate in GL(V )". Moreover, in the characteristic zero case, we show how the character of G on V determines this generic symmetry group. We apply our theory to classify all affine symmetry groups of vertex-transitive polytopes, thereby answering a question of Babai (1977).2010 Mathematics Subject Classification. Primary 52B15, Secondary 05E15, 20B25, 20C15.

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Any Finite Group is the Group of Some Binary, Convex Polytope

Cited by 7 publications

References 13 publications

Prescribing Symmetries and Automorphisms for Polytopes

Prescribing Symmetries and Automorphisms for Polytopes

Prescribed Riemannian Symmetries

Classification of affine symmetry groups of orbit polytopes

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