Polytopes with Preassigned Automorphism Groups

Schulte, Egon; Williams, Gordon

doi:10.1007/s00454-015-9710-1

Cited by 13 publications

(20 citation statements)

References 17 publications

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“…Chiral polyhedra are arguably the most important class of 2-orbit complexes. A (geometric) polyhedron K is chiral if G(K) has exactly two orbits on the flags such that any two adjacent flags are in distinct orbits (Schulte, 2004(Schulte, , 2005. This notion of chirality for polyhedra is different from the standard notion of chirality used in crystallography, but is inspired by it.…”

Section: Chiral Polyhedramentioning

confidence: 99%

“…The chiral polyhedra in E 3 fall into six infinite families (Schulte, 2004(Schulte, , 2005, each with two or one free parameters depending on whether the classification is up to congruence or similarity, respectively. Each chiral polyhedron is a non-planar "irreducible" apeirohedron (with an affinely irrreducible symmetry group), so in particular there are no finite, planar, The six families comprise three families of apeirohedra with finite skew faces and three families of apeirohedra with infinite helical faces.…”

Section: Chiral Polyhedramentioning

confidence: 99%

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Polyhedra, complexes, nets and symmetry

Schulte¹

2014

Acta Cryst Sect A

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Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial and algebraic properties. They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zigzag or helical. A polyhedron or complex is regular if its geometric symmetry group is transitive on the flags (incident vertex-edge-face triples). There are 48 regular polyhedra (18 finite polyhedra and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all infinite, which are not polyhedra. Their edge graphs are nets well known to crystallographers and they are identified explicitly. There are also six infinite families of chiral apeirohedra, which have two orbits on the flags such that adjacent flags lie in different orbits.

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Section: Chiral Polyhedramentioning

confidence: 99%

Section: Chiral Polyhedramentioning

confidence: 99%

Polyhedra, complexes, nets and symmetry

Schulte¹

2014

Acta Cryst Sect A

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“…A combinatorial automorphism of a convex polytope is a permutation of its set of vertices that maps the set of vertices of any face to the set of vertices of some face. Recently, Schulte & Williams (2015) established that any finite group is isomorphic to the group of combinatorial automorphisms of some convex polytope. Their proof is rather long and involved.…”

Section: Introductionmentioning

confidence: 99%

“…

For any given finite group, Schulte and Williams (2015) establish the existence of a convex polytope whose combinatorial automorphisms form a group isomorphic to the given group. We provide here a shorter proof for a stronger result: the convex polytope we build for the given finite group is binary, and even combinatorial in the sense of Naddef and Pulleyblank (1981); the diameter of its skeleton is at most 2; any combinatorial automorphism of the polytope is induced by some isometry of the space; any automorphism of the skeleton is a combinatorial automorphism.

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confidence: 99%

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

Doignon

2017

Discrete Comput Geom

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“…Specifically, a short argument of Isaacs [12] can be modified to show that every finite group is the symmetry group of a polytope with at most two orbits on the vertices. 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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Polytopes with Preassigned Automorphism Groups

Abstract: We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be realized as a convex polytope.

Cited by 13 publications

References 17 publications

Polyhedra, complexes, nets and symmetry

Polyhedra, complexes, nets and symmetry

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

Classification of affine symmetry groups of orbit polytopes

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