Reduction of orbits of finite Coxeter groups of non-crystallographic type

Grabowiecka, Zofia; Patera, J.; Szajewska, Marzena

doi:10.1063/1.5032210

Cited by 4 publications

(6 citation statements)

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“…The decomposition of vertices of the polytopes (1, 1, 1), (1, 0, 0), (0, 1, 0) and (0, 0, 1) of the Coxeter group H 3 , by means of the branching rules, has been published recently (Grabowiecka et al, 2018). It is closely related to finding the branching rules of a reflection group, where a similar process is performed with the use of a projection matrix.…”

Section: Discussionmentioning

confidence: 99%

“…It is closely related to finding the branching rules of a reflection group, where a similar process is performed with the use of a projection matrix. The decomposition of vertices of the polytopes (1, 1, 1), (1, 0, 0), (0, 1, 0) and (0, 0, 1) of the Coxeter group H 3 , by means of the branching rules, has been published recently (Grabowiecka et al, 2018).…”

Section: Discussionmentioning

confidence: 99%

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The polytopes of the H ₃ group with 60 vertices and their orbit decompositions

Bourret

Grabowiecka

2019

Acta Cryst Sect A

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The goal of this article is to compare the geometrical structure of polytopes with 60 vertices, generated by the finite Coxeter group H 3 , i.e. an icosahedral group in three dimensions. The method of decorating a Coxeter-Dynkin diagram is used to easily read the structure of the reflection-generated polytopes. The decomposition of the vertices of the polytopes into a sum of orbits of subgroups of H 3 is given and presented as a 'pancake structure'.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

The polytopes of the H ₃ group with 60 vertices and their orbit decompositions

Bourret

Grabowiecka

2019

Acta Cryst Sect A

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“…For any finite reflection group G, the branching rules providing the symmetry reduction were determined in several papers [for instance, see Grabowiecka et al (2018) and the references therein]. Applying a symmetry-breaking mechanism to DðÞ and VðÞ polytopes, their structures can be extended into tube-like structures.…”

Section: Figure 32mentioning

confidence: 99%

On symmetry breaking of dual polyhedra of non-crystallographic group H ₃

Myronova

2021

Acta Cryst Sect A

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The study of the polyhedra described in this paper is relevant to the icosahedral symmetry in the assembly of various spherical molecules, biomolecules and viruses. A symmetry-breaking mechanism is applied to the family of polytopes {\cal V}_{H_{3}}(\lambda) constructed for each type of dominant point λ. Here a polytope {\cal V}_{H_{3}}(\lambda) is considered as a dual of a {\cal D}_{H_{3}}(\lambda) polytope obtained from the action of the Coxeter group H 3 on a single point \lambda\in{\bb R}^{3}. The H 3 symmetry is reduced to the symmetry of its two-dimensional subgroups H 2, A 1 × A 1 and A 2 that are used to examine the geometric structure of {\cal V}_{H_{3}}(\lambda) polytopes. The latter is presented as a stack of parallel circular/polygonal orbits known as the `pancake' structure of a polytope. Inserting more orbits into an orbit decomposition results in the extension of the {\cal V}_{H_{3}}(\lambda) structure into various nanotubes. Moreover, since a {\cal V}_{H_{3}}(\lambda) polytope may contain the orbits obtained by the action of H 3 on the seed points (a, 0, 0), (0, b, 0) and (0, 0, c) within its structure, the stellations of flat-faced {\cal V}_{H_{3}}(\lambda) polytopes are constructed whenever the radii of such orbits are appropriately scaled. Finally, since the fullerene C20 has the dodecahedral structure of {\cal V}_{H_{3}}(a,0,0), the construction of the smallest fullerenes C24, C26, C28, C30 together with the nanotubes C20+6N , C20+10N is presented.

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“…For crystallographic reflection groups, the branching rules are determined for the rank up to n = 8 (for instance, see [25,26] and references therein). Recently, the branching rules have been formulated for the non-crystallographic reflection groups as well [27].…”

Section: Introductionmentioning

confidence: 99%

Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

2020

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The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H2 and H3. The geometrical structures of nested polytopes are exemplified.

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Reduction of orbits of finite Coxeter groups of non-crystallographic type

Cited by 4 publications

References 34 publications

The polytopes of the H ₃ group with 60 vertices and their orbit decompositions

The polytopes of the H ₃ group with 60 vertices and their orbit decompositions

On symmetry breaking of dual polyhedra of non-crystallographic group H ₃

Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

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Reduction of orbits of finite Coxeter groups of non-crystallographic type

Cited by 4 publications

References 34 publications

The polytopes of the H 3 group with 60 vertices and their orbit decompositions

The polytopes of the H 3 group with 60 vertices and their orbit decompositions

On symmetry breaking of dual polyhedra of non-crystallographic group H 3

Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

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Product

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The polytopes of the H ₃ group with 60 vertices and their orbit decompositions

The polytopes of the H ₃ group with 60 vertices and their orbit decompositions

On symmetry breaking of dual polyhedra of non-crystallographic group H ₃