Construction of graphene, nanotubes and polytopes using finite reflection groups

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Université de Montréal

Construction of graphene, nanotubes and polytopes

using finite reflection groups

par

Zofia Grabowiecka

Département de mathématiques et de statistique Faculté des arts et des sciences

Thèse présentée à la Faculté des études supérieures et postdoctorales en vue de l’obtention du grade de

Philosophiæ Doctor (Ph.D.) en mathématiques

Orientation mathématiques appliquées

novembre 2019

c

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Université de Montréal

Faculté des études supérieures et postdoctorales

Cette thèse intitulée

Construction of graphene, nanotubes and polytopes

using finite reflection groups

présentée par

Zofia Grabowiecka

a été évaluée par un jury composé des personnes suivantes :

Yvan Saint–Aubin (président-rapporteur) Jiří Patera (directeur de recherche) Marzena Szajewska (co-directeur) Luc Vinet (membre du jury) Pierre–Philippe Dechant (examinateur externe) William Witczak–Krempa

(représentant du doyen de la FESP)

Thèse acceptée le : October 4th, 2019

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Sommaire

Le but de cette thèse est d’étudier les structures obtenues à partir des groupes de réflexion finis. Ce travail consiste en quatre articles publiés, un article soumis et un article en prépa-ration dont les résultats partiels constituent un chapitre de cette thèse.

Dans le premier article, nous présentons une réduction des orbites des groupes de Coxeter finis vers leurs sous-groupes. Nous utilisons des matrices de projection, c’est-à-dire, des applications qui transforment les racines simples d’un groupe de réflexion en les racines simples du sous-groupe associé. Les résultats présentés dans ce papier se concentrent sur les groupes finis de réflexion non crystallographiques. De plus, nous utilisons les polytopes engendrés par le groupe non crystallographique H3 pour illustrer les lois de ramification (branching rules), c’est-à-dire une réduction des orbites des groupes finis de Coxeter. Dans le deuxième article, nous étudions les polytopes avec 60 sommets engendrés par le groupe non crystallographique H3. Nous utilisons la méthode de décoration des diagrammes de Coxeter–Dynkin pour décrire leurs structures en détails et décomposer les sommets en somme des orbits de symétries de dimension inférieure. Le troisième article compare deux notations utilisées pour décrire le polyèdre engendré par le groupe de réflexion. Il s’agit du symbole de Schläfli et de la notation des points dominants. Nous y présentons les avantages de chaque méthode, expliquons les deux approches et nous les illustrons par des exemples. Dans le quatrième article, nous nous concentrons sur le graphène, c’est-à-dire un pavement d’hexagones sur le plan, qui possède de remarquables propriétés quand les sommets sont modélisés par des atomes de carbone. Dans ce travail, nous présentons différentes méthodes pour obtenir du graphène à partir de réseaux (lattices) et des orbites de dimension 3 des groupes finis de réflexion. De plus, une technique de coloriage des hexagones au moyen d’un nombre fini de couleurs est donnée avec une méthode systématique pour raffiner le graphène. Dans le cinquième article, nous utilisons des

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fonctions spéciales et les transformations de Fourier pour traiter les données échantillonnées sur un réseau de carrés du groupe de Lie SU (2)×SU (2), relié au groupe de symétrie A1×A1.

Mots clefs : groupe de Coxeter, groupe de réflexion fini, systèmes de racine non crystal-lographique, graphène, matrice de projection, orbite de décomposition, polytope convexe, nanotubes.

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Summary

The goal of this thesis is to study structures obtained from finite reflection groups. The work is contained in four published papers, one submitted article and a research paper currently in preparation, with partial results presented as a chapter of this thesis.

In the first article, we present a reduction of the orbits of finite Coxeter groups to their subgroups. We use projection matrices, that is, mappings that transform the simple roots of a reflection group to the simple roots of the appropriate subgroup. The results presented in this paper focus on non-crystallographic finite reflection groups. Moreover, we use polytopes generated by the non-crystallographic group H3 to illustrate the obtained branching rules, i.e., reductions of orbits of the finite Coxeter groups. In the second article, we study polytopes with 60 vertices, generated by the non-crystallographic group H3. We use a method of decoration of the Coxeter–Dynkin diagram to describe their structure in detail, and decompose their vertices into sums of orbits of lower-dimensional symmetries. The third article compares two notations used to describe polyhedra generated by reflection groups, namely the Schläfli symbol, and the dominant point notation. Here, we present the advantages of each method, we explain the two approaches, and we illustrate them through examples. In the fourth article, we focus on graphene, i.e., a hexagonal tiling of the plane that possesses remarkable properties when the vertices are modelled with carbon atoms. In this work, we present different methods to obtain graphene from lattices and three-dimensional orbits of finite reflection groups. Moreover, a technique to colour the hexagons by a finite number of colours is provided, along with a systematic method to refine the graphene. In the fifth article, we use special functions and Fourier transforms to process data sampled on a square lattice of the Lie group SU (2) × SU (2), related to the A1 × A1 symmetry group.

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Keywords: Coxeter group, finite reflection group, non-crystallographic root system, graphene, projection matrix, orbit decomposition, convex polytope, nanotube.

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4.6. H2 structure of H3 polytopes . . . 74 4.6.1. The (1, 0, 1) polytope . . . 75 4.6.2. The (0, 1, 1) polytope . . . 76 4.7. A2 structure of H3 polytopes . . . 77 4.7.1. The (1, 0, 1) polytope . . . 77 4.7.2. The (0, 1, 1) polytope . . . 78 4.8. A1× A1 structure of H3 polytopes . . . 78 4.8.1. The (1, 0, 1) polytope . . . 79 4.8.2. The (0, 1, 1) polytope . . . 80 4.9. Concluding remarks . . . 80

4.10. Vertices of considered polytopes in the ω-basis.. . . 82

4.11. Vertices of considered polytopes and their orbit decomposition in the Cartesian coordinates. . . 83

Chapter 5. The decoration of a Coxeter–Dynkin diagram and the Schläfli symbol as two methods to describe polytopes generated by finite reflection groups . . . . 87

5.2. Polytopes . . . 90

5.3. Schläfli symbols . . . 91

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5.4. Decoration of the diagram . . . 93

5.5. Concluding remarks . . . 96

Chapter 6. Group theoretical methods to construct graphene . . . . 99

6.2. Construction of the graphene sheet . . . 101

6.2.1. The root system A2 and graphene . . . 102

6.2.2. Reflections generating the affine Weyl group of A2. . . 103

6.2.3. The root system G2 and graphene . . . 104

6.2.4. 3-dimensional hexagons and graphene . . . 106

6.3. Colouring graphene . . . 108

6.4. Refinement of graphene . . . 110

Acknowledgements . . . 113

Chapter 7. Decomposition matrices for the square lattices of the Lie groups SU (2) × SU (2) . . . 115

7.2. Weyl groups of A1 and A1× A1. . . 117

7.3. The data points in A1× A1. . . 117

7.4. The special functions of A1× A1. . . 118

7.5. Splitting data into congruence classes . . . 120

7.6. Fourier decomposition of functions on FM,M0. . . 122

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Chapter 8. Conclusion and future work . . . 127 Bibliography . . . 129 Chapter A. Contributions . . . A-i Chapter B. Corrections to published papers . . . B-i

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List of Tables

1.1 Number of roots and orders of finite reflection groups. . . 7

2.1 The number of orbits and the number of pancakes for all the considered polytopes of the A3 group and the related subsymmetries. . . 19 2.2 The number of orbits and the number of pancakes for all the considered polytopes

of the B3 group and the related symmetries. . . 25 2.3 The number of orbits and the number of pancakes for all the considered polytopes

of the C3 group and the related symmetries. . . 30 4.1 Numbers of vertices of polytopes arising from the H3 group. . . 70 4.2 A table showing the number of orbits and the number of pancakes for all the

considered polytopes and related symmetries. . . 81 5.1 A list of polytopes related to the reflection group A3. The table provides the seed

point, the number of vertices, the Schläfli symbol and the name of the polytope. 91 5.2 A list of polytopes related to the reflection group H3. The table provides the seed

point, the number of vertices, the Schläfli symbol and the name of the polytope. 91 7.1 Orders of the orbits of εxi and the stabilizers Stab(λ) for A1 × A1. There is an

assumption that s0, s1, s00, s 0 1 ∈ Z>0 and t0, t1, t00, t 0 1 ∈ Z>0. . . 120 xv

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List of Figures

1.1 The irreducible root systems in two dimensions. The simple roots are labelled by

α1 and α2. H2 is used as an example for the infinite family. . . 8 1.2 The Dynkin diagrams of finite reflection groups of crystallographic type. Short

roots are marked by black nodes. Number of lines specifies the angle between the simple roots. . . 9 1.3 The Dynkin diagrams of non-crystallographic finite reflection groups. Number 5

indicates the angle between the simple roots is 4π/5. . . . 9 1.4 The decorations of the Dynkin diagram in three dimensions, with the seed point

(1, 1, 0). Notice that we consider the diagrams of the A3, B3, C3 and H3 group simultaneously in the picture . . . 12

1.5 An illustration of how orbits of a lower symmetry are positioned in a polytope, and how they are presented as a pancake structure on the example of the (1, 0, 1) polytope of the H3 Coxeter group. . . 13

2.1 Polytopes generated by the Coxeter group A3 labelled by their dominant points. 16 2.2 The (1, 1, 1) polytope and the pancake structure related to the A2 symmetry.

Pancakes are shown parallel to a plane spanned by the ω2 and ω3, and oriented in the α1 direction. A column of the α1 coordinate of a pancake and the number of points in each orbit is presented. . . 17 2.3 The (1, 1, 1) polytope and the pancake structure related to the A1× A1 symmetry.

Pancakes are shown parallel to a plane spanned by the ω1 and ω3, and oriented in the α2 direction. A column of the α2 coordinate of a pancake and the number of points in each orbit is presented. . . 18

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2.4 Polytopes generated by the Coxeter group B3 labelled by their dominant points. 20 2.5 Polytopes generated by the Coxeter group C3 labelled by their dominant points. 27

2.6 A nanotube constructed from the (0, 1, 1) polytope of the B3 group. . . 31

2.7 A nanotube constructed from the (1, 0, 1) polytope of the A3 group. . . 32

2.8 A nanotube obtained from the (1, 1, 1) polytope of the A3 group. . . 33

2.9 A fragment of the graphene sheet and the armchair nanotube.. . . 33

2.10 A nanotube obtained from the (1, 1, 0) polytope of the B3group in the α1direction. 34 2.11 A nanotube obtained form the (1, 1, 0) polytope of the B3group in the α2direction. 35 3.1 The extended Coxeter diagrams of types H2(m), H3 and H4 are shown. The diagrams without an additional point α0 correspond to the Coxeter diagrams. The circular nodes stand for the simple roots αi, i = 1, . . . , 4, α0 = −P i αi. The value m above a line indicates the angle π −_mπ, between the roots, or equivalently, the angle _mπ between the reflection mirrors. The most frequently occurring value, m = 3, is not shown on the diagram. The absence of a direct link between two nodes implies that the corresponding simple roots, as well as the mirrors, are orthogonal. . . 41

3.2 The mapping P transforms the simple roots α1, . . . , α4 of A4 to the simple roots β1, β2 of H2, or a τ -multiple of the simple roots of H2.. . . 45

3.3 The mapping P transforms the simple roots α1, . . . , α6 of D6 to the simple roots β1, . . . , β3 of H3, or a τ -multiple of the simple roots of H3. . . 45

3.4 The mapping P transforms the simple roots α1, . . . , α8 of E8 to the simple roots β1, . . . , β4 of H4, or a τ -multiple of the simple roots of H4. . . 46

3.5 The mapping P transforms the simple roots α1, α2, α3 of H3 to the simple roots β1, β2 of extended H2. . . 47

3.6 The mapping P transforms the simple roots α0, . . . , α4 of extended H4 to the simple roots β1, . . . , β4 of A1× H3 and the simple roots γ1. . . γ4 of H2. . . 49

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3.7 The mapping P transforms the simple roots α1, α2 of H2 to the simple root β1 of

A1. . . 50 3.8 The mapping P transforms the simple roots α0, . . . , α3 of extended H3 to the

simple roots β1, β2, β3 of A1× A1× A1 and the simple roots γ1, γ2 of A2. . . 50 3.9 Truncated icosidodecahedron (1, 1, 1). . . 53 3.10 (a) The polytope is formed by orbits of H2 group inside of the (1, 1, 1) polytope.

(b) The polytope viewed in the direction parallel to the H2 plane spanned by ω2 and ω3. Horizontal segments are projections of the H2 orbits. The number in a row shows the coordinate of the H2 orbits. (c) The polytope projected to the H2 plane spanned by ω2 and ω3. . . 54 3.11 (a) The polytope is formed by orbits of A2 group inside of the (1, 1, 1) polytope.

(b) The polytope viewed in the direction parallel to the A2 plane spanned by ω1 and ω2. Horizontal segments are projections of the A2 orbits. The number in a row shows the coordinate of the A2 orbits. (c) The polytope viewed from the top. 54 3.12 A1 × A1× A1 reduction of the (1, 1, 1) polytope. The orbits of A1 × A1× A1 are

cuboids. . . 55 3.13 Icosahedron (1, 0, 0). . . 55 3.14 (a) The polytope is formed by orbits of the H2group inside of the (1, 0, 0) polytope.

(b) The polytope viewed in the direction parallel to the H2 plane spanned by ω2 and ω3. Horizontal segments are projections of the H2 orbits. The number in a row shows the coordinate of the H2 orbits. (c) The polytope projected to the H2 plane spanned by ω2 and ω3. . . 56 3.15 (a) The polytope is formed by orbits of the A2 group inside of the (1, 0, 0) polytope.

rectangles. . . 57

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3.17 Icosidodecahedron (0, 1, 0). . . 57 3.18 (a) The polytope is formed by orbits of the H2group inside of the (0, 1, 0) polytope.

(b) The polytope viewed in the direction parallel to the H2 plane spanned by ω2 and ω3. Horizontal segments are projections of the H2 orbits. The number in a row shows the coordinate of the H2 orbits. (c) The polytope projected to the H2 plane spanned by ω2 and ω3. . . 58 3.19 (a) The polytope is formed by orbits of the A2 group inside of the (0, 1, 0) polytope.

rectangles. . . 59 3.21 Dodecahedron (0, 0, 1). . . 59 3.22 (a) The polytope is formed by orbits of H2 group inside of the (0, 0, 1) polytope.

(b) The polytope viewed in the direction parallel to the H2 plane spanned by ω2 and ω3. Horizontal segments are projections of the H2 orbits. The number in a row shows the coordinate of the H2 orbits. (c) The polytope projected to the H2 plane spanned by ω2 and ω3. . . 60 3.23 (a) The polytope is formed by orbits of A2 group inside of the (0, 0, 1) polytope.

rectangles. . . 61 4.1 The Coxeter-Dynkin diagrams of the H3 group and its subgroups H2, A2 and

A1 × A1. The diagrams of the subgroups are obtained by deleting a node from the H3 diagram. The simple roots are represented by the nodes. No direct link indicates the angle between the roots is π/2, one line stands for angle 2π/3, a

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number 5 indicates that the angle between the simple roots is 4π/5 (between reflecting mirrors it is π/5). . . 66 4.2 From left to right, the (1, 1, 0), (2, 1, 0) and (1, 2, 0) polytopes of the H3 group.

The structure of the polytopes is the same, but the lengths of edges depend on the seed point. . . 68 4.3 The polytopes of H3 group with 60 vertices. From left to right, the seed points

are respectively (1, 1, 0), (1, 0, 1) and (0, 1, 1). . . 70 4.4 Decorations of the Coxeter-Dynkin diagram, of the H3 group, for the polytope

with the seed point (1, 1, 0). . . 71 4.5 Decorations of the Coxeter-Dynkin diagram, of the H3 group, for the polytope

with the seed point (1, 0, 1). . . 72 4.6 Decorations of the Coxeter-Dynkin diagram, of the H3 group, for the polytope

with the seed point (0, 1, 1). . . 73 4.7 An illustration of how orbits of the lower symmetry are positioned in a polytope,

and how they are presented as a pancake structure. . . 74 4.8 The (1, 0, 1) polytope and its pancake structure related to the H2 symmetry.

Pancakes are shown parallel to a plane spanned by the ω2 and ω3, and oriented in the α1 direction. A column of the α1 coordinate of a pancake and the number of points in each orbit is presented. . . 76 4.9 The (0, 1, 1) polytope and its pancake structure related to the H2 symmetry.

Pancakes are shown parallel to a plane spanned by the ω2 and ω3, and oriented in the α1 direction. A column of the α1 coordinate of a pancake and the number of points in each orbit is presented. . . 77 4.10 The (1, 0, 1) polytope and its pancake structure related to the A2 symmetry.

Pancakes are shown parallel to a plane spanned by the ω1 and ω2, and oriented in the α3 direction. A column of the α3 coordinate of a pancake and the number of points in each orbit is presented. . . 78

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4.11 The (0, 1, 1) polytope and its pancake structure related to the A2 symmetry. Pancakes are shown parallel to a plane spanned by the ω1 and ω2, and oriented in the α3 direction. A column of the α3 coordinate of a pancake and the number of points in each orbit is presented. . . 79

4.12 The (1, 0, 1) polytope and its pancake structure related to the A1× A1 symmetry. Pancakes are shown parallel to a plane spanned by the ω1 and ω3, and oriented in the α2 direction. A column of the α2 coordinate of a pancake and the number of points in each orbit is presented. . . 80 4.13 The (0, 1, 1) polytope and the pancake structure related to the A1× A1 symmetry.

Pancakes are shown parallel to a plane spanned by the ω1 and ω3, and oriented in the α2 direction. A column of the α2 coordinate of a pancake and the number of points in each orbit is presented. . . 81 5.1 Coxeter–Dynkin diagrams of the Coxeter groups A3 and H3. The simple roots

are represented by three nodes. No direct link indicates that the angle between the roots is π/2, one line stands for the angle 2π/3, and a number 5 indicates the angle between the simple roots is 4π/5. . . 89

5.2 An illustration of a truncation operation. A tetrahedron {3, 3} becomes a truncated tetrahedron t0,1{3, 3}. . . 92

5.3 An illustration of a rectification operation. A dodecahedron {5, 3} becomes an icosidodecahedron t1{5, 3}. . . 93 5.4 An illustration of a cantellation operation. A dodecahedron {5, 3} becomes a

rhombicosidodecahedron t0,2{5, 3}. . . 93 5.5 An illustration of an omnitruncation operation. A dodecahedron {5, 3} becomes

a truncated icosidodecahedron t0,1,2{5, 3}. . . 93

5.6 The (1, 1, 0) polytopes of the A3 and H3 reflection groups. In the A3 case we have a truncated tetrahedron t0,1{3, 3}, and in the H3 case we have a truncated icosahedron t0,1{3, 5}. . . 95

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5.7 The decorations of the Coxeter–Dynkin diagram of the A3 and H3 group for a polytope with the seed point (1, 1, 0). The decoration method is independent of the choice of the Coxeter group, the number 5 on the diagram applies to the H3 group only. Listed here are the type of a face, related decoration step, the number of times that face appears int a polytope, and what kind of a face it is. . . 96 6.1 Fragment of the root lattice of A2 where the Brillouin zones are shown and points

of Q are centers of obtained hexagons. . . 102 6.2 A fragment of the P lattice of A2 is shown. Points of congruence class K1 are

marked by circles, points of congruence class K2 are marked by black dots, and points of congruence class K0 are marked by rectangles. . . 103 6.3 Hexagon H and its neighbours, each shown with the reflections used to obtain

them. Notation rijH shows the reflections needed to get the rijH hexagon. First,

we apply ri, then rj to H. . . 104

6.4 Nonplanar hexagons with vertices in the coordinates of the ω-bases of A3, B3 and

C3. . . 106 6.5 Examples of fragments of graphene with 2-colouring of the hexagons. . . 110 6.6 Examples of fragments of graphene with 3-colouring of hexagons.. . . 111 6.7 Fragment of the refinement of a graphene sheet. . . 112 7.1 The fragments of the A1× A1 lattice for M = 2, M0 = 5 and M = 4, M0 = 5. . . 118 7.2 The set of points on the fundamental region F for M = 2, M0 = 4. . . 119

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Remerciements

I would like to express my gratitude to my advisor, Prof. Jiří Patera, for his guidance and support throughout my Ph. D. studies. He gave me an opportunity that changed my life and I will be always grateful for that.

My sincere thanks go to my co-supervisor of research, Dr. Marzena Szajewska, for her insightful comments and encouragements. This thesis would not have been completed without her.

My gratitude goes to the commitee members for accepting the task and the insightful comments on my work, with special thanks to Yvan Saint–Aubin and Pierre–Philippe Dechant.

I would like to thank the Département de Mathématiques et de Statistique (bourse d’admission) and the Faculté des Études Supérieures et Postdoctorales (bourse d’exemption des droits supplémentaires de scolarité pour les étudiants internationaux).

Finally, I would like to thank my beloved family and friends. I cannot imagine finishing my studies without having them in my life.

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Introduction

The discovery of graphene and other carbon structures, i.e., fullerenes and nanotubes, is one of the significant accomplishments of science in recent decades [73, 58, 35]. Carbon based materials have remarkable properties, such as electric and thermal conductivity, strength, flexibility, transparency and lightness. For this reason, they have been intensively studied and investigated by physicists, biologists, chemists, engineers and others, leading to applications in various devices and materials [1, 32, 100]. Graphene, the fullerene C60 and three types of carbon nanotubes, ie., armchair, zigzag and chiral, can be mathematically modelled and constructed from the finite Coxeter groups, i.e., reflection groups. Indeed, the fullerene

C60 is modelled by the truncated icosahedron, that is the polytope (1, 1, 0) of the non-crystallographic Coxeter group H3. A graphene sheet is a layer of carbon atoms arranged in the hexagonal tiling, sometimes called a honeycomb lattice. This two-dimensional structure can be obtained, for example, from the lattices of the Coxeter groups A2 or G2. Thus, we were motivated to continue the research in this area, and look for more structures that are obtained from lattices and orbits of finite reflection groups.

The purpose of this work is to describe and to study polytopes, nanotubes and two-dimensional tilings related to the finite Coxeter groups. We consider polytopes as orbits of finite reflection groups, which makes it possible to decompose them as a sum of two-dimensional orbits. Such decomposition allows a polytope to be cut in half, and to insert more orbits of a certain type, leading to a nanotube-like structure. Moreover, the cylindrical part of a nanotube can be unrolled, revealing a lattice, or a two-dimensional tiling.

In Chapter 1 we recall some useful notations and facts about the finite Coxeter groups, i.e., finite reflection groups and root systems of crystallographic and non-crystallographic type. The crystallographic reflection groups are Weyl groups of semi-simple Lie algebras,

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therefore we use the related notation. We introduce the method of decorating a Coxeter– Dynkin diagram and a method of decomposing a three-dimensional polytope into a sum of orbits of two-dimensional symmetry groups. Chapter 2 contains the results of the orbit decomposition procedure for all the polytopes of the crystallographic groups A3, B3 and

C3. Furthermore, we present developments in the ongoing research, i.e., nanotubes and the related two-dimensional tilings constructed from those polytopes. Chapters 3 and 4 enclose results of orbit decomposition for the polytopes of the non-crystallographic Coxeter group

H3. Moreover, in Chapter 3 we present a general method of finding such decompositions involving the non-crystallographic groups H2, H3 and H4. Chapter 5 describes the Schläfli symbol notation of polytopes. We compare it to the method of decorating a Coxeter–Dynkin diagram and the dominant point notation. In Chapter 6 we present the construction of the graphene sheet from lattices of crystallographic Coxeter groups, as well as from projections of three-dimensional hexagonal orbits onto a plane. Moreover, we describe the colouring and the refinement of the hexagons of the graphene sheet. Chapter 7 describes a method of processing data given on a square lattice of the group A1× A1. Finally, in Chapter 8 we discuss possibilities for future research.

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Chapter 1 Preliminaries

1.1. Root systems and reflection groups

Consider E being a fixed n-dimensional euclidean space over R endowed with a standard scalar product (·, ·). Any non-zero vector α ∈ E determines a reflection rα and a

correspond-ing reflectcorrespond-ing hyperplane Pα = {β ∈ E : (β, α) = 0} that passes through the origin. The

reflection is given by the formula

rα(β) = β − 2(β, α)

(α, α) α, (1.1.1)

where α, β ∈ E. Such a transformation fixes the points of Pα and sends vectors that are

orthogonal to Pα into its negatives. Moreover, the reflection rα is of order 2, it is orthogonal

(preserves the scalar product) and the non-zero vectors that are proportional to α yield the same reflection (rα = rcα, c ∈ R). We introduce the notation

hβ, αi = 2(β, α)

(α, α) , (1.1.2)

and the reflection formula (1.1.1) becomes

rα(β) = β − hβ, αiα. (1.1.3)

Notice that the h·, ·i product is linear only in the first variable. Indeed,

hbβ + cγ, αi =2(bβ + cγ, α) (α, α) = 2(bβ, α) + 2(cγ, α) (α, α) = 2b(β, α) (α, α) + 2c(γ, α) (α, α) = bhβ, αi + chγ, αi.

In our considerations, a reflection group is defined and specified by the simple roots of the corresponding root system, therefore we recall some facts and definitions about those objects.

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Let Φ = {αi : i = 1, . . . , k} be a subset of E. Φ is called a root system in E if it satisfies

the following conditions :

(1) Φ is finite, spans E and 0 6∈ Φ,

(2) if α ∈ Φ, then the only multiplicities of α in Φ are {+α, −α}, (3) if α ∈ Φ, then Φ is invariant under the reflection rα.

Crystallographic root systems satisfy an extra condition : (4) if α, β ∈ Φ, then hβ, αi ∈ Z.

A subset ∆ ⊂ Φ is called the basis of simple roots of a root system when : (1) ∆ is a basis of E,

(2) each β ∈ Φ can be presented as a linear combination of elements of ∆ such that all the coefficients are either non-negative or non-positive integers, i.e.,

β = X

α∈∆

kαα, (1.1.4)

where all kα ∈ Z≤0 or kα ∈ Z≥0.

The elements of the root system Φ, that have all kα ≥ 0 in the above decomposition (1.1.4),

are called the positive roots, and the roots with kα ≤ 0 are the negative roots of the system.

The value P

α∈∆kα is the height of a root. The elements α1, . . . , αn ∈ ∆ are called the

simple roots, and we refer to this basis as the α−basis of the root system. Furthermore, the simple roots are normal vectors to the reflecting mirrors (reflecting hyperplanes). The rank

n of the root system is equal to the dimension of the space E, i.e, n = dim E. A vector γ ∈ E is regular when it does not lie on a reflecting hyperplane Pα, α ∈ Φ. Moreover, we

define the positive side of a hyperplane as

Φ+(γ) = {α ∈ Φ : (γ, α) > 0}. (1.1.5)

An element α ∈ Φ+_{(γ) is decomposable when α = β}

1+ β2 for β1, β2 ∈ Φ+(γ), otherwise it is called indecomposable. For a regular γ ∈ E we have that Φ = Φ+_{(γ) ∪ −Φ}+_{(γ) and a subset} ∆(γ) ⊂ Φ+_{(γ) of indecomposable roots is a base of a root system Φ. Furthermore, reflecting} hyperplanes Pα, α ∈ Φ, partition E into so-called Weyl chambers and then a regular γ ∈ E

belongs to precisely one of such chambers.

A root system is called irreducible if it is not empty and cannot be presented as a union of two mutually orthogonal sets. The roots are not necessarily of the same length, but they are

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proportional. When all the roots are of the same length, they are called long. Otherwise we distinguish short and long roots. We use the standard convention, i.e., (α, α) = 2 when α is a long root. We introduce the notation rαi = ri.

The reflections r1, . . . , rn, determined by the simple roots α1, . . . , αn∈ ∆ give rise to a finite

Coxeter group W [20, 21]. In other words,

W = hrα : α ∈ ∆i ⊂ GL(E), (1.1.6)

where GL(E) is a general linear group of E.

To specify the relations in the group it suffices to provide the relations between its generators. Therefore, it is enough to give the information about the reflecting mirrors, or the related simple roots, more specifically, about the angles between them. This is usually done by indicating a Cartan matrix or a Dynkin diagram of the considered group or the root system. This information is also provided by the numbers mij, i.e., the lowest numbers such that

(rirj)mij = 1, i, j = 1, . . . , n, (1.1.7)

in particular mii = 2 for all i = 1, . . . , n. The number mij specifies that the angle between

the simple roots αi and αj is equal to π −_mπ

ij. Equivalently, the angle between the reflecting

mirrors ri and rj is equal to _mπ_ij.

Example 1. The irreducible Coxeter groups in three dimensions are defined as follows :

A3 = hr1, r2, r3 : r12 = r 2 2 = r 2 3 = 1, (r1r2)3 = (r1r3)2 = (r2r3)3 = 1i (1.1.8) B3 = hr1, r2, r3 : r12 = r22 = r23 = 1, (r1r2)3 = (r1r3)2 = (r2r3)4 = 1i (1.1.9) C3 = hr1, r2, r3 : r21 = r22 = r23 = 1, (r1r2)3 = (r1r3)2 = (r2r3)4 = 1i (1.1.10) H3 = hr1, r2, r3 : r21 = r22 = r23 = 1, (r1r2)3 = (r1r3)2 = (r2r3)5 = 1i (1.1.11) A Cartan matrix or a Dynkin diagram give the relative angles between the simple roots as well as their lengths.

A Cartan matrix of a root system is defined as

C(W) = (Cij) = (hαi, αji), i, j = 1, . . . , n, (1.1.12)

where αi, αj are the simple roots. The entries Cij satisfy the following conditions :

(1) Cii= 2 for i = 1, . . . , n,

(2) Cij ≤ 0 for i 6= j,

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(3) Cij = 0 iff Cji= 0.

The condition (2) is related to the fact that the angle between the simple roots is obtuse. Recall that for two vectors α, β ∈ E we have that

(α, β) = ||α|| · ||β|| cos θ, (1.1.13)

where θ is the angle between them. As such,

hαi, αji = 2(αi, αj) (αj, αj) = 2||αi|| · ||αj|| cos θ ||αj||2 = 2 ||αi|| ||αj|| cos θ. (1.1.14)

Since cos θ ≤ 0 for π₂ ≤ θ ≤ 3π

2 , the corresponding entry Cij is non-positive. In particular,

Cij = 0 when the simple roots αi and αj are orthogonal, and Cii = 2. In the crystallographic

case those entries are sometimes called Cartan integers. Moreover, the entries Cij specify

if the roots are of the same length (symmetric), or if they are of different lengths and proportional.

Along with the α−basis of simple roots, we introduce the ω−basis of fundamental weights or fundamental dominant weights. The relation between the two bases is given by the duality formula

2(ωi, αj)

(αj, αj)

= δij, i, j = 1, . . . , n. (1.1.15)

A matrix specifying relations between the fundamental weights is called a quadratic form matrix [13] and is defined as follows :

Cq(W) = (C_ijq) = (hωi, ωji), i, j = 1, . . . , n. (1.1.16)

Note that for the root systems that have only long roots, the quadratic form matrix is simply an inverse of the Cartan matrix. The matrices C and Cq _{provide us with the information}

about the simple roots and fundamental weights, but more importantly, they are used to switch from one basis to another, namely :

αi = n X j=1 Cijωj, ωj = n X k=1 C_jkq αk, i, j, k = 1, . . . , n. (1.1.17)

The ω−basis is particularly useful when working with polytopes, i.e., orbits of the Coxeter group, as we can easily identify the dominant point of an orbit, that is a unique point that has non-negative coordinates in the ω−basis. Moreover, due to the duality relation (1.1.15), the reflection formula (1.1.1) is simplified. Indeed,

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g rank |Φ| |W| g rank |Φ| |W| An n ≥ 1 n(n + 1) (n + 1)! F4 4 48 27· 32 Bn n ≥ 3 2n2 n! · 2n E6 6 72 27· 34· 5 Cn n ≥ 2 2n2 n! · 2n E7 7 126 210· 34· 5 · 7 Dn n ≥ 4 2n2− 2n n! · 2n−1 E8 8 240 214· 35· 52· 7 G2 2 12 12 H2 2 10 10 H3 3 30 120 H4 4 120 1202 I2(m) 2 2m 2m

Table 1.1. Number of roots and orders of finite reflection groups.

All finite irreducible root systems are classified, see for example [12, 47, 16]. In Table 1.1, we list those root systems along with their rank, order and the order of the corresponding Coxeter group W. There is an infinite number of reflection groups, or root systems, in two dimensions denoted as I2(m). The crystallographic cases are those where m = 3, 4, 6, i.e., A2,

C2 and G2 respectively. For m = 5, 7, 8, 9, . . . the related groups are non-crystallographic. By convention we write H2 for I2(5) and use it as a representative of the non-crystallographic case. The number m specifies the angle between the simple roots as in (1.1.7). The irreducible root systems in two dimensions are presented in Fig. 1.1.

The Cartan matrices for crystallographic root systems are well known, see for example [47], the quadratic form matrices are found in [13]. The Cartan and quadratic form matrices for the non-crystallographic cases [16] are as follows :

C(H2) =    2 −τ −τ 2   , C q_(H 2) = 1 3 − τ    2 τ τ 2   , (1.1.19) C(H3) =       2 −1 0 −1 2 −τ 0 −τ 2       , Cq(H3) = 1 2       2 + τ 2 + 2τ 1 + 2τ 2 + 2τ 4 + 4τ 2 + 4τ 1 + 2τ 2 + 4τ 3 + 3τ       , (1.1.20) 7

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Figure 1.1. The irreducible root systems in two dimensions. The simple roots are labelled by α1 and α2.

H2 is used as an example for the infinite family.

C(H4) =           2 −1 0 0 −1 2 −1 0 0 −1 2 −τ 0 0 −τ 2           , Cq(H4) =           2 + 2τ 3 + 4τ 4 + 6τ 3 + 5τ 3 + 4τ 6 + 8τ 8 + 12τ 6 + 10τ 4 + 6τ 8 + 12τ 12 + 18τ 9 + 15τ 3 + 5τ 6 + 10τ 9 + 15τ 8 + 12τ           , (1.1.21) where τ = 1+ √ 5

2 is the golden ratio, i.e., the positive root of the equation x

2 _{= x + 1.}

The Dynkin diagram of a root system provides the same information as the Cartan matrix, but in a different form. The simple roots are represented by the nodes of a diagram. The number of lines connecting the nodes indicates the angle between them. Indeed, when there is no line connecting the nodes, the corresponding angle is π

2, one line corresponds to 2π

3 , two lines to 3π₄ , and three lines to 5π₆ . In the non-crystallographic case the number ’5’ over a line indicates the angle to be 4π₅ . The colour of a node specifies that the related root is long

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(white node) or short (black node). We present the Dynkin diagrams for the crystallographic cases in Fig. 1.2 and for the non-crystallographic cases in Fig. 1.3.

An g g g . . . g n ≥ 1 Bn g g . . . g w n ≥ 3 Cn w w . . . w g n ≥ 2 Dn g g g g g . . . _{n ≥ 4} E6 g g g g g g E7 g g g g g g g E8 g g g g g g g g F4 g g w w G2 w g

Figure 1.2. The Dynkin diagrams of finite reflection groups of crystallographic type. Short roots are marked by black nodes. Number of lines specifies the angle between the simple roots.

H4 g g g g5 H3 g g g5 H2 g g5

Figure 1.3. The Dynkin diagrams of non-crystallographic finite reflection groups. Number 5 indicates the angle between the simple roots is 4π/5.

The Cartan matrices and the Dynkin diagrams contain the same information about the system under consideration and we use them interchangeably. Indeed, matrices are necessary for changing the coordinates between the α− and the ω−basis, and diagrams are used to perform the decoration method.

The crystallographic root systems have a related lattice structure. The root lattice is defined as Q = ( _n X i=1 aiαi | ai ∈ Z ) , (1.1.22)

where α1, . . . , αn are the simple roots. Similarly, we define the weight lattice, i.e.,

P = ( _n X i=1 biωi, | bi ∈ Z ) , (1.1.23)

where ω1, . . . , ωn are the fundamental weights.

Lattices, or more generally, tessellations, are used in data decomposition and processing, [10, 11, G4], and in the nanotube construction [5] -[8] and Section 2.4. More precisely, once a nanotube is constructed from a polytope, its cylindrical part can be cut lengthwise and unrolled onto a plane. The results vary depending on the considered case, but, in general,

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there is a two-dimensional structure arising in such a way. Some of the results are, in fact, lattices of the root systems. Moreover, in our work, we use the root and weight lattices to obtain a graphene sheet; details of the construction are presented in [G3].

There are three classes of Coxeter groups, namely finite, affine and hyperbolic. Moreover, a finite Coxeter group can be extended into an affine Coxeter group of infinite order [31, 16]. A standard approach is to construct the affine reflection and add it to the group generators. The affine reflecting hyperplane does not pass through the origin, but rather between the origin and a unique point α0 =Pni=1miαi, called the highest root (sometimes denoted as ξ).

The explicit formulas for the highest roots are found in [47] for the crystallographic cases, and in [16] for the non-crystallographic cases. The affine reflection formula is the following :

rα0(β) = α0+ β − hβ, α0iα0 (1.1.24)

where β ∈ E. The affine reflection can be used to produce a lattice of a crystallographic root system (see [G3]). The affine Dynkin diagrams show us the position of the highest root in the system and are used to determine the projection matrices as shown in [G5].

1.2. Polytopes and their orbit decomposition

In our considerations, we focus on the polytopes in three dimensions. By a polytope we mean a polyhedron, i.e., a three-dimensional solid with flat two-dimensional faces. It is the smallest convex hull of a collection of points in three-dimensional space. In our case those are the points of an orbit of a reflection group and, simultaneously, the vertices of a polytope. The vertices are equidistant from the origin and the polytopes are spherical and centered at the origin. Moreover, the polytopes considered here are vertex-transitive, i.e., the local structure is the same around each vertex.

Thinking of the vertices of polytopes as the points of an orbit offers some advantages. We can label and distinguish the polytopes by their dominant points in the ω−basis of the related Coxeter group. We can compute the number of vertices of a polytope from its dominant point. Indeed, let

λ = n

X

i=1

λiωi (1.2.1)

be a vector in E. Its orbit under the action of a Coxeter group W is the following

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in particular, |W(0)| = 1 for every group W. The stabilizer of λ in the group W is a set defined as

StabW(λ) = {w ∈ W : wλ = λ}. (1.2.3)

When λ is a dominant point, the number of points of the corresponding orbit is given by

|W(λ)| = |W|

|StabW(λ)|; (1.2.4)

in particular, |W(λ)| = |W| when λ is strongly dominant, i.e., all the coefficients λi in

(1.2.1) are positive, in other words, the point does not lie on any reflecting hyperplane. The stabilizer is easily found using the decoration method which we explain below.

The method of decorating a Dynkin diagram was developed in [67] and [15]. The idea is the following : given a connected Dynkin diagram, we label its nodes by 3 symbols in a specific way. This allows us to decode the information about the dimension, type and number of times a facet occurs in a polyhedron. The method works well in all finite dimensions and all finite reflection groups, but in this work we limit ourselves to dimension three.

The procedure starts by choosing the reflection group and a dominant point, referred to as a seed point. On the Dynkin diagram of the chosen group we decorate the nodes corresponding to non-negative values (of the dominant point) with and the nodes corresponding to zeros with ♦. This step is also called an initial decoration and it is followed by recursive decoration steps

— replace one with ,

— replace every ♦ linked to by a .

Each step of a decoration process represents a face f of a polytope.

The number of gives the dimension q = 0, . . . , n − 1 of a face fq, i.e., faces f0 are vertices,

f1 are edges and f2 are two-dimensional. The upper index is used to distinguish between multiple types of faces of the same dimension. The decoration method allows one to easily find a stabilizer of a face, thus to compute how many times a given face appears in a polytope. The stabilizer is given by the product of two subgroups :

StabW(fq) = W(D) × W(V ), (1.2.5)

where W(D) is the symmetry group of the face, and W(V ) fixes the face pointwise. Those subgroups are easily read from the decorations. Namely, W(D) is labelled by and W(V )

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by ♦. Therefore, the number of times the face occurs in a polytope is given by :

N (fq) =

|W|

|W(D)| · |W(V )|. (1.2.6)

Fig. 1.4 presents the decoration steps for the dominant point (1, 1, 0) and the number of faces of the related polytope depending on the choice of the Coxeter group, namely A3, B3, C3 and H3. Notice that the decoration steps do not depend on the Coxeter group. The choice of the group becomes important only when reading the information from the decoration steps. Furthermore, we can determine how many faces of each type meet in a face of lower dimen-sion. Put m < n, the formula

|StabW(fm)|

|StabW(fm) ∩ StabW(fn)|

(1.2.7)

specifies how many faces fn are meeting in a face fm. This is particularly useful to determine

a local structure around a vertex of a polytope. Further details on the decoration method are found in [15] and Section 5.4.

face type decoration number of faces A3(m = 3) B3(m = 4) C3(m = 4) H3(m = 5) f0

♦

m 12 24 24 60 f₁1

♦

m 6 12 12 30 f₁2

m 12 24 24 60 f1 2

m 4 8 8 20 f2 2

m 4 6 6 12

Figure 1.4. The decorations of the Dynkin diagram in three dimensions, with the seed point (1, 1, 0). Notice that we consider the diagrams of the A3, B3, C3 and H3 group simultaneously in the picture

Once we have all the vertices of the polytope, i.e., all points of the orbit, in the ω−basis, they can be decomposed into a sum of orbits of reflection groups of lower symmetry. In the case of the group A3, there are two possibilities, either the A2 or the A1× A1 symmetry. In the case of the B3 and the C3 group, the vertices can be decomposed as sums of orbits of the A2, the A1× A1or the C2 group. In the case of the H3 group, the possible choices are the A2, the

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polytope as a stack of two-dimensional polygons, i.e., orbits of two-dimensional reflection groups. Moreover, due to the duality relation of the α− and the ω−basis (1.1.15), we can present such a decomposition sorted by the αi value and call it a pancake structure of the

polytope.

To find the orbit decomposition, the dominant points of the orbits of lower symmetry groups need to be determined. That means, from the points in the {ω1, ω2, ω3} coordinates of the considered group, we choose the two coordinates specified by the lower symmetry with the same sign. For example, assume we are looking for the A2 decomposition of a B3 polytope, then we would choose the points with ω1 and ω2 of the same sign as those are representing the A2 subgroup of the B3 group. By taking the points with those coordinates non-negative, we obtain the dominant points of the orbits of lower symmetry. Now we can change the remaining coordinate, i.e., the one unused in the previous step, from the ωi to the αi value.

This makes it possible to sort the dominant points, i.e, the two-dimensional orbits, by the

αi value, providing us with the pancake structure. Fig. 1.5 presents the pancake structure

of the (1, 0, 1) polytope of the H3 group with respect to the H2 symmetry. Notice that we examine all the points of the original orbit only in the first step, after that we consider only the dominant points of the two-dimensional orbits; therefore the amount of points to check is significantly smaller.

6

Figure 1.5. An illustration of how orbits of a lower symmetry are positioned in a polytope, and how they are presented as a pancake structure on the example of the (1, 0, 1) polytope of the H3 Coxeter group.

The orbit decomposition of the polytopes generated by the reflection groups A3, B3 and

C3 are presented in the following chapter. The orbit decomposition of the polytopes of the

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H3 group is given in [G1] and in [G5] where the results were obtained with the use of the projection matrices, a more general method of decomposing orbits of reflection groups.

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Chapter 2 Polytopes of finite Coxeter groups and related

nanotubes

In this chapter we present the results obtained in the research related to the polytopes of the crystallographic groups in three dimensions, namely A3, B3 and C3. In Sections 2.1-2.3, we present the decomposition of vertices of the polytopes into the sum of orbits of two-dimensional reflection groups. We use the method explained in Section 1.2. Note that we consider convex polytopes and use the dominant point notation.

2.1. Polytopes of the Coxeter group A

3

In this section, a polytope is an orbit of the Coxeter group A3. All the possible types of polytopes are presented in Fig. 2.1. There are two reflection subgroups of the group A3 in two dimensions, namely the groups A2 and A1× A1. Therefore, we can decompose an orbit of the A3 group into a sum of orbits of the A2 or the A1× A1 group. We present those results by listing the dominant points of those orbits sorted by the appropriate αi value, namely α1 for the A2 decomposition, and α2 for the A1× A1 decomposition.

The polytope could be oriented in the appropriate direction of the simple root αi, and

then the corresponding decomposition is illustrated as a stack of two-dimensional polygons positioned along the αi direction, a so-called ‘pancake structure’.

The Cartan matrix and the quadratic form matrix of the A3 group are the following :

C(A3) =       2 −1 0 −1 2 −1 0 −1 2       , Cq(A3) = 1 4       3 2 1 2 4 2 1 2 3       . (2.1.1)

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(a) (1, 0, 0) (b) (0, 1, 0) (c) (0, 1, 1) (d) (1, 0, 1) (e) (1, 1, 1)

Figure 2.1. Polytopes generated by the Coxeter group A3 labelled by their dominant points.

The matrices used to change the coordinates are the following :

Cα1(A3) =       3/4 0 0 1/2 1 0 1/4 0 1       , Cα2(A3) =       1 1/2 0 0 1 0 0 1/2 1       , Cα3(A3) =       1 0 1/4 0 1 1/2 0 0 3/4       , (2.1.2) where Cα1(A3) provides the change from the {ω1, ω2, ω3} to the {α1, ω2, ω3} coordinates in

the following way :

(a, b, c)       3/4 0 0 1/2 1 0 1/4 0 1       = 3 4a + 1 2b + 1 4c, b, c . (2.1.3)

Similarly, we use Cα2(A3) and Cα3(A3) to change to {ω1, α2, ω3} and {ω1, ω2, α3}, allowing to

sort the orbits of lower symmetry groups with respect to the αi coordinate of their dominant

points.

The A3 group is highly symmetric; that means the pairs of polytopes (1, 0, 0) and (0, 0, 1), as well as (1, 1, 0) and (0, 1, 1) have the same structure, but are differently oriented in space. As such, we consider only one of each pair. Moreover, to find the orbits of the A2 symmetry we use the Cα1(A3) matrix, and results are given relative to {α1, ω2, ω3}.

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The general (a, b, c) orbit of the A3 group consists of 24 points : (a, b, c), (−a, a + b, c), (a + b, −b, b + c), (a, b + c, −c), (b, −a − b, a + b + c), (−a, a + b + c, −c), (−a − b, a, b + c), (a + b, c, −b − c), (a + b + c, −b − c, b), (b, c, −a − b − c), (c, −b − c, −a), (−b, −a, a + b + c), (−a − b − c, a, b), (−c, −b, −a), (a + b + c, −c, −b), (b + c, −c, −a − b), (−b, b + c, −a − b − c), (−a − b − c, a + b, −b), (−b − c, −a, a + b), (c, −a − b − c, a), (b + c, −a − b − c, a + b), (−b − c, b, −a − b), (c, −a − b, a), (−a − b, a + b + c, −b − c),

(2.1.4) where a, b, c ≥ 0. 3 2 6 1 2 6 −1 2 6 −3 2 6 6α1

Figure 2.2. The (1, 1, 1) polytope and the pancake structure related to the A2 symmetry. Pancakes are shown parallel to a plane spanned by the ω2 and ω3, and oriented in the α1 direction. A column of the α1 coordinate of a pancake and the number of points in each orbit is presented.

The decomposition procedure yields the following results :

the (1, 1, 1) polytope: 24 vertices decompose as

— four orbits with the A2 symmetry, sorted by the α1 value :

3 2, 1, 1 , 1 2, 2, 1 , −1 2, 1, 2 , −3 2, 1, 1 . (2.1.5)

Each orbit has six elements, see Fig 2.2.

— six orbits with the A1× A1 symmetry, sorted by the α2 value:

(1, 2, 1), (2, 1, 2), (1, 0, 3), (3, 0, 1), (2, −1, 2), (1, −2, 1). (2.1.6)

Each orbit has 4 elements. Notice that there are two separate orbits with the same

α2 value, see Fig 2.3.

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2 4 1 4 0 4+4 -1 4 -2 4 6α2

Figure 2.3. The (1, 1, 1) polytope and the pancake structure related to the A1× A1 symmetry. Pancakes are shown parallel to a plane spanned by the ω1 and ω3, and oriented in the α2 direction. A column of the

α2coordinate of a pancake and the number of points in each orbit is presented.

— three orbits with the A2 symmetry, sorted by the α1 value :

₅ 4, 1, 0 , ₁ 4, 2, 0 , −3 4, 1, 1 . (2.1.7)

The number of elements in each orbit is respectively 3, 3 and 6. — four orbits with the A1× A1 symmetry, sorted by the α2 value :

1,3 2, 0 , 2,1 2, 1 , 1, −1 2, 2 , 0, −3 2, 1 . (2.1.8)

The orbits have 2, 4, 4 and 2 elements respectively.

— three orbits with the A2 symmetry, sorted by the α1 value :

(1, 0, 1), (0, 1, 1), (−1, 1, 0). (2.1.9)

They have 3, 6 and 3 elements respectively.

— four orbits with the A1× A1 symmetry, sorted by the α2 value :

(1, 1, 1), (0, 0, 2), (2, 0, 0), (1, −1, 1). (2.1.10)

They have 4, 2, 2 and 4 elements respectively.

— two orbits with the A2 symmetry, sorted by the α1 value :

₁ 2, 1, 0 , −1 2, 0, 1 . (2.1.11)

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Symmetry group

Polytope vertices A2 A1× A1

orbits pancakes orbits pancakes

(1, 1, 1) 24 4 4 6 5

(1, 1, 0) 12 3 3 4 4

(1, 0, 1) 12 3 3 4 3

(0, 1, 0) 6 2 2 3 3

(1, 0, 0) 4 2 2 2 2

Table 2.1. The number of orbits and the number of pancakes for all the considered polytopes of the A3 group and the related subsymmetries.

Both orbits have 3 elements.

— three orbits with the A1× A1 symmetry, sorted by the α2:

(0, 1, 0), (1, 0, 1), (0, −1, 0). (2.1.12)

The middle orbit has 4 elements. The other two are degenerate orbits consisting of a single point.

— two orbits with the A2 symmetry, sorted by the α1 value :

₃ 4, 0, 0 , −1 4, 1, 0 . (2.1.13)

The first one is just one point, the second one has 3 elements. — two orbits with the A1× A1 symmetry, sorted by the α2 value :

1,1 2, 0 , 0, −1 2, 1 . (2.1.14)

Both orbits have 2 elements. The results are summarized in Table 2.1.

2.2. Polytopes of the Coxeter group B

3

In this section we present the decomposition of orbits of the B3 group. We use the same method as for the A3 group. All the types of polytopes generated by the B3 group are presented in Fig. 2.4.

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The Cartan matrix and the quadratic form matrix of the B3 group are the following : C(B3) =       2 −1 0 −1 2 −2 0 −1 2       , Cq(B3) = 1 2       2 2 1 2 4 2 1 2 3/2       . (2.2.1)

In this case there are three different subgroups, namely B2 = C2, A1× A1 and A2, and the following matrices are used to change the coordinates respectively :

Cα1(B3) =       1 0 0 1 1 0 1/2 0 1       , Cα2(B3) =       1 1 0 0 2 0 0 1 1       , Cα3(B3) =       1 0 1/2 0 1 1 0 0 3/4       . (2.2.2) (a) (1, 0, 0) (b) (0, 1, 0) (c) (0, 0, 1) (d) (1, 1, 0) (e) (1, 0, 1) (f) (0, 1, 1) (g) (1, 1, 1)

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Notice, that although the groups B3 and C3 are different, their subgroups B2 and C2 are the same. The general orbit of B3 group has 48 points :

(a, b, c), (−a − b − c, −b, 2b + c), (−a − b − c, b + c, −2b − c), (a + b, −b, 2b + c), (b + c, a, −2a − 2b − c), (a + b + c, −a − 2b − c, 2b + c), (b, a, −2a − 2b − c), (b, −a − b, 2a + 2b + c), (b + c, a + b, −2a − 2b − c), (−a, −b, −c), (−a, a + b + c, −c), (b + c, −a − b − c, 2a + 2b + c), (−a, −b − c, c), (a + 2b + c, −a − b, −c), (a + b, −a − 2b − c, 2b + c), (−a − b, b, −2b − c), (−a − b, −b − c, 2b + c), (b, a + b + c, −2a − 2b − c), (−a − b, a, 2b + c), (−b − c, −a, 2a + 2b + c), (b, −a − 2b − c, 2a + 2b + c), (−a − 2b − c, b, c), (a + 2b + c, −a − b − c, c), (−b − c, a + 2b + c, −2a − 2b − c), (a + b, −a, −2b − c), (−b, −a, 2a + 2b + c), (−a − b, a + 2b + c, −2b − c), (a + b, b + c, −2b − c), (a + b + c, −a, −2b − c), (b + c, −a − 2b − c, 2a + 2b + c), (−a, a + b, c), (a + b + c, −b − c, 2b + c), (−b − c, a + b + c, −2a − 2b − c), (a + 2b + c, −b − c, c), (−a − 2b − c, a + b, c), (−a − 2b − c, a + b + c, −c), (a, −a − b, −c), (−b, a + b, −2a − 2b − c), (−b, −a − b − c, 2a + 2b + c), (a + b + c, b, −2b − c), (a + 2b + c, −b, −c), (−a − b − c, a + 2b + c, −2b − c), (a, −a − b − c, −c), (−a − b − c, a, 2b + c), (−b − c, −a − b, 2a + 2b + c), (a, b + c, −c), (−a − 2b − c, b + c, −c), (−b, a + 2b + c, −2a − 2b − c),

(2.2.3) where a, b, c ≥ 0.

The decomposition procedure yields the following results :

— six orbits with the C2 symmetry, sorted by the α1 value :

5 2, 1, 1 , 3 2, 2, 1 , 1 2, 1, 3 , −1 2, 1, 3 , −3 2, 2, 1 , −5 2, 1, 1 . (2.2.4)

Each orbit has 8 elements.

— twelve orbits with the A1× A1 symmetry, sorted by the α2 value :

(1, 4, 1), (2, 3, 3), (1, 2, 5), (3, 2, 3), (2, 1, 5), (4, 1, 1) (4, −1, 1), (2, −1, 5), (3, −2, 3), (1, −2, 5), (2, −3, 3), (1, −4, 1).

(2.2.5)

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— eight orbits with the A2 symmetry, sorted by the α3 value : 1, 1,9 4 , 1, 2,7 4 , 2, 2,3 4 , 3, 1,1 4 , 1, 3, −1 4 , 2, 2, −3 4 , 2, 1, −7 4 , 1, 1, −9 4 . (2.2.6) Each orbit has 6 elements.

— five orbits with the C2 symmetry

(2, 1, 0), (1, 2, 0), (0, 1, 2), (−1, 2, 0), (−2, 1, 0). (2.2.7)

They have 4, 4, 8, 4 and 4 elements respectively. — eight orbits with the A1× A1 symmetry

(1, 3, 0), (2, 2, 2), (1, 1, 4), (3, 1, 0), (3, −1, 0), (1, −1, 4), (2, −2, 2), (1, −3, 0). (2.2.8)

They have 2, 4, 4, 2, 2, 4, 4 and 2 elements respectively. — four orbits with the A2 symmetry

1, 1, 3 2 , 2, 1,1 2 , 1, 2, −1 2 , 1, 1, −3 2 . (2.2.9)

— four orbits with the C2 symmetry:

₃ 2, 1, 1 , ₁ 2, 0, 3 , −1 2, 0, 3 , −3 2, 1, 1 . (2.2.10)

They have 8, 4, 4 and 8 elements respectively. — seven orbits with the A1× A1 symmetry

(1, 2, 1), (0, 1, 3), (2, 1, 1), (1, 0, 3), (2, −1, 1), (0, −1, 3), (1, −2, 1). (2.2.11)

They have 4, 2, 4, 4, 4, 2 and 4 elements respectively. — six orbits with the A2 symmetry

1, 0,5 4 , 1, 1,3 4 , 2, 0,1 4 , 0, 2, −1 4 , 1, 1, −3 4 , 0, 1, −5 4 . (2.2.12)

They have 3, 6, 3, 3, 6 and 3 elements respectively.

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— four orbits with the C2 symmetry 3 2, 1, 1 , 1 2, 0, 3 , −1 2, 0, 3 , −3 2, 1, 1 . (2.2.13)

They have 8, 4, 4 and 8 elements respectively. — seven orbits with the A1× A1 symmetry

(0, 3, 1), (1, 2, 3), (2, 1, 3), (3, 0, 1), (2, −1, 3), (1, −2, 3), (0, −3, −1). (2.2.14)

They have 2, 4, 4, 4, 4, 4 and 2 elements respectively. — six orbits with the A2 symmetry

0, 1,7 4 , 0, 2,5 4 , 1, 2,1 4 , 2, 1, −1 4 , 2, 0, −5 4 , 1, 0, −7 4 . (2.2.15)

They have 3, 3, 6, 6, 3 and 3 elements respectively.

— 3 orbits with the C2 symmetry:

(1, 0, 0), (0, 1, 0), (−1, 0, 0) (2.2.16)

They have 1, 4 and 1 elements respectively. — three orbits with the A1× A1 symmetry

(1, 1, 0), (0, 0, 1), (1, −1, 0). (2.2.17)

All orbits have 2 elements.

— two orbits with the A2 symmetry

1, 0,1 2 , 0, 1, −1 2 . (2.2.18)

Both orbits have 3 elements.

— three orbits with the C2 symmetry :

(1, 1, 0), (0, 0, 2), (−1, 1, 0). (2.2.19)

Construction of graphene, nanotubes and polytopes using finite reflection groups

Université de Montréal

Construction of graphene, nanotubes and polytopes

using finite reflection groups

Zofia Grabowiecka

Université de Montréal

Construction of graphene, nanotubes and polytopes

using finite reflection groups

Zofia Grabowiecka

Sommaire

Summary

Contents

List of Tables

List of Figures

Remerciements

Introduction

Chapter 1

Preliminaries

1.1. Root systems and reflection groups

1.2. Polytopes and their orbit decomposition

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Chapter 2

Polytopes of finite Coxeter groups and related

nanotubes

2.1. Polytopes of the Coxeter group A

2.2. Polytopes of the Coxeter group B

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