Hyperbolic Geometry and Coxeter Groups
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Hyperbolic Reflection Groups
Uspekhi Mat. N auk 40:1 (1985), 29 66 Russian Math. Surveys 40:1 (1985), 31 75 Hyperbolic reflection groups E.B. Vinberg CONTENTS Introduction 31 Chapter I. Acute angled polytopes in Lobachevskii spaces 39 § 1. The G ram matrix of a convex polytope 39 §2. The existence theorem for an acute angled polytope with given Gram 42 matrix §3. Determination of the combinatorial structure of an acute angled 46 polytope from its Gram matrix §4. Criteria for an acute angled polytope to be bounded and to have finite 50 volume Chapter II. Crystallographic reflection groups in Lobachevskii spaces 57 §5. The language of Coxeter schemes. Construction of crystallographic 57 reflection groups §6. The non existence of discrete reflection groups with bounded 67 fundamental polytope in higher dimensional Lobachevskii spaces References 71 Introduction 1. Let X" be the « dimensional Euclidean space Ε", the « dimensional sphere S", or the « dimensional Lobachevskii space Λ". A convex polytope PCX" bounded by hyperplanes //,·, i G / , is said to be a Coxeter polytope if for all /, / G / , i Φ j, the hyperplanes //, and Hj are disjoint or form a dihedral angle of π/rijj, wh e r e «,y G Ζ and η,γ > 2. (We have in mind a dihedral angle containing P.) If Ρ is a Coxeter polytope, then the group Γ of motions of X" generated by the reflections Λ, in the hyperplanes Hj is discrete, and Ρ is a fundamental polytope for it. This means that the polytopes yP, y G Γ, do not have pairwise common interior points and cover X"; that is, they form a tessellation for X". -
The Magic Square of Reflections and Rotations
THE MAGIC SQUARE OF REFLECTIONS AND ROTATIONS RAGNAR-OLAF BUCHWEITZ†, ELEONORE FABER, AND COLIN INGALLS Dedicated to the memories of two great geometers, H.M.S. Coxeter and Peter Slodowy ABSTRACT. We show how Coxeter’s work implies a bijection between complex reflection groups of rank two and real reflection groups in O(3). We also consider this magic square of reflections and rotations in the framework of Clifford algebras: we give an interpretation using (s)pin groups and explore these groups in small dimensions. CONTENTS 1. Introduction 2 2. Prelude: the groups 2 3. The Magic Square of Rotations and Reflections 4 Reflections, Take One 4 The Magic Square 4 Historical Note 8 4. Quaternions 10 5. The Clifford Algebra of Euclidean 3–Space 12 6. The Pin Groups for Euclidean 3–Space 13 7. Reflections 15 8. General Clifford algebras 15 Quadratic Spaces 16 Reflections, Take two 17 Clifford Algebras 18 Real Clifford Algebras 23 Cl0,1 23 Cl1,0 24 Cl0,2 24 Cl2,0 25 Cl0,3 25 9. Acknowledgements 25 References 25 Date: October 22, 2018. 2010 Mathematics Subject Classification. 20F55, 15A66, 11E88, 51F15 14E16 . Key words and phrases. Finite reflection groups, Clifford algebras, Quaternions, Pin groups, McKay correspondence. R.-O.B. and C.I. were partially supported by their respective NSERC Discovery grants, and E.F. acknowl- edges support of the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 789580. All three authors were supported by the Mathematisches Forschungsinstitut Oberwolfach through the Leibniz fellowship program in 2017, and E.F and C.I. -
COXETER GROUPS (Unfinished and Comments Are Welcome)
COXETER GROUPS (Unfinished and comments are welcome) Gert Heckman Radboud University Nijmegen [email protected] October 10, 2018 1 2 Contents Preface 4 1 Regular Polytopes 7 1.1 ConvexSets............................ 7 1.2 Examples of Regular Polytopes . 12 1.3 Classification of Regular Polytopes . 16 2 Finite Reflection Groups 21 2.1 NormalizedRootSystems . 21 2.2 The Dihedral Normalized Root System . 24 2.3 TheBasisofSimpleRoots. 25 2.4 The Classification of Elliptic Coxeter Diagrams . 27 2.5 TheCoxeterElement. 35 2.6 A Dihedral Subgroup of W ................... 39 2.7 IntegralRootSystems . 42 2.8 The Poincar´eDodecahedral Space . 46 3 Invariant Theory for Reflection Groups 53 3.1 Polynomial Invariant Theory . 53 3.2 TheChevalleyTheorem . 56 3.3 Exponential Invariant Theory . 60 4 Coxeter Groups 65 4.1 Generators and Relations . 65 4.2 TheTitsTheorem ........................ 69 4.3 The Dual Geometric Representation . 74 4.4 The Classification of Some Coxeter Diagrams . 77 4.5 AffineReflectionGroups. 86 4.6 Crystallography. .. .. .. .. .. .. .. 92 5 Hyperbolic Reflection Groups 97 5.1 HyperbolicSpace......................... 97 5.2 Hyperbolic Coxeter Groups . 100 5.3 Examples of Hyperbolic Coxeter Diagrams . 108 5.4 Hyperbolic reflection groups . 114 5.5 Lorentzian Lattices . 116 3 6 The Leech Lattice 125 6.1 ModularForms ..........................125 6.2 ATheoremofVenkov . 129 6.3 The Classification of Niemeier Lattices . 132 6.4 The Existence of the Leech Lattice . 133 6.5 ATheoremofConway . 135 6.6 TheCoveringRadiusofΛ . 137 6.7 Uniqueness of the Leech Lattice . 140 4 Preface Finite reflection groups are a central subject in mathematics with a long and rich history. The group of symmetries of a regular m-gon in the plane, that is the convex hull in the complex plane of the mth roots of unity, is the dihedral group of order 2m, which is the simplest example of a reflection Dm group. -
Binary Icosahedral Group and 600-Cell
Article Binary Icosahedral Group and 600-Cell Jihyun Choi and Jae-Hyouk Lee * Department of Mathematics, Ewha Womans University 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-2-3277-3346 Received: 10 July 2018; Accepted: 26 July 2018; Published: 7 August 2018 Abstract: In this article, we have an explicit description of the binary isosahedral group as a 600-cell. We introduce a method to construct binary polyhedral groups as a subset of quaternions H via spin map of SO(3). In addition, we show that the binary icosahedral group in H is the set of vertices of a 600-cell by applying the Coxeter–Dynkin diagram of H4. Keywords: binary polyhedral group; icosahedron; dodecahedron; 600-cell MSC: 52B10, 52B11, 52B15 1. Introduction The classification of finite subgroups in SLn(C) derives attention from various research areas in mathematics. Especially when n = 2, it is related to McKay correspondence and ADE singularity theory [1]. The list of finite subgroups of SL2(C) consists of cyclic groups (Zn), binary dihedral groups corresponded to the symmetry group of regular 2n-gons, and binary polyhedral groups related to regular polyhedra. These are related to the classification of regular polyhedrons known as Platonic solids. There are five platonic solids (tetrahedron, cubic, octahedron, dodecahedron, icosahedron), but, as a regular polyhedron and its dual polyhedron are associated with the same symmetry groups, there are only three binary polyhedral groups(binary tetrahedral group 2T, binary octahedral group 2O, binary icosahedral group 2I) related to regular polyhedrons. -
Reflection Groups 3
Lecture Notes Reflection Groups Dr Dmitriy Rumynin Anna Lena Winstel Autumn Term 2010 Contents 1 Finite Reflection Groups 3 2 Root systems 6 3 Generators and Relations 14 4 Coxeter group 16 5 Geometric representation of W (mij) 21 6 Fundamental chamber 28 7 Classification 34 8 Crystallographic Coxeter groups 43 9 Polynomial invariants 46 10 Fundamental degrees 54 11 Coxeter elements 57 1 Finite Reflection Groups V = (V; h ; i) - Euclidean Vector space where V is a finite dimensional vector space over R and h ; i : V × V −! R is bilinear, symmetric and positiv definit. n n P Example: (R ; ·): h(αi); (βi)i = αiβi i=1 Gram-Schmidt theory tells that for all Euclidean vector spaces, there exists an isometry (linear n bijective and 8 x; y 2 V : T (x) · T (y) = hx; yi) T : V −! R . In (V; h ; i) you can • measure length: jjxjj = phx; xi hx;yi • measure angles: xyc = arccos jjx||·||yjj • talk about orthogonal transformations O(V ) = fT 2 GL(V ): 8 x; y 2 V : hT x; T yi = hx; yig ≤ GL(V ) T 2 GL(V ). Let V T = fv 2 V : T v = vg the fixed points of T or the 1-eigenspace. Definition. T 2 GL(V ) is a reflection if T 2 O(V ) and dim V T = dim V − 1. Lemma 1.1. Let T be a reflection, x 2 (V T )? = fv : 8 w 2 V T : hv; wi = 0g; x 6= 0. Then 1. T (x) = −x hx;zi 2. 8 z 2 V : T (z) = z − 2 hx;xi · x Proof. -
Computing Invariants of Hyperbolic Coxeter Groups
LMS J. Comput. Math. 18 (1) (2015) 754{773 C 2015 Author doi:10.1112/S1461157015000273 CoxIter { Computing invariants of hyperbolic Coxeter groups R. Guglielmetti Abstract CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and whether it provides the Euler characteristic and the combinatorial structure of the associated fundamental polyhedron. The aim of this paper is to present the theoretical background for the program. The source code is available online as supplementary material with the published article and on the author's website (http://coxiter.rgug.ch). Supplementarymaterialsareavailablewiththisarticle. Introduction Let Hn be the hyperbolic n-space, and let Isom Hn be the group of isometries of Hn. For a given discrete hyperbolic Coxeter group Γ < Isom Hn and its associated fundamental polyhedron P ⊂ Hn, we are interested in geometrical and combinatorial properties of P . We want to know whether P is compact, has finite volume and, if the answer is yes, what its volume is. We also want to find the combinatorial structure of P , namely, the number of vertices, edges, 2-faces, and so on. Finally, it is interesting to find out whether Γ is arithmetic, that is, if Γ is commensurable to the reflection group of the automorphism group of a quadratic form of signature (n; 1). Most of these questions can be answered by studying finite and affine subgroups of Γ, but this involves a huge number of computations. -
Regular Elements of Finite Reflection Groups
Inventiones math. 25, 159-198 (1974) by Springer-Veriag 1974 Regular Elements of Finite Reflection Groups T.A. Springer (Utrecht) Introduction If G is a finite reflection group in a finite dimensional vector space V then ve V is called regular if no nonidentity element of G fixes v. An element ge G is regular if it has a regular eigenvector (a familiar example of such an element is a Coxeter element in a Weyl group). The main theme of this paper is the study of the properties and applications of the regular elements. A review of the contents follows. In w 1 we recall some known facts about the invariant theory of finite linear groups. Then we discuss in w2 some, more or less familiar, facts about finite reflection groups and their invariant theory. w3 deals with the problem of finding the eigenvalues of the elements of a given finite linear group G. It turns out that the explicit knowledge of the algebra R of invariants of G implies a solution to this problem. If G is a reflection group, R has a very simple structure, which enables one to obtain precise results about the eigenvalues and their multiplicities (see 3.4). These results are established by using some standard facts from algebraic geometry. They can also be used to give a proof of the formula for the order of finite Chevalley groups. We shall not go into this here. In the case of eigenvalues of regular elements one can go further, this is discussed in w4. One obtains, for example, a formula for the order of the centralizer of a regular element (see 4.2) and a formula for the eigenvalues of a regular element in an irreducible representation (see 4.5). -
Petrie Schemes
Canad. J. Math. Vol. 57 (4), 2005 pp. 844–870 Petrie Schemes Gordon Williams Abstract. Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grunbaum–Dress¨ polyhedra, pos- sess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes. 1 Introduction Historically, polyhedra have been conceived of either as closed surfaces (usually topo- logical spheres) made up of planar polygons joined edge to edge or as solids enclosed by such a surface. In recent times, mathematicians have considered polyhedra to be convex polytopes, simplicial spheres, or combinatorial structures such as abstract polytopes or incidence complexes. A Petrie polygon of a polyhedron is a sequence of edges of the polyhedron where any two consecutive elements of the sequence have a vertex and face in common, but no three consecutive edges share a commonface. For the regular polyhedra, the Petrie polygons form the equatorial skew polygons. Petrie polygons may be defined analogously for polytopes as well. Petrie polygons have been very useful in the study of polyhedra and polytopes, especially regular polytopes. -
Arxiv:1705.01294V1
Branes and Polytopes Luca Romano email address: [email protected] ABSTRACT We investigate the hierarchies of half-supersymmetric branes in maximal supergravity theories. By studying the action of the Weyl group of the U-duality group of maximal supergravities we discover a set of universal algebraic rules describing the number of independent 1/2-BPS p-branes, rank by rank, in any dimension. We show that these relations describe the symmetries of certain families of uniform polytopes. This induces a correspondence between half-supersymmetric branes and vertices of opportune uniform polytopes. We show that half-supersymmetric 0-, 1- and 2-branes are in correspondence with the vertices of the k21, 2k1 and 1k2 families of uniform polytopes, respectively, while 3-branes correspond to the vertices of the rectified version of the 2k1 family. For 4-branes and higher rank solutions we find a general behavior. The interpretation of half- supersymmetric solutions as vertices of uniform polytopes reveals some intriguing aspects. One of the most relevant is a triality relation between 0-, 1- and 2-branes. arXiv:1705.01294v1 [hep-th] 3 May 2017 Contents Introduction 2 1 Coxeter Group and Weyl Group 3 1.1 WeylGroup........................................ 6 2 Branes in E11 7 3 Algebraic Structures Behind Half-Supersymmetric Branes 12 4 Branes ad Polytopes 15 Conclusions 27 A Polytopes 30 B Petrie Polygons 30 1 Introduction Since their discovery branes gained a prominent role in the analysis of M-theories and du- alities [1]. One of the most important class of branes consists in Dirichlet branes, or D-branes. D-branes appear in string theory as boundary terms for open strings with mixed Dirichlet-Neumann boundary conditions and, due to their tension, scaling with a negative power of the string cou- pling constant, they are non-perturbative objects [2]. -
Regular Polyhedra Through Time
Fields Institute I. Hubard Polytopes, Maps and their Symmetries September 2011 Regular polyhedra through time The greeks were the first to study the symmetries of polyhedra. Euclid, in his Elements showed that there are only five regular solids (that can be seen in Figure 1). In this context, a polyhe- dron is regular if all its polygons are regular and equal, and you can find the same number of them at each vertex. Figure 1: Platonic Solids. It is until 1619 that Kepler finds other two regular polyhedra: the great dodecahedron and the great icosahedron (on Figure 2. To do so, he allows \false" vertices and intersection of the (convex) faces of the polyhedra at points that are not vertices of the polyhedron, just as the I. Hubard Polytopes, Maps and their Symmetries Page 1 Figure 2: Kepler polyhedra. 1619. pentagram allows intersection of edges at points that are not vertices of the polygon. In this way, the vertex-figure of these two polyhedra are pentagrams (see Figure 3). Figure 3: A regular convex pentagon and a pentagram, also regular! In 1809 Poinsot re-discover Kepler's polyhedra, and discovers its duals: the small stellated dodecahedron and the great stellated dodecahedron (that are shown in Figure 4). The faces of such duals are pentagrams, and are organized on a \convex" way around each vertex. Figure 4: The other two Kepler-Poinsot polyhedra. 1809. A couple of years later Cauchy showed that these are the only four regular \star" polyhedra. We note that the convex hull of the great dodecahedron, great icosahedron and small stellated dodecahedron is the icosahedron, while the convex hull of the great stellated dodecahedron is the dodecahedron. -
Platonic Solids Generate Their Four-Dimensional Analogues
This is a repository copy of Platonic solids generate their four-dimensional analogues. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/85590/ Version: Accepted Version Article: Dechant, Pierre-Philippe orcid.org/0000-0002-4694-4010 (2013) Platonic solids generate their four-dimensional analogues. Acta Crystallographica Section A : Foundations of Crystallography. pp. 592-602. ISSN 1600-5724 https://doi.org/10.1107/S0108767313021442 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ 1 Platonic solids generate their four-dimensional analogues PIERRE-PHILIPPE DECHANT a,b,c* aInstitute for Particle Physics Phenomenology, Ogden Centre for Fundamental Physics, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, United Kingdom, bPhysics Department, Arizona State University, Tempe, AZ 85287-1604, United States, and cMathematics Department, University of York, Heslington, York, YO10 5GG, United Kingdom. E-mail: [email protected] Polytopes; Platonic Solids; 4-dimensional geometry; Clifford algebras; Spinors; Coxeter groups; Root systems; Quaternions; Representations; Symmetries; Trinities; McKay correspondence Abstract In this paper, we show how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone. -
REFLECTION GROUPS and COXETER GROUPS by Kouver
REFLECTION GROUPS AND COXETER GROUPS by Kouver Bingham A Senior Honors Thesis Submitted to the Faculty of The University of Utah In Partial Fulfillment of the Requirements for the Honors Degree in Bachelor of Science In Department of Mathematics Approved: Mladen Bestviva Dr. Peter Trapa Supervisor Chair, Department of Mathematics ( Jr^FeraSndo Guevara Vasquez Dr. Sylvia D. Torti Department Honors Advisor Dean, Honors College July 2014 ABSTRACT In this paper we give a survey of the theory of Coxeter Groups and Reflection groups. This survey will give an undergraduate reader a full picture of Coxeter Group theory, and will lean slightly heavily on the side of showing examples, although the course of discussion will be based on theory. We’ll begin in Chapter 1 with a discussion of its origins and basic examples. These examples will illustrate the importance and prevalence of Coxeter Groups in Mathematics. The first examples given are the symmetric group <7„, and the group of isometries of the ^-dimensional cube. In Chapter 2 we’ll formulate a general notion of a reflection group in topological space X, and show that such a group is in fact a Coxeter Group. In Chapter 3 we’ll introduce the Poincare Polyhedron Theorem for reflection groups which will vastly expand our understanding of reflection groups thereafter. We’ll also give some surprising examples of Coxeter Groups that section. Then, in Chapter 4 we’ll make a classification of irreducible Coxeter Groups, give a linear representation for an arbitrary Coxeter Group, and use this complete the fact that all Coxeter Groups can be realized as reflection groups with Tit’s Theorem.