On Maximal Finite Irreducible Subgroups of GL(N, Z) V
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The Classification and the Conjugacy Classesof the Finite Subgroups of The
Algebraic & Geometric Topology 8 (2008) 757–785 757 The classification and the conjugacy classes of the finite subgroups of the sphere braid groups DACIBERG LGONÇALVES JOHN GUASCHI Let n 3. We classify the finite groups which are realised as subgroups of the sphere 2 braid group Bn.S /. Such groups must be of cohomological period 2 or 4. Depend- ing on the value of n, we show that the following are the maximal finite subgroups of 2 Bn.S /: Z2.n 1/ ; the dicyclic groups of order 4n and 4.n 2/; the binary tetrahedral group T ; the binary octahedral group O ; and the binary icosahedral group I . We give geometric as well as some explicit algebraic constructions of these groups in 2 Bn.S / and determine the number of conjugacy classes of such finite subgroups. We 2 also reprove Murasugi’s classification of the torsion elements of Bn.S / and explain 2 how the finite subgroups of Bn.S / are related to this classification, as well as to the 2 lower central and derived series of Bn.S /. 20F36; 20F50, 20E45, 57M99 1 Introduction The braid groups Bn of the plane were introduced by E Artin in 1925[2;3]. Braid groups of surfaces were studied by Zariski[41]. They were later generalised by Fox to braid groups of arbitrary topological spaces via the following definition[16]. Let M be a compact, connected surface, and let n N . We denote the set of all ordered 2 n–tuples of distinct points of M , known as the n–th configuration space of M , by: Fn.M / .p1;:::; pn/ pi M and pi pj if i j : D f j 2 ¤ ¤ g Configuration spaces play an important roleˆ in several branches of mathematics and have been extensively studied; see Cohen and Gitler[9] and Fadell and Husseini[14], for example. -
Platonic Solids Generate Their Four-Dimensional Analogues.', Acta Crystallographica Section A., 69 (6)
Durham Research Online Deposited in DRO: 26 January 2014 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Dechant, Pierre-Philippe (2013) 'Platonic solids generate their four-dimensional analogues.', Acta crystallographica section A., 69 (6). pp. 592-602. Further information on publisher's website: http://dx.doi.org/10.1107/S0108767313021442 Publisher's copyright statement: Additional information: Published on behalf of the International Union of Crystallography. Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk electronic reprint Acta Crystallographica Section A Foundations of Crystallography ISSN 0108-7673 Platonic solids generate their four-dimensional analogues Pierre-Philippe Dechant Acta Cryst. (2013). A69, 592–602 Copyright c International Union of Crystallography Author(s) of this paper may load this reprint on their own web site or institutional repository provided that this cover page is retained. Republication of this article or its storage in electronic databases other than as specified above is not permitted without prior permission in writing from the IUCr. -
The Fundamental Group of SO(N) Via Quotients of Braid Groups Arxiv
The Fundamental Group of SO(n) Via Quotients of Braid Groups Ina Hajdini∗ and Orlin Stoytchevy July 21, 2016 Abstract ∼ We describe an algebraic proof of the well-known topological fact that π1(SO(n)) = Z=2Z. The fundamental group of SO(n) appears in our approach as the center of a certain finite group defined by generators and relations. The latter is a factor group of the braid group Bn, obtained by imposing one additional relation and turns out to be a nontrivial central extension by Z=2Z of the corresponding group of rotational symmetries of the hyperoctahedron in dimension n. 1 Introduction. n The set of all rotations in R forms a group denoted by SO(n). We may think of it as the group of n × n orthogonal matrices with unit determinant. As a topological space it has the structure of a n2 smooth (n(n − 1)=2)-dimensional submanifold of R . The group structure is compatible with the smooth one in the sense that the group operations are smooth maps, so it is a Lie group. The space SO(n) when n ≥ 3 has a fascinating topological property—there exist closed paths in it (starting and ending at the identity) that cannot be continuously deformed to the trivial (constant) path, but going twice along such a path gives another path, which is deformable to the trivial one. For example, if 3 you rotate an object in R by 2π along some axis, you get a motion that is not deformable to the trivial motion (i.e., no motion at all), but a rotation by 4π is deformable to the trivial motion. -
Convex Polytopes and Tilings with Few Flag Orbits
Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals. -
[Math.RT] 2 Feb 2005 Ler for Algebra Eti Nt Ru Hc Cso H Peeado Matrices
CERTAIN EXAMPLES OF DEFORMED PREPROJECTIVE ALGEBRAS AND GEOMETRY OF THEIR *-REPRESENTATIONS ANTON MELLIT λ Abstract. We consider algebras eiΠ (Q)ei obtained from deformed preprojective algebra λ of affine type Π (Q) and an idempotent ei for certain concrete value of the vector λ which corresponds to the traces of −1 ∈ SU(2, C) in irreducible representations of finite subgroups of SU(2, C). We give a certain realization of these algebras which allows us to construct the C∗-enveloping algebras for them. Some well-known results, including description of four projections with sum 2 happen to be a particular case of this picture. 1. Introduction Let us take the *-algebra, generated by self-adjoint projections a1, a2, a3, a4 with relation a1 + a2 + a3 + a4 = 2: ∗ A := C a1, a2, a3, a4 ai = a ,σ(ai) 0, 1 , a1 + a2 + a3 + a4 = 2 . D4 h | i ⊂ { } i Here and belowf σ(a) denotes the spectrum of self-adjoint element a and writing σ(a) ⊂ x1,x2,...xk for real numbers x1,x2,...,xk we mean that there is a relation { } (a x1)(a x2) . (a xk) = 0. − − − Classification of irreducible representations of A is well known (see [OS]). All irreducible D4 representations, except some finite number of exceptional cases, form a two dimensional family f of irreducible representations in dimension 2. The paper [M1] was devoted to the classification of irreducible representations of the algebra ∗ A := C a1, a2, a3 ai = a ,σ(ai) 0, 1, 2 , a1 + a2 + a3 = 3 . E6 h | i ⊂ { } i After that, the classificationf problem for irreducible representations of the *-algebra ∗ A := C a1, a2, a3 ai = a ,σ(a1) 0, 1, 2, 3 ,σ(a2 ) 0, 1, 2, 3 ,σ(a3 ) 0, 2 , E7 h | i ⊂ { } ⊂ { } ⊂ { } f a1 + a2 + a3 = 4 i was solved in [O]. -
The Classification and the Conjugacy Classes of the Finite Subgroups of the Sphere Braid Groups Daciberg Gonçalves, John Guaschi
The classification and the conjugacy classes of the finite subgroups of the sphere braid groups Daciberg Gonçalves, John Guaschi To cite this version: Daciberg Gonçalves, John Guaschi. The classification and the conjugacy classes of the finite subgroups of the sphere braid groups. Algebraic and Geometric Topology, Mathematical Sciences Publishers, 2008, 8 (2), pp.757-785. 10.2140/agt.2008.8.757. hal-00191422 HAL Id: hal-00191422 https://hal.archives-ouvertes.fr/hal-00191422 Submitted on 26 Nov 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The classification and the conjugacy classes of the finite subgroups of the sphere braid groups DACIBERG LIMA GONC¸ALVES Departamento de Matem´atica - IME-USP, Caixa Postal 66281 - Ag. Cidade de S˜ao Paulo, CEP: 05311-970 - S˜ao Paulo - SP - Brazil. e-mail: [email protected] JOHN GUASCHI Laboratoire de Math´ematiques Nicolas Oresme UMR CNRS 6139, Universit´ede Caen BP 5186, 14032 Caen Cedex, France. e-mail: [email protected] 21st November 2007 Abstract Let n ¥ 3. We classify the finite groups which are realised as subgroups of the sphere braid 2 Õ group Bn ÔS . -
Quaternionic Root Systems and Subgroups of the $ Aut (F {4}) $
Quaternionic Root Systems and Subgroups of the Aut(F4) Mehmet Koca∗ and Muataz Al-Barwani† Department of Physics, College of Science, Sultan Qaboos University, PO Box 36, Al-Khod 123, Muscat, Sultanate of Oman Ramazan Ko¸c‡ Department of Physics, Faculty of Engineering University of Gaziantep, 27310 Gaziantep, Turkey (Dated: November 12, 2018) Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions and octonions. Starting with the roots and weights of SU(2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4), SP (2) ≈ SO(5), SO(8), SO(9), F4 and E8 in terms of the discrete elements of the division algebras. The roots themselves display the group structures besides the octonionic roots of E8 which form a closed octonion algebra. The automorphism group Aut(F4) of the Dynkin diagram of F4 of order 2304, the largest crystallographic group in 4-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F4 .The Weyl groups of many Lie algebras, such as, G2, SO(7), SO(8), SO(9), SU(3)XSU(3) and SP (3) × SU(2) have been constructed as the subgroups of Aut(F4) .We have also classified the other non-parabolic subgroups of Aut(F4) which are not Weyl groups. Two subgroups of orders192 with different conju- gacy classes occur as maximal subgroups in the finite subgroups of the Lie group G2 of orders 12096 and 1344 and proves to be useful in their constructions. -
Quaternionic Kleinian Modular Groups and Arithmetic Hyperbolic Orbifolds Over the Quaternions
Università degli Studi di Trieste ar Ts Archivio della ricerca – postprint Quaternionic Kleinian modular groups and arithmetic hyperbolic orbifolds over the quaternions 1 1 Juan Pablo Díaz · Alberto Verjovsky · 2 Fabio Vlacci Abstract Using the rings of Lipschitz and Hurwitz integers H(Z) and Hur(Z) in the quater- nion division algebra H, we define several Kleinian discrete subgroups of PSL(2, H).We define first a Kleinian subgroup PSL(2, L) of PSL(2, H(Z)). This group is a general- ization of the modular group PSL(2, Z). Next we define a discrete subgroup PSL(2, H) of PSL(2, H) which is obtained by using Hurwitz integers. It contains as a subgroup PSL(2, L). In analogy with the classical modular case, these groups act properly and discontinuously on the hyperbolic quaternionic half space. We exhibit fundamental domains of the actions of these groups and determine the isotropy groups of the fixed points and 1 1 describe the orb-ifold quotients HH/PSL(2, L) and HH/PSL(2, H) which are quaternionic versions of the classical modular orbifold and they are of finite volume. Finally we give a thorough study of their descriptions by Lorentz transformations in the Lorentz–Minkowski model of hyperbolic 4-space. Keywords Modular groups · Arithmetic hyperbolic 4-manifolds · 4-Orbifolds · Quaternionic hyperbolic geometry Mathematics Subject Classification (2000) Primary 20H10 · 57S30 · 11F06; Secondary 30G35 · 30F45 Dedicated to Bill Goldman on occasion of his 60th birthday. B Alberto Verjovsky [email protected] Juan Pablo Díaz [email protected] Fabio Vlacci [email protected]fi.it 1 Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Av. -
Tetrahedral Symmetry and the Graviton 1 Introduction
International Mathematical Forum, 2, 2007, no. 31, 1537 - 1551 Tetrahedral Symmetry and the Graviton J. A. de Wet Box 514, Plettenberg Bay, 6600, South Africa Abstract We attempt to give a purely geometric interpretation of the Standard Model and trace of the graviton. The concept of a ”point” particle is replaced by fundamental symmetries assumed to characterize physical space. In particular the construction rests on Equation (1) which is an irreducible representation (or IR) of the Dirac ring, in line with some of Einstein’s last ideas that view ”particles” not as objects sitting in space-time but rather as parts of space-time itself.Essentially (1), governing the nucleon,imposes tetrahedral symmetry and consequently the Clebsch cubic of Fig.4 that carries the famous 27 lines that will be identified with E6 and the Standard Model.A section is the Cayley cubic of Fig.6 that shows a circle belonging to a photon γ. To find the other half of the Standard Model we study E7 , homomorphic to the binary octahedral group,, that introduces the product γ.γ with spin 2 thought to represent the graviton. Mathematics Subject Classification: 14J30, 22E70, 81V05, 81V35 Keywords: Standard Model, Quarks, Gluon Condensate, Clebsch Cubic, Strong Interaction, Graviton 1 Introduction In his book [14] Penrose, Section 22.1,refers to a point particle as a myste- rious spread out wavy thing in the quantum formalism. Such a ”spread out wave” may account for the Uncertainty Principle and for quantum entangle- ment where entangled ”particles” ,described by the wave function,may inter- act at a distance. -
Symmetry of the Pyritohedron and Lattices
SQU Journal for Science, 2016, 21(2), 139-149 © 2016 Sultan Qaboos University Symmetry of the Pyritohedron and Lattices Nazife O. Koca*, Aida Y. Al-Mukhaini, Mehmet Koca, and Amal J. Al-Qanobi Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, PC 123 Al-Khoud, Muscat, Sultanate of Oman. *Email: [email protected]. ABSTRACT: The pyritohedron consisting of twelve identical but non regular pentagonal faces and its dual pseudoicosahedron that possess the pyritohedral (Th) symmetry play an essential role in understanding the crystallographic structures with the pyritohedral symmetry. The pyritohedral symmetry takes a simpler form in terms of quaternionic representation. We discuss the 3D crystals with the pyritohedral symmetry which can be derived from the Coxeter-Dynkin diagram of D3. Keywords: Pseudoicosahedron; Pyritohedron; Lattice; Coxeter groups and Quaternions. تماثل متعدد اﻷوجه ذو التركيب المشابه لمعدن البايرايت والشبيكات الفراغية نظيفة كوجا، عايدة يوسف المخيني، محمد كوجا وأمل جمعه القنوبي مستخلص: إن متعدد اﻷوجه ذو التركيب المشابه لمعدن البايرايت المحتوي على اثني عشر وجها متماثﻻ كل منها هو مضلع خماسي غير منتظم ونظيره شبيه ذي العشرين وجها المنتظم والذي يتميز بتناظر اﻷوجه الموجود في معدن البايرايت يلعبان دورا أساسيا في فهم تراكيب البلورات ذات التماثل الموجود في معدن البايرايت. يأخذ تماثل معدن البايرايت صورة بسيطة باستخدام التمثيل المعتمد على العدد التخيلي الرباعي. تناقش هذه الورقة البلورات ثﻻثية اﻷبعاد والتي لها تماثل شبيه بمعدن البايرايت والتي يمكن اشتقاقها من مخطط كوكستر- دنكن 퐷3. كلمات مفتاحية: شبيه ذي العشرين وجها المنتظم، متعدد اﻷوجه ذو التركيب المشابه لمعدن البايرايت، الشبيكة الفراغية، مجموعات كوكستر، الرباعيات. 1. Introduction ymmetry describes the periodic repetition of structural features. -
Representation Theory and the Mckay Correspondence
Representation Theory and the McKay Correspondence Bachelorthesis Ralf Bendermacher 4038053 [email protected] Supervisor: Dr. J. Stienstra Utrecht University July 15, 2014 1 Contents 1 introduction 3 2 Representation Theory 4 2.1 Representations . .4 2.2 The group algebra . .5 2.3 CG-modules . .6 2.3.1 CG-submodules . .7 2.3.2 Irreducible CG-modules . .7 2.3.3 The regular CG-module . .7 2.3.4 CG-homomorphisms . .8 2.4 Characters . 10 2.5 Conjugacy classes . 15 3 Unitary matrices 16 4 Complex numbers 17 4.1 isomorphism SO(2) and U(1) . 17 5 Quaternions 18 5.1 H as a unital ring . 18 5.2 Representation of H ............................. 20 5.3 Rotations . 21 5.4 Composition of rotations . 24 6 The Platonic Solids 26 6.1 Symmetry groups of the solids . 26 6.2 Dual solids . 27 6.3 Tetrahedron . 27 6.4 The Cube and the Octahedron . 28 6.5 The Dodecahedron and the Icosahedron . 30 6.6 Groups . 30 7 Binary groups 32 8 The character tables of A4, S4 and A5 35 9 The symmetry groups 36 9.1 Octahedral group . 36 9.2 Tetrahedral group . 39 9.3 Icosahedral group . 41 9.4 Note on the Coxeter-Dynkin diagrams . 44 2 1 introduction We start by giving an introduction to representation theory and then apply it to the symmetry groups belonging to the 5 platonic solids; the tetrahedron, the cube, the oc- tahedron, the dodecahedron and the icosahedron. These 5 solids constitute 3 important symmetry groups, namely the tetrahedral group, octahedral group and the icosahedral 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.