Coxeter Groups and Artin Groups Day 2: Symmetric Spaces and Simple Groups
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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. -
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. -
Geometries, the Principle of Duality, and Algebraic Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Expo. Math. 24 (2006) 195–234 www.elsevier.de/exmath Geometries, the principle of duality, and algebraic groups Skip Garibaldi∗, Michael Carr Department of Mathematics & Computer Science, Emory University, Atlanta, GA 30322, USA Received 3 June 2005; received in revised form 8 September 2005 Abstract J. Tits gave a general recipe for producing an abstract geometry from a semisimple algebraic group. This expository paper describes a uniform method for giving a concrete realization of Tits’s geometry and works through several examples. We also give a criterion for recognizing the auto- morphism of the geometry induced by an automorphism of the group. The E6 geometry is studied in depth. ᭧ 2005 Elsevier GmbH. All rights reserved. MSC 2000: Primary 22E47; secondary 20E42, 20G15 Contents 1. Tits’s geometry P ........................................................................197 2. A concrete geometry V , part I..............................................................198 3. A concrete geometry V , part II .............................................................199 4. Example: type A (projective geometry) .......................................................203 5. Strategy .................................................................................203 6. Example: type D (orthogonal geometry) ......................................................205 7. Example: type E6 .........................................................................208 -
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]. -
On Dynkin Diagrams, Cartan Matrices
Physics 220, Lecture 16 ? Reference: Georgi chapters 8-9, a bit of 20. • Continue with Dynkin diagrams and the Cartan matrix, αi · αj Aji ≡ 2 2 : αi The j-th row give the qi − pi = −pi values of the simple root αi's SU(2)i generators acting on the root αj. Again, we always have αi · µ 2 2 = qi − pi; (1) αi where pi and qi are the number of times that the weight µ can be raised by Eαi , or lowered by E−αi , respectively, before getting zero. Applied to µ = αj, we know that qi = 0, since E−αj jαii = 0, since αi − αj is not a root for i 6= j. 0 2 Again, we then have AjiAij = pp = 4 cos θij, which must equal 0,1,2, or 3; these correspond to θij = π=2, 2π=3, 3π=4, and 5π=6, respectively. The Dynkin diagram has a node for each simple root (so the number of nodes is 2 2 r =rank(G)), and nodes i and j are connected by AjiAij lines. When αi 6= αj , sometimes it's useful to darken the node for the smaller root. 2 2 • Another example: constructing the roots for C3, starting from α1 = α2 = 1, and 2 α3 = 2, i.e. the Cartan matrix 0 2 −1 0 1 @ −1 2 −1 A : 0 −2 2 Find 9 positive roots. • Classify all simple, compact Lie algebras from their Aji. Require 3 properties: (1) det A 6= 0 (since the simple roots are linearly independent); (2) Aji < 0 for i 6= j; (3) AijAji = 0,1, 2, 3. -
Semi-Simple Lie Algebras and Their Representations
i Semi-Simple Lie Algebras and Their Representations Robert N. Cahn Lawrence Berkeley Laboratory University of California Berkeley, California 1984 THE BENJAMIN/CUMMINGS PUBLISHING COMPANY Advanced Book Program Menlo Park, California Reading, Massachusetts ·London Amsterdam Don Mills, Ontario Sydney · · · · ii Preface iii Preface Particle physics has been revolutionized by the development of a new “paradigm”, that of gauge theories. The SU(2) x U(1) theory of electroweak in- teractions and the color SU(3) theory of strong interactions provide the present explanation of three of the four previously distinct forces. For nearly ten years physicists have sought to unify the SU(3) x SU(2) x U(1) theory into a single group. This has led to studies of the representations of SU(5), O(10), and E6. Efforts to understand the replication of fermions in generations have prompted discussions of even larger groups. The present volume is intended to meet the need of particle physicists for a book which is accessible to non-mathematicians. The focus is on the semi-simple Lie algebras, and especially on their representations since it is they, and not just the algebras themselves, which are of greatest interest to the physicist. If the gauge theory paradigm is eventually successful in describing the fundamental particles, then some representation will encompass all those particles. The sources of this book are the classical exposition of Jacobson in his Lie Algebras and three great papers of E.B. Dynkin. A listing of the references is given in the Bibliography. In addition, at the end of each chapter, references iv Preface are given, with the authors’ names in capital letters corresponding to the listing in the bibliography. -
The Mckay Correspondence, the Coxeter Element and Representation Theory Astérisque, Tome S131 (1985), P
Astérisque BERTRAM KOSTANT The McKay correspondence, the Coxeter element and representation theory Astérisque, tome S131 (1985), p. 209-255 <http://www.numdam.org/item?id=AST_1985__S131__209_0> © Société mathématique de France, 1985, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Société Mathématique de France Astérisque, hors série, 1985, p. 209-255 THE McKAY CORRESPONDENCE, THE COXETER ELEMENT AND REPRESENTATION THEORY BY Bertram KOSTANT* 1.1. Introduction. — A fundamental question in harmonie analysis is : given a locally compact group £?, a closed subgroup H, and an irreducible representation n of G, how does the restriction ir\H decompose. Consider this question in the following very basic situation : G = SU(2) and H — T is any finite subgroup of SU(2). The classification of all finite subgroups T of SU(2) 2 2 is classical and well known. See e.g. [12] or [13]. Let S(C ) = n=0 (c ) 2 be the symmetric algebra over C and let nn be the representation of SU(2) n 2 2 on S (C ) induced by its action on C . One knows nn is irreducible for all n and the set of equivalence classes {TTn}, n = 0,..., defines the unitary dual of SU(2). -
Algebraic Groups of Type D4, Triality and Composition Algebras
1 Algebraic groups of type D4, triality and composition algebras V. Chernousov1, A. Elduque2, M.-A. Knus, J.-P. Tignol3 Abstract. Conjugacy classes of outer automorphisms of order 3 of simple algebraic groups of classical type D4 are classified over arbitrary fields. There are two main types of conjugacy classes. For one type the fixed algebraic groups are simple of type G2; for the other type they are simple of type A2 when the characteristic is different from 3 and are not smooth when the characteristic is 3. A large part of the paper is dedicated to the exceptional case of characteristic 3. A key ingredient of the classification of conjugacy classes of trialitarian automorphisms is the fact that the fixed groups are automorphism groups of certain composition algebras. 2010 Mathematics Subject Classification: 20G15, 11E57, 17A75, 14L10. Keywords and Phrases: Algebraic group of type D4, triality, outer au- tomorphism of order 3, composition algebra, symmetric composition, octonions, Okubo algebra. 1Partially supported by the Canada Research Chairs Program and an NSERC research grant. 2Supported by the Spanish Ministerio de Econom´ıa y Competitividad and FEDER (MTM2010-18370-C04-02) and by the Diputaci´onGeneral de Arag´on|Fondo Social Europeo (Grupo de Investigaci´on de Algebra).´ 3Supported by the F.R.S.{FNRS (Belgium). J.-P. Tignol gratefully acknowledges the hospitality of the Zukunftskolleg of the Universit¨atKonstanz, where he was in residence as a Senior Fellow while this work was developing. 2 Chernousov, Elduque, Knus, Tignol 1. Introduction The projective linear algebraic group PGLn admits two types of conjugacy classes of outer automorphisms of order two. -
Special Unitary Group - Wikipedia
Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Special unitary group In mathematics, the special unitary group of degree n, denoted SU( n), is the Lie group of n×n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U( n), consisting of all n×n unitary matrices. As a compact classical group, U( n) is the group that preserves the standard inner product on Cn.[nb 1] It is itself a subgroup of the general linear group, SU( n) ⊂ U( n) ⊂ GL( n, C). The SU( n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1] The simplest case, SU(1) , is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+ I, − I}. [nb 2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. Contents Properties Lie algebra Fundamental representation Adjoint representation The group SU(2) Diffeomorphism with S 3 Isomorphism with unit quaternions Lie Algebra The group SU(3) Topology Representation theory Lie algebra Lie algebra structure Generalized special unitary group Example Important subgroups See also 1 of 10 2/22/2018, 8:54 PM Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Remarks Notes References Properties The special unitary group SU( n) is a real Lie group (though not a complex Lie group). -
View This Volume's Front and Back Matter
Functions of Several Complex Variables and Their Singularities Functions of Several Complex Variables and Their Singularities Wolfgang Ebeling Translated by Philip G. Spain Graduate Studies in Mathematics Volume 83 .•S%'3SL"?|| American Mathematical Society s^s^^v Providence, Rhode Island Editorial Board David Cox (Chair) Walter Craig N. V. Ivanov Steven G. Krantz Originally published in the German language by Friedr. Vieweg & Sohn Verlag, D-65189 Wiesbaden, Germany, as "Wolfgang Ebeling: Funktionentheorie, Differentialtopologie und Singularitaten. 1. Auflage (1st edition)". © Friedr. Vieweg & Sohn Verlag | GWV Fachverlage GmbH, Wiesbaden, 2001 Translated by Philip G. Spain 2000 Mathematics Subject Classification. Primary 32-01; Secondary 32S10, 32S55, 58K40, 58K60. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-83 Library of Congress Cataloging-in-Publication Data Ebeling, Wolfgang. [Funktionentheorie, differentialtopologie und singularitaten. English] Functions of several complex variables and their singularities / Wolfgang Ebeling ; translated by Philip Spain. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 83) Includes bibliographical references and index. ISBN 0-8218-3319-7 (alk. paper) 1. Functions of several complex variables. 2. Singularities (Mathematics) I. Title. QA331.E27 2007 515/.94—dc22 2007060745 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
Symmetry and Particle Physics, 4
Michaelmas Term 2008, Mathematical Tripos Part III Hugh Osborn Symmetry and Particle Physics, 4 1. The generators of the Lie algebra of SU(3) are Ei± for i = 1, 2, 3 and H1,H2 where [H1,H2] = 0, † [E1+,E2+] = E3+,[E1+,E3+] = [E2+,E3+] = 0 and Ei− = Ei+ . Ei±,Hi obey, for each i, the SU(2) algebra [E+,E−] = H, [H, E±] = ±2E± if H3 = H1 + H2. Also we have [H1,E2±] = ∓E2±, [H2,E1±] = ∓E1±. A state |ψi has weight [r1, r2] if H1|ψi = r1|ψi,H2|ψi = r2|ψi. |n1, n2ihw is a highest weight state with weight [n1, n2] satisfying Ei+|n1, n2ihw = 0. Let (l,k) i l−i k−i |ψi i = (E3−) (E1−) (E2−) |n1, n2ihw . Show that these all have weight [n1 + k − 2l, n2 − 2k + l]. If l ≥ k explain why, in order to construct a basis, it is necessary to restrict i = 0, . , k except if k > n2 when i = k − n2, . , k. For SU(2) generators E±,H obtain n n−1 [E+, (E−) ] = n(E−) (H − n + 1) . From the SU(3) commutators [E3+,E1−] = −E2+,[E3+,E2−] = E1+,[E2+,E3−] = E1− and [E2+,E1−] = [E1+,E2−] = 0 obtain also l−i l−i−1 k−i k−i−1 [E3+, (E1−) ] = − (l − i)(E1−) E2+ , [E3+, (E2−) ] = (k − i)(E2−) E1+ , i i−1 l−i [E2+, (E3−) ] = i E1−(E3−) , [E2+(E1−) ] = 0 . Hence obtain (l,k) (l−1,k−1) (l−1,k−1) E3+|ψi i = i(n1 + n2 − k − l + i + 1)|ψi−1 i − (l − i)(k − i)(n2 − k + i + 1)|ψi i , (l,k) (l−1,k−1) (l−1,k−1) E2+|ψi i = E1− i |ψi−1 i + (k − i)(n2 − k + i + 1)|ψi i . -
Representing the Sporadic Archimedean Polyhedra As Abstract Polytopes$
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Discrete Mathematics 310 (2010) 1835–1844 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Representing the sporadic Archimedean polyhedra as abstract polytopesI Michael I. Hartley a, Gordon I. Williams b,∗ a DownUnder GeoSolutions, 80 Churchill Ave, Subiaco, 6008, Western Australia, Australia b Department of Mathematics and Statistics, University of Alaska Fairbanks, PO Box 756660, Fairbanks, AK 99775-6660, United States article info a b s t r a c t Article history: We present the results of an investigation into the representations of Archimedean Received 29 August 2007 polyhedra (those polyhedra containing only one type of vertex figure) as quotients of Accepted 26 January 2010 regular abstract polytopes. Two methods of generating these presentations are discussed, Available online 13 February 2010 one of which may be applied in a general setting, and another which makes use of a regular polytope with the same automorphism group as the desired quotient. Representations Keywords: of the 14 sporadic Archimedean polyhedra (including the pseudorhombicuboctahedron) Abstract polytope as quotients of regular abstract polyhedra are obtained, and summarised in a table. The Archimedean polyhedron Uniform polyhedron information is used to characterize which of these polyhedra have acoptic Petrie schemes Quotient polytope (that is, have well-defined Petrie duals). Regular cover ' 2010 Elsevier B.V. All rights reserved. Flag action Exchange map 1. Introduction Much of the focus in the study of abstract polytopes has been on the regular abstract polytopes. A publication of the first author [6] introduced a method for representing any abstract polytope as a quotient of regular polytopes.