Math 249B. Root Systems for Split Classical Groups
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Matroid Automorphisms of the F4 Root System
∗ Matroid automorphisms of the F4 root system Stephanie Fried Aydin Gerek Gary Gordon Dept. of Mathematics Dept. of Mathematics Dept. of Mathematics Grinnell College Lehigh University Lafayette College Grinnell, IA 50112 Bethlehem, PA 18015 Easton, PA 18042 [email protected] [email protected] Andrija Peruniˇci´c Dept. of Mathematics Bard College Annandale-on-Hudson, NY 12504 [email protected] Submitted: Feb 11, 2007; Accepted: Oct 20, 2007; Published: Nov 12, 2007 Mathematics Subject Classification: 05B35 Dedicated to Thomas Brylawski. Abstract Let F4 be the root system associated with the 24-cell, and let M(F4) be the simple linear dependence matroid corresponding to this root system. We determine the automorphism group of this matroid and compare it to the Coxeter group W for the root system. We find non-geometric automorphisms that preserve the matroid but not the root system. 1 Introduction Given a finite set of vectors in Euclidean space, we consider the linear dependence matroid of the set, where dependences are taken over the reals. When the original set of vectors has `nice' symmetry, it makes sense to compare the geometric symmetry (the Coxeter or Weyl group) with the group that preserves sets of linearly independent vectors. This latter group is precisely the automorphism group of the linear matroid determined by the given vectors. ∗Research supported by NSF grant DMS-055282. the electronic journal of combinatorics 14 (2007), #R78 1 For the root systems An and Bn, matroid automorphisms do not give any additional symmetry [4]. One can interpret these results in the following way: combinatorial sym- metry (preserving dependence) and geometric symmetry (via isometries of the ambient Euclidean space) are essentially the same. -
Platonic Solids Generate Their Four-Dimensional Analogues
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. Via the Cartan-Dieudonne´ theorem, the reflective symmetries of the arXiv:1307.6768v1 [math-ph] 25 Jul 2013 Platonic solids generate rotations. In a Clifford algebra framework, the space of spinors gen- erating such three-dimensional rotations has a natural four-dimensional Euclidean structure. The spinors arising from the Platonic Solids can thus in turn be interpreted as vertices in four- dimensional space, giving a simple construction of the 4D polytopes 16-cell, 24-cell, the F4 root system and the 600-cell. In particular, these polytopes have ‘mysterious’ symmetries, that are almost trivial when seen from the three-dimensional spinorial point of view. In fact, all these induced polytopes are also known to be root systems and thus generate rank-4 Coxeter PREPRINT: Acta Crystallographica Section A A Journal of the International Union of Crystallography 2 groups, which can be shown to be a general property of the spinor construction. -
Classifying the Representation Type of Infinitesimal Blocks of Category
Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. Platt (Under the Direction of Brian D. Boe) Abstract Let g be a simple Lie algebra over the field C of complex numbers, with root system Φ relative to a fixed maximal toral subalgebra h. Let S be a subset of the simple roots ∆ of Φ, which determines a standard parabolic subalgebra of g. Fix an integral weight ∗ µ µ ∈ h , with singular set J ⊆ ∆. We determine when an infinitesimal block O(g,S,J) := OS of parabolic category OS is nonzero using the theory of nilpotent orbits. We extend work of Futorny-Nakano-Pollack, Br¨ustle-K¨onig-Mazorchuk, and Boe-Nakano toward classifying the representation type of the nonzero infinitesimal blocks of category OS by considering arbitrary sets S and J, and observe a strong connection between the theory of nilpotent orbits and the representation type of the infinitesimal blocks. We classify certain infinitesimal blocks of category OS including all the semisimple infinitesimal blocks in type An, and all of the infinitesimal blocks for F4 and G2. Index words: Category O; Representation type; Verma modules Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. Platt B.A., Utah State University, 1999 M.S., Utah State University, 2001 M.A, The University of Georgia, 2006 A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Athens, Georgia 2008 c 2008 Kenyon J. Platt All Rights Reserved Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. -
Strategies for Growing a High‐Quality Root System, Trunk, and Crown in a Container Nursery
Strategies for Growing a High‐Quality Root System, Trunk, and Crown in a Container Nursery Companion publication to the Guideline Specifications for Nursery Tree Quality Table of Contents Introduction Section 1: Roots Defects ...................................................................................................................................................................... 1 Liner Development ............................................................................................................................................. 3 Root Ball Management in Larger Containers .......................................................................................... 6 Root Distribution within Root Ball .............................................................................................................. 10 Depth of Root Collar ........................................................................................................................................... 11 Section 2: Trunk Temporary Branches ......................................................................................................................................... 12 Straight Trunk ...................................................................................................................................................... 13 Section 3: Crown Central Leader ......................................................................................................................................................14 Heading and Re‐training -
Introuduction to Representation Theory of Lie Algebras
Introduction to Representation Theory of Lie Algebras Tom Gannon May 2020 These are the notes for the summer 2020 mini course on the representation theory of Lie algebras. We'll first define Lie groups, and then discuss why the study of representations of simply connected Lie groups reduces to studying representations of their Lie algebras (obtained as the tangent spaces of the groups at the identity). We'll then discuss a very important class of Lie algebras, called semisimple Lie algebras, and we'll examine the repre- sentation theory of two of the most basic Lie algebras: sl2 and sl3. Using these examples, we will develop the vocabulary needed to classify representations of all semisimple Lie algebras! Please email me any typos you find in these notes! Thanks to Arun Debray, Joakim Færgeman, and Aaron Mazel-Gee for doing this. Also{thanks to Saad Slaoui and Max Riestenberg for agreeing to be teaching assistants for this course and for many, many helpful edits. Contents 1 From Lie Groups to Lie Algebras 2 1.1 Lie Groups and Their Representations . .2 1.2 Exercises . .4 1.3 Bonus Exercises . .4 2 Examples and Semisimple Lie Algebras 5 2.1 The Bracket Structure on Lie Algebras . .5 2.2 Ideals and Simplicity of Lie Algebras . .6 2.3 Exercises . .7 2.4 Bonus Exercises . .7 3 Representation Theory of sl2 8 3.1 Diagonalizability . .8 3.2 Classification of the Irreducible Representations of sl2 .............8 3.3 Bonus Exercises . 10 4 Representation Theory of sl3 11 4.1 The Generalization of Eigenvalues . -
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 -
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. -
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. -
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. -
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. -
Contents 1 Root Systems
Stefan Dawydiak February 19, 2021 Marginalia about roots These notes are an attempt to maintain a overview collection of facts about and relationships between some situations in which root systems and root data appear. They also serve to track some common identifications and choices. The references include some helpful lecture notes with more examples. The author of these notes learned this material from courses taught by Zinovy Reichstein, Joel Kam- nitzer, James Arthur, and Florian Herzig, as well as many student talks, and lecture notes by Ivan Loseu. These notes are simply collected marginalia for those references. Any errors introduced, especially of viewpoint, are the author's own. The author of these notes would be grateful for their communication to [email protected]. Contents 1 Root systems 1 1.1 Root space decomposition . .2 1.2 Roots, coroots, and reflections . .3 1.2.1 Abstract root systems . .7 1.2.2 Coroots, fundamental weights and Cartan matrices . .7 1.2.3 Roots vs weights . .9 1.2.4 Roots at the group level . .9 1.3 The Weyl group . 10 1.3.1 Weyl Chambers . 11 1.3.2 The Weyl group as a subquotient for compact Lie groups . 13 1.3.3 The Weyl group as a subquotient for noncompact Lie groups . 13 2 Root data 16 2.1 Root data . 16 2.2 The Langlands dual group . 17 2.3 The flag variety . 18 2.3.1 Bruhat decomposition revisited . 18 2.3.2 Schubert cells . 19 3 Adelic groups 20 3.1 Weyl sets . 20 References 21 1 Root systems The following examples are taken mostly from [8] where they are stated without most of the calculations. -
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).