Matroid Automorphisms of the F4 Root System
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JMM 2017 Student Poster Session Abstract Book
Abstracts for the MAA Undergraduate Poster Session Atlanta, GA January 6, 2017 Organized by Eric Ruggieri College of the Holy Cross and Chasen Smith Georgia Southern University Organized by the MAA Committee on Undergraduate Student Activities and Chapters and CUPM Subcommittee on Research by Undergraduates Dear Students, Advisors, Judges and Colleagues, If you look around today you will see over 300 posters and nearly 500 student presenters, representing a wide array of mathematical topics and ideas. These posters showcase the vibrant research being conducted as part of summer programs and during the academic year at colleges and universities from across the United States and beyond. It is so rewarding to see this session, which offers such a great opportunity for interaction between students and professional mathematicians, continue to grow. The judges you see here today are professional mathematicians from institutions around the world. They are advisors, colleagues, new Ph.D.s, and administrators. We have acknowledged many of them in this booklet; however, many judges here volunteered on site. Their support is vital to the success of the session and we thank them. We are supported financially by Tudor Investments and Two Sigma. We are also helped by the mem- bers of the Committee on Undergraduate Student Activities and Chapters (CUSAC) in some way or other. They are: Dora C. Ahmadi; Jennifer Bergner; Benjamin Galluzzo; Kristina Cole Garrett; TJ Hitchman; Cynthia Huffman; Aihua Li; Sara Louise Malec; Lisa Marano; May Mei; Stacy Ann Muir; Andy Nieder- maier; Pamela A. Richardson; Jennifer Schaefer; Peri Shereen; Eve Torrence; Violetta Vasilevska; Gerard A. -
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. -
European Journal of Combinatorics a Survey on Spherical Designs And
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector European Journal of Combinatorics 30 (2009) 1392–1425 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc A survey on spherical designs and algebraic combinatorics on spheres Eiichi Bannai a, Etsuko Bannai b a Faculty of Mathematics, Graduate School, Kyushu University, Hakozaki 6-10-1, Higashu-ku, Fukuoka, 812-8581, Japan b Misakigaoka 2-8-21, Maebaru-shi, Fukuoka, 819-1136, Japan article info a b s t r a c t Article history: This survey is mainly intended for non-specialists, though we try to Available online 12 February 2009 include many recent developments that may interest the experts as well. We want to study ``good'' finite subsets of the unit sphere. To consider ``what is good'' is a part of our problem. We start with n−1 n the definition of spherical t-designs on S in R : After discussing some important examples, we focus on tight spherical t-designs on Sn−1: Tight t-designs have good combinatorial properties, but they rarely exist. So, we are interested in the finite subsets on Sn−1, which have properties similar to tight t-designs from the various viewpoints of algebraic combinatorics. For example, rigid t-designs, universally optimal t-codes (configurations), as well as finite sets which admit the structure of an association scheme, are among them. We will discuss various results on the existence and the non-existence of special spherical t-designs, as well as general spherical t-designs, and their constructions. -
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. -
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. -
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 . -
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. -
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. -
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. -
Math 210C. Size of Fundamental Group 1. Introduction Let (V,Φ) Be a Nonzero Root System. Let G Be a Connected Compact Lie Group
Math 210C. Size of fundamental group 1. Introduction Let (V; Φ) be a nonzero root system. Let G be a connected compact Lie group that is 0 semisimple (equivalently, ZG is finite, or G = G ; see Exercise 4(ii) in HW9) and for which the associated root system is isomorphic to (V; Φ) (so G 6= 1). More specifically, we pick a maximal torus T and suppose an isomorphism (V; Φ) ' (X(T )Q; Φ(G; T ))) is chosen. Where can X(T ) be located inside V ? It certainly has to contain the root lattice Q := ZΦ determined by the root system (V; Φ). Exercise 1(iii) in HW7 computes the index of this inclusion of lattices: X(T )=Q is dual to the finite abelian group ZG. The root lattice is a \lower bound" on where the lattice X(T ) can sit inside V . This handout concerns the deeper \upper bound" on X(T ) provided by the Z-dual P := (ZΦ_)0 ⊂ V ∗∗ = V of the coroot lattice ZΦ_ (inside the dual root system (V ∗; Φ_)). This lattice P is called the weight lattice (the French word for \weight" is \poids") because of its relation with the representation theory of any possibility for G (as will be seen in our discussion of the Weyl character formula). To explain the significance of P , recall from HW9 that semisimplicity of G implies π1(G) is a finite abelian group, and more specifically if f : Ge ! G is the finite-degree universal cover and Te := f −1(T ) ⊂ Ge (a maximal torus of Ge) then π1(G) = ker f = ker(f : Te ! T ) = ker(X∗(Te) ! X∗(T )) is dual to the inclusion X(T ) ,! X(Te) inside V . -
Linear Algebraic Groups
Clay Mathematics Proceedings Volume 4, 2005 Linear Algebraic Groups Fiona Murnaghan Abstract. We give a summary, without proofs, of basic properties of linear algebraic groups, with particular emphasis on reductive algebraic groups. 1. Algebraic groups Let K be an algebraically closed field. An algebraic K-group G is an algebraic variety over K, and a group, such that the maps µ : G × G → G, µ(x, y) = xy, and ι : G → G, ι(x)= x−1, are morphisms of algebraic varieties. For convenience, in these notes, we will fix K and refer to an algebraic K-group as an algebraic group. If the variety G is affine, that is, G is an algebraic set (a Zariski-closed set) in Kn for some natural number n, we say that G is a linear algebraic group. If G and G′ are algebraic groups, a map ϕ : G → G′ is a homomorphism of algebraic groups if ϕ is a morphism of varieties and a group homomorphism. Similarly, ϕ is an isomorphism of algebraic groups if ϕ is an isomorphism of varieties and a group isomorphism. A closed subgroup of an algebraic group is an algebraic group. If H is a closed subgroup of a linear algebraic group G, then G/H can be made into a quasi- projective variety (a variety which is a locally closed subset of some projective space). If H is normal in G, then G/H (with the usual group structure) is a linear algebraic group. Let ϕ : G → G′ be a homomorphism of algebraic groups. Then the kernel of ϕ is a closed subgroup of G and the image of ϕ is a closed subgroup of G.