Algebraic Groups of Type D4, Triality, and Composition Algebras
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Algebraic Graph Theory: Automorphism Groups and Cayley Graphs
Algebraic Graph Theory: Automorphism Groups and Cayley graphs Glenna Toomey April 2014 1 Introduction An algebraic approach to graph theory can be useful in numerous ways. There is a relatively natural intersection between the fields of algebra and graph theory, specifically between group theory and graphs. Perhaps the most natural connection between group theory and graph theory lies in finding the automorphism group of a given graph. However, by studying the opposite connection, that is, finding a graph of a given group, we can define an extremely important family of vertex-transitive graphs. This paper explores the structure of these graphs and the ways in which we can use groups to explore their properties. 2 Algebraic Graph Theory: The Basics First, let us determine some terminology and examine a few basic elements of graphs. A graph, Γ, is simply a nonempty set of vertices, which we will denote V (Γ), and a set of edges, E(Γ), which consists of two-element subsets of V (Γ). If fu; vg 2 E(Γ), then we say that u and v are adjacent vertices. It is often helpful to view these graphs pictorially, letting the vertices in V (Γ) be nodes and the edges in E(Γ) be lines connecting these nodes. A digraph, D is a nonempty set of vertices, V (D) together with a set of ordered pairs, E(D) of distinct elements from V (D). Thus, given two vertices, u, v, in a digraph, u may be adjacent to v, but v is not necessarily adjacent to u. This relation is represented by arcs instead of basic edges. -
How Symmetric Are Real-World Graphs? a Large-Scale Study
S S symmetry Article How Symmetric Are Real-World Graphs? A Large-Scale Study Fabian Ball * and Andreas Geyer-Schulz Karlsruhe Institute of Technology, Institute of Information Systems and Marketing, Kaiserstr. 12, 76131 Karlsruhe, Germany; [email protected] * Correspondence: [email protected]; Tel.: +49-721-608-48404 Received: 22 December 2017; Accepted: 10 January 2018; Published: 16 January 2018 Abstract: The analysis of symmetry is a main principle in natural sciences, especially physics. For network sciences, for example, in social sciences, computer science and data science, only a few small-scale studies of the symmetry of complex real-world graphs exist. Graph symmetry is a topic rooted in mathematics and is not yet well-received and applied in practice. This article underlines the importance of analyzing symmetry by showing the existence of symmetry in real-world graphs. An analysis of over 1500 graph datasets from the meta-repository networkrepository.com is carried out and a normalized version of the “network redundancy” measure is presented. It quantifies graph symmetry in terms of the number of orbits of the symmetry group from zero (no symmetries) to one (completely symmetric), and improves the recognition of asymmetric graphs. Over 70% of the analyzed graphs contain symmetries (i.e., graph automorphisms), independent of size and modularity. Therefore, we conclude that real-world graphs are likely to contain symmetries. This contribution is the first larger-scale study of symmetry in graphs and it shows the necessity of handling symmetry in data analysis: The existence of symmetries in graphs is the cause of two problems in graph clustering we are aware of, namely, the existence of multiple equivalent solutions with the same value of the clustering criterion and, secondly, the inability of all standard partition-comparison measures of cluster analysis to identify automorphic partitions as equivalent. -
The J-Invariant, Tits Algebras and Triality
The J-invariant, Tits algebras and triality A. Qu´eguiner-Mathieu, N. Semenov, K. Zainoulline Abstract In the present paper we set up a connection between the indices of the Tits algebras of a semisimple linear algebraic group G and the degree one indices of its motivic J-invariant. Our main technical tools are the second Chern class map and Grothendieck's γ-filtration. As an application we provide lower and upper bounds for the degree one indices of the J-invariant of an algebra A with orthogonal involution σ and describe all possible values of the J-invariant in the trialitarian case, i.e., when degree of A equals 8. Moreover, we establish several relations between the J-invariant of (A; σ) and the J-invariant of the corresponding quadratic form over the function field of the Severi-Brauer variety of A. MSC: Primary 20G15, 14C25; Secondary 16W10, 11E04. Keywords: linear algebraic group, torsor, Tits algebra, triality, algebra with involution, Chow motive. Introduction The notion of a Tits algebra was introduced by Jacques Tits in his celebrated paper on irreducible representations [Ti71]. This invariant of a linear algebraic group G plays a crucial role in the computation of the K-theory of twisted flag varieties by Panin [Pa94] and in the index reduction formulas by Merkurjev, Panin and Wadsworth [MPW96]. It has important applications to the classifi- cation of linear algebraic groups, and to the study of the associated homogeneous varieties. Another invariant of a linear algebraic group, the J-invariant, has been recently defined in [PSZ08]. It extends the J-invariant of a quadratic form which was studied during the last decade, notably by Karpenko, Merkurjev, Rost and Vishik. -
On the Group-Theoretic Properties of the Automorphism Groups of Various Graphs
ON THE GROUP-THEORETIC PROPERTIES OF THE AUTOMORPHISM GROUPS OF VARIOUS GRAPHS CHARLES HOMANS Abstract. In this paper we provide an introduction to the properties of one important connection between the theories of groups and graphs, that of the group formed by the automorphisms of a given graph. We provide examples of important results in graph theory that can be understood through group theory and vice versa, and conclude with a treatment of Frucht's theorem. Contents 1. Introduction 1 2. Fundamental Definitions, Concepts, and Theorems 2 3. Example 1: The Orbit-Stabilizer Theorem and its Application to Graph Automorphisms 4 4. Example 2: On the Automorphism Groups of the Platonic Solid Skeleton Graphs 4 5. Example 3: A Tight Bound on the Product of the Chromatic Number and Independence Number of Vertex-Transitive Graphs 6 6. Frucht's Theorem 7 7. Acknowledgements 9 8. References 9 1. Introduction Groups and graphs are two highly important kinds of structures studied in math- ematics. Interestingly, the theory of groups and the theory of graphs are deeply connected. In this paper, we examine one particular such connection: that which emerges from the observation that the automorphisms of any given graph form a group under composition. In section 2, we provide a framework for understanding the material discussed in the paper. In sections 3, 4, and 5, we demonstrate how important results in group theory illuminate some properties of automorphism groups, how the geo- metric properties of particular embeddings of graphs can be used to determine the structure of the automorphism groups of all embeddings of those graphs, and how the automorphism group can be used to determine fundamental truths about the structure of the graph. -
Construction of Strongly Regular Graphs Having an Automorphism Group of Composite Order
Volume 15, Number 1, Pages 22{41 ISSN 1715-0868 CONSTRUCTION OF STRONGLY REGULAR GRAPHS HAVING AN AUTOMORPHISM GROUP OF COMPOSITE ORDER DEAN CRNKOVIC´ AND MARIJA MAKSIMOVIC´ Abstract. In this paper we outline a method for constructing strongly regular graphs from orbit matrices admitting an automorphism group of composite order. In 2011, C. Lam and M. Behbahani introduced the concept of orbit matrices of strongly regular graphs and developed an algorithm for the construction of orbit matrices of strongly regu- lar graphs with a presumed automorphism group of prime order, and construction of corresponding strongly regular graphs. The method of constructing strongly regular graphs developed and employed in this paper is a generalization of that developed by C. Lam and M. Behba- hani. Using this method we classify SRGs with parameters (49,18,7,6) having an automorphism group of order six. Eleven of the SRGs with parameters (49,18,7,6) constructed in that way are new. We obtain an additional 385 new SRGs(49,18,7,6) by switching. Comparing the con- structed graphs with previously known SRGs with these parameters, we conclude that up to isomorphism there are at least 727 SRGs with parameters (49,18,7,6). Further, we show that there are no SRGs with parameters (99,14,1,2) having an automorphism group of order six or nine, i.e. we rule out automorphism groups isomorphic to Z6, S3, Z9, or E9. 1. Introduction One of the main problems in the theory of strongly regular graphs (SRGs) is constructing and classifying SRGs with given parameters. -
Graph Automorphism Groups
Graph Automorphism Groups Robert A. Beeler, Ph.D. East Tennessee State University February 23, 2018 Robert A. Beeler, Ph.D. (East Tennessee State University)Graph Automorphism Groups February 23, 2018 1 / 1 What is a graph? A graph G =(V , E) is a set of vertices, V , together with as set of edges, E. For our purposes, each edge will be an unordered pair of distinct vertices. a e b d c V (G)= {a, b, c, d, e} E(G)= {ab, ae, bc, be, cd, de} Robert A. Beeler, Ph.D. (East Tennessee State University)Graph Automorphism Groups February 23, 2018 2 / 1 Graph Automorphisms A graph automorphism is simply an isomorphism from a graph to itself. In other words, an automorphism on a graph G is a bijection φ : V (G) → V (G) such that uv ∈ E(G) if and only if φ(u)φ(v) ∈ E(G). Note that graph automorphisms preserve adjacency. In layman terms, a graph automorphism is a symmetry of the graph. Robert A. Beeler, Ph.D. (East Tennessee State University)Graph Automorphism Groups February 23, 2018 3 / 1 An Example Consider the following graph: a d b c Robert A. Beeler, Ph.D. (East Tennessee State University)Graph Automorphism Groups February 23, 2018 4 / 1 An Example (Part 2) One automorphism simply maps every vertex to itself. This is the identity automorphism. a a d b d b c 7→ c e =(a)(b)(c)(d) Robert A. Beeler, Ph.D. (East Tennessee State University)Graph Automorphism Groups February 23, 2018 5 / 1 An Example (Part 3) One automorphism switches vertices a and c. -
Graphs with Given Automorphism Group and Few Edge Orbits
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Europ. 1. Combinatorics (1991) U, 185-203 Graphs with Given Automorphism Group and Few Edge Orbits LAsZLO BABAl, ALBERT J. GOODMAN AND LAsZLO LovAsz A graph X is said to represent the group G with k edge (vertex) orbits if the automorphism group of X is isomorphic to G and it acts with k orbits on the set of edges (vertices, resp.) of X. It is known that (with three exceptions) every finite group can be represented with ..:2 vertex orbits (Babai, 1974). In this paper we build a framework to study the minimum number of edge orbits. We show that surprisingly large classes of groups admit a representation with a bounded number of edge-orbits. These include the groups generated by a bounded number of abelian subgroups; in particular, any direct product of finite simple groups. The case of abelian groups is related to work of Brenner, and of Gelfand and Ponomarev. 1. INTRODUCTION We say that a graph X represents the group G if its automorphism group Aut(X) is isomorphic to G. An old theorem of Frucht [16J says that every finite group is represented by some finite graph. It is natural to ask how small or how symmetrical such a representation can be. The first measure of size is the number of vertices; the second, the number of edges. The number of orbits lends itself as a measure of symmetry. Let G have n elements. Then it is known [2J that G can be represented with at most two orbits on the vertices, with the exceptions of the cyclic groups of orders 3, 4 and 5. -
[Math.RA] 22 Apr 2002 Lsia Rus Hscetsadsicinbtentecascll Classical the Between the Distinction for a Available Only Creates Are This Which Groups, Groups
28 February 2002 MAGIC SQUARES AND MATRIX MODELS OF LIE ALGEBRAS C. H. BARTON AND A. SUDBERY Abstract. This paper is concerned with the description of excep- tional simple Lie algebras as octonionic analogues of the classical matrix Lie algebras. We review the Tits-Freudenthal construction of the magic square, which includes the exceptional Lie algebras as the octonionic case of a construction in terms of a Jordan alge- bra of hermitian 3 3 matrices (Tits) or various plane and other geometries (Freudenthal).× We present alternative constructions of the magic square which explain its symmetry, and show explicitly how the use of split composition algebras leads to analogues of the matrix Lie algebras su(3), sl(3) and sp(6). We adapt the magic square construction to include analogues of su(2), sl(2) and sp(4) for all real division algebras. Contents 1. Introduction 1 2. Algebras: Notation 3 3. The Tits construction 8 4. Symmetrical constructions of the n =3magicsquare 12 4.1. The triality algebra Tri K and Der H3(K) 12 4.2. The Vinberg construction 19 4.3. The triality construction 21 5. The rows of the magic square 22 6. Magic squares of n n matrices 26 × arXiv:math/0203010v2 [math.RA] 22 Apr 2002 6.1. The Santander-Herranz construction 27 7. Maximal compact subalgebras 28 8. The n =2magicsquare 34 Appendix A. Matrix identities 40 References 43 1. Introduction Semisimple Lie groups and Lie algebras are normally discussed in terms of their root systems, which makes possible a unified treatment and emphasises the common features of their underlying structures. -
Groups, Triality, and Hyperquasigroups
GROUPS, TRIALITY, AND HYPERQUASIGROUPS JONATHAN D. H. SMITH Abstract. Hyperquasigroups were recently introduced to provide a more symmetrical approach to quasigroups, and a far-reaching implementation of triality (S3-action). In the current paper, various connections between hyper- quasigroups and groups are examined, on the basis of established connections between quasigroups and groups. A new graph-theoretical characterization of hyperquasigroups is exhibited. Torsors are recognized as hyperquasigroups, and group representations are shown to be equivalent to linear hyperquasi- groups. The concept of orthant structure is introduced, as a tool for recovering classical information from a hyperquasigroup. 1. Introduction This paper is concerned with the recently introduced algebraic structures known as hyperquasigroups, which are intended as refinements of quasigroups. A quasi- group (Q; ·) was first understood as a set Q with a binary multiplication, denoted by · or mere juxtaposition, such that in the equation x · y = z ; knowledge of any two of x; y; z specifies the third uniquely. To make a distinction with subsequent concepts, it is convenient to describe a quasigroup in this original sense as a combinatorial quasigroup (Q; ·). A loop (Q; ·; e) is a combinatorial quasi- group (Q; ·) with an identity element e such that e·x = x = x·e for all x in Q. The body of the multiplication table of a combinatorial quasigroup of finite order n is a Latin square of order n, an n × n square array in which each row and each column contains each element of an n-element set exactly once. Conversely, each Latin square becomes the body of the multiplication table of a combinatorial quasigroup if distinct labels from the square are attached to the rows and columns of the square. -
Arxiv:1801.02942V2 [Math.OA]
THE PETERSEN GRAPH HAS NO QUANTUM SYMMETRY SIMON SCHMIDT Abstract. In 2007, Banica and Bichon asked whether the well-known Petersen graph has quantum symmetry. In this article, we show that the Petersen graph has no quantum symmetry, i.e. the quantum automorphism group of the Petersen graph is its usual automorphism group, the symmetric group S5. Introduction The study of graph automorphisms is an important branch of graph theory. The automorphism group of a finite graph on n vertices without multiple edges is given by Gaut(Γ) := {σ ∈ Sn | σε = εσ} ⊆ Sn, where ε is the adjacency matrix of the graph and Sn denotes the symmetric group. As a generalization of this concept, quantum automorphism groups of finite graphs were introduced in the framework of compact matrix quantum groups. In 2005, + ∗ Banica [1] defined the quantum automorphism group Gaut(Γ) based on the C - algebra + + C(Gaut(Γ)) := C(Sn )/huε = εui, + where Sn denotes the quantum symmetric group defined by Wang [10]. There is also another definition given by Bichon in [5], which is a quantum subgroup of Banica’s quantum automorphism group. + In [2], Banica and Bichon computed Gaut(Γ) for all vertex-transitive graphs Γ of order up to eleven, except the Petersen graph. The Petersen graph (see figure 1) is a arXiv:1801.02942v2 [math.OA] 17 Jan 2018 strongly regular graph on ten vertices and it often appears as a counter-example to conjectures in graph theory. For more information concerning the Petersen graph, we refer to the book [7]. We say that a graph has no quantum symmetry if the quantum automorphism group of Banica coincides with the usual automorphism group of the graph, as defined in [2]. -
A Note on Standard Composition Algebras of Types II and III
A note on standard composition algebras of types II and III Beites,∗ P. D. and Nicol´as,y A. P. March 28, 2016 Abstract: Some identities satisfied by certain standard composition al- gebras, of types II and III, are studied and become candidates for the cha- racterization of the mentioned types. Composition algebras of arbitrary dimension, over a field F with char(F ) 6= 2 and satisfying the identity x2y = n(x)y are shown to be standard composition algebras of type II. As a consequence, the identity yx2 = n(x)y characterizes the type III. Keywords: standard composition algebra, identity 2010 Mathematics Subject Classification: 17A75, 15A21 1 Introduction Among composition algebras, the well known ones are those with identity, that is, Hurwitz algebras. These over a field of characteristic different from 2, by the generalized Hurwitz Theorem in [5], are isomorphic either to the base field, a separable quadratic extension of the base field, a generalized quater- nion algebra or a generalized Cayley algebra. In particular, the dimension of any Hurwitz algebra is 1; 2; 4 or 8. This restriction on the dimension was also proved by Chevalley using Clifford algebras (for more details, see [3]). In general, the classification of all finite dimensional composition alge- bras is still an open problem. Nevertheless, the imposition of one or more additional conditions on these algebras, in the form of identities, can afford such a classification. For instance, in [8], Okubo proved that, over fields of characteristic different from 2, any finite dimensional composition algebra ∗corresponding author, Centro de Matem´aticae Aplica¸c~oesand Departamento de Matem´atica,Universidade da Beira Interior, Portugal yInstituto de Investigaci´onen Matem´aticasand Departamento de Matem´aticaApli- cada, Universidad de Valladolid, Spain 1 satisfying the flexible identity (xy)x = x(yx) is either a form of a Hurwitz algebra, a form of a para-Hurwitz algebra or an Okubo algebra. -
Triality and Algebraic Groups of Type $^ 3D 4$
3 TRIALITY AND ALGEBRAIC GROUPS OF TYPE D4 MAX-ALBERT KNUS AND JEAN-PIERRE TIGNOL 3 Abstract. We determine which simple algebraic groups of type D4 over arbitrary fields of characteristic different from 2 admit outer automorphisms of order 3, and classify these automorphisms up to conjugation. The criterion is formulated in terms of a representation of the group by automorphisms of a trialitarian algebra: outer automorphisms of order 3 exist if and only if the algebra is the endomorphism algebra of an induced cyclic composition; their conjugacy classes are in one-to-one correspondence with isomorphism classes of symmetric compositions from which the induced cyclic composition stems. 1. Introduction Let G0 be an adjoint Chevalley group of type D4 over a field F . Since the auto- morphism group of the Dynkin diagram of type D4 is isomorphic to the symmetric group S3, there is a split exact sequence of algebraic groups / Int / π / / (1) 1 G0 Aut(G0) S3 1. Thus, Aut(G0) =∼ G0 ⋊ S3; in particular G0 admits outer automorphisms of order 3, which we call trialitarian automorphisms. Adjoint algebraic groups of type D4 1 over F are classified by the Galois cohomology set H (F, G0 ⋊ S3) and the map induced by π in cohomology 1 1 π∗ : H (F, G0 ⋊ S3) → H (F, S3) associates to any group G of type D4 the isomorphism class of a cubic ´etale F - 1 2 algebra L. The group G is said to be of type D4 if L is split, of type D4 if L =∼ F ×∆ 3 for some quadratic separable field extension ∆/F , of type D4 if L is a cyclic field 6 extension of F and of type D4 if L is a non-cyclic field extension.