Notes on Classical Groups
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View This Volume's Front and Back Matter
http://dx.doi.org/10.1090/gsm/039 Selected Titles in This Series 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2001 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2001 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. -
On Absolute Points of Correlations in PG $(2, Q^ N) $
On absolute points of correlations in PG(2,qn) Jozefien D’haeseleer ∗ Department of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University Gent, Flanders, Belgium [email protected] Nicola Durante Dipartimento di Matematica e Applicazioni ”Caccioppoli” Universit`adi Napoli ”Federico II”, Complesso di Monte S. Angelo Napoli, Italy [email protected] Submitted: Aug 1, 2019; Accepted: May 8, 2020; Published: TBD c The authors. Released under the CC BY-ND license (International 4.0). Abstract Let V bea(d+1)-dimensional vector space over a field F. Sesquilinear forms over V have been largely studied when they are reflexive and hence give rise to a (possibly degenerate) polarity of the d-dimensional projective space PG(V ). Everything is known in this case for both degenerate and non-degenerate reflexive forms if F is either R, C or a finite field Fq. In this paper we consider degenerate, non- F3 reflexive sesquilinear forms of V = qn . We will see that these forms give rise to degenerate correlations of PG(2,qn) whose set of absolute points are, besides cones, m the (possibly degenerate) CF -sets studied in [8]. In the final section we collect some results from the huge work of B.C. Kestenband regarding what is known for the set of the absolute points of correlations in PG(2,qn) induced by a non-degenerate, F3 non-reflexive sesquilinear form of V = qn . Mathematics Subject Classifications: 51E21,05B25 arXiv:2005.05698v1 [math.CO] 12 May 2020 1 Introduction and definitions Let V and W be two vector spaces over the same field F. -
[Math.GR] 9 Jul 2003 Buildings and Classical Groups
Buildings and Classical Groups Linus Kramer∗ Mathematisches Institut, Universit¨at W¨urzburg Am Hubland, D–97074 W¨urzburg, Germany email: [email protected] In these notes we describe the classical groups, that is, the linear groups and the orthogonal, symplectic, and unitary groups, acting on finite dimen- sional vector spaces over skew fields, as well as their pseudo-quadratic gen- eralizations. Each such group corresponds in a natural way to a point-line geometry, and to a spherical building. The geometries in question are pro- jective spaces and polar spaces. We emphasize in particular the rˆole played by root elations and the groups generated by these elations. The root ela- tions reflect — via their commutator relations — algebraic properties of the underlying vector space. We also discuss some related algebraic topics: the classical groups as per- mutation groups and the associated simple groups. I have included some remarks on K-theory, which might be interesting for applications. The first K-group measures the difference between the classical group and its subgroup generated by the root elations. The second K-group is a kind of fundamental group of the group generated by the root elations and is related to central extensions. I also included some material on Moufang sets, since this is an in- arXiv:math/0307117v1 [math.GR] 9 Jul 2003 teresting topic. In this context, the projective line over a skew field is treated in some detail, and possibly with some new results. The theory of unitary groups is developed along the lines of Hahn & O’Meara [15]. -
Hartley's Theorem on Representations of the General Linear Groups And
Turk J Math 31 (2007) , Suppl, 211 – 225. c TUB¨ ITAK˙ Hartley’s Theorem on Representations of the General Linear Groups and Classical Groups A. E. Zalesski To the memory of Brian Hartley Abstract We suggest a new proof of Hartley’s theorem on representations of the general linear groups GLn(K)whereK is a field. Let H be a subgroup of GLn(K)andE the natural GLn(K)-module. Suppose that the restriction E|H of E to H contains aregularKH-module. The theorem asserts that this is then true for an arbitrary GLn(K)-module M provided dim M>1andH is not of exponent 2. Our proof is based on the general facts of representation theory of algebraic groups. In addition, we provide partial generalizations of Hartley’s theorem to other classical groups. Key Words: subgroups of classical groups, representation theory of algebraic groups 1. Introduction In 1986 Brian Hartley [4] obtained the following interesting result: Theorem 1.1 Let K be a field, E the standard GLn(K)-module, and let M be an irre- ducible finite-dimensional GLn(K)-module over K with dim M>1. For a finite subgroup H ⊂ GLn(K) suppose that the restriction of E to H contains a regular submodule, that ∼ is, E = KH ⊕ E1 where E1 is a KH-module. Then M contains a free KH-submodule, unless H is an elementary abelian 2-groups. His proof is based on deep properties of the duality between irreducible representations of the general linear group GLn(K)and the symmetric group Sn. -
Morita Equivalence and Isomorphisms Between General Linear Groups
Morita Equivalence and Isomorphisms between General Linear Groups by LOK Tsan-ming , f 〜 旮 // (z 织.'U f/ ’; 乂.: ,...‘. f-- 1 K SEP :;! : \V:丄:、 -...- A thesis Submitted to the Graduate School of The Chinese University of Hong Kong (Division of Mathematics) In Partial Fulfiilinent of the Requirement the Degree of Master of Philosophy(M. Phil.) HONG KONG June, 1994 I《)/ l入 G:升 16 1 _ j / L- b ! ? ^ LL [I l i Abstract In this thesis, we generalize Bolla's theorem. We replace progenerators in Bolla's theorem by quasiprogenerators. Then, we use this result to describe the isomorphisms between general linear groups over rings. Some characterizations of these isomorphisms are obtained. We also prove some results about the endomorphism ring of infinitely generated projective module. The relationship between the annihilators and closed submodules of infinitely generated projective module is investigated. Acknowledgement I would like to express my deepest gratitude to Dr. K. P. Shum for his energetic and stimulating guidance. His supervision has helped me a lot during the preparation of this thesis. LOK Tsan-ming The Chinese University of Hong Kong June, 1994 The Chinese University of Hong Kong Graduate School The undersigned certify that we have read a thesis, entitled “Morita Equivalence and Isomorphisms between General Linear Groups" submitted to the Graduate School by LOK Tsan-ming ( ) in Partial fulfillment of the requirements for the degree of Master of Philosophy in Mathematics. We recommend that it be accepted. Dr. K. P. Shum, Supervisor Dr. K. W. Leung ( ) ( ) Dr. C. W. Leung Prof. Yonghua Xu, External Examiner ( ) ("I今永等)各^^L^ Prof. -
Majorana Spinors
MAJORANA SPINORS JOSE´ FIGUEROA-O'FARRILL Contents 1. Complex, real and quaternionic representations 2 2. Some basis-dependent formulae 5 3. Clifford algebras and their spinors 6 4. Complex Clifford algebras and the Majorana condition 10 5. Examples 13 One dimension 13 Two dimensions 13 Three dimensions 14 Four dimensions 14 Six dimensions 15 Ten dimensions 16 Eleven dimensions 16 Twelve dimensions 16 ...and back! 16 Summary 19 References 19 These notes arose as an attempt to conceptualise the `symplectic Majorana{Weyl condition' in 5+1 dimensions; but have turned into a general discussion of spinors. Spinors play a crucial role in supersymmetry. Part of their versatility is that they come in many guises: `Dirac', `Majorana', `Weyl', `Majorana{Weyl', `symplectic Majorana', `symplectic Majorana{Weyl', and their `pseudo' counterparts. The tra- ditional physics approach to this topic is a mixed bag of tricks using disparate aspects of representation theory of finite groups. In these notes we will attempt to provide a uniform treatment based on the classification of Clifford algebras, a work dating back to the early 60s and all but ignored by the theoretical physics com- munity. Recent developments in superstring theory have made us re-examine the conditions for the existence of different kinds of spinors in spacetimes of arbitrary signature, and we believe that a discussion of this more uniform approach is timely and could be useful to the student meeting this topic for the first time or to the practitioner who has difficulty remembering the answer to questions like \when do symplectic Majorana{Weyl spinors exist?" The notes are organised as follows. -
148. Symplectic Translation Planes by Antoni
Lecture Notes of Seminario Interdisciplinare di Matematica Vol. 2 (2003), pp. 101 - 148. Symplectic translation planes by Antonio Maschietti Abstract1. A great deal of important work on symplectic translation planes has occurred in the last two decades, especially on those of even order, because of their link with non-linear codes. This link, which is the central theme of this paper, is based upon classical groups, particularly symplectic and orthogonal groups. Therefore I have included an Appendix, where standard notation and basic results are recalled. 1. Translation planes In this section we give an introductory account of translation planes. Compre- hensive textbooks are for example [17] and [3]. 1.1. Notation. We will use linear algebra to construct interesting geometrical structures from vector spaces, with special regard to vector spaces over finite fields. Any finite field has prime power order and for any prime power q there is, up to isomorphisms, a unique finite field of order q. This unique field is commonly de- noted by GF (q)(Galois Field); but also other symbols are usual, such as Fq.If q = pn with p a prime, the additive structure of F is that of an n dimensional q − vector space over Fp, which is the field of integers modulo p. The multiplicative group of Fq, denoted by Fq⇤, is cyclic. Finally, the automorphism group of the field Fq is cyclic of order n. Let A and B be sets. If f : A B is a map (or function or else mapping), then the image of x A will be denoted! by f(x)(functional notation). -
Spacetime May Have Fractal Properties on a Quantum Scale 25 March 2009, by Lisa Zyga
Spacetime May Have Fractal Properties on a Quantum Scale 25 March 2009, By Lisa Zyga values at short scales. More than being just an interesting idea, this phenomenon might provide insight into a quantum theory of relativity, which also has been suggested to have scale-dependent dimensions. Benedetti’s study is published in a recent issue of Physical Review Letters. “It is an old idea in quantum gravity that at short scales spacetime might appear foamy, fuzzy, fractal or similar,” Benedetti told PhysOrg.com. “In my work, I suggest that quantum groups are a valid candidate for the description of such a quantum As scale decreases, the number of dimensions of k- spacetime. Furthermore, computing the spectral Minkowski spacetime (red line), which is an example of a dimension, I provide for the first time a link between space with quantum group symmetry, decreases from four to three. In contrast, classical Minkowski spacetime quantum groups/noncommutative geometries and (blue line) is four-dimensional on all scales. This finding apparently unrelated approaches to quantum suggests that quantum groups are a valid candidate for gravity, such as Causal Dynamical Triangulations the description of a quantum spacetime, and may have and Exact Renormalization Group. And establishing connections with a theory of quantum gravity. Image links between different topics is often one of the credit: Dario Benedetti. best ways we have to understand such topics.” In his study, Benedetti explains that a spacetime with quantum group symmetry has in general a (PhysOrg.com) -- Usually, we think of spacetime scale-dependent dimension. Unlike classical as being four-dimensional, with three dimensions groups, which act on commutative spaces, of space and one dimension of time. -
Chapter IX. Tensors and Multilinear Forms
Notes c F.P. Greenleaf and S. Marques 2006-2016 LAII-s16-quadforms.tex version 4/25/2016 Chapter IX. Tensors and Multilinear Forms. IX.1. Basic Definitions and Examples. 1.1. Definition. A bilinear form is a map B : V V C that is linear in each entry when the other entry is held fixed, so that × → B(αx, y) = αB(x, y)= B(x, αy) B(x + x ,y) = B(x ,y)+ B(x ,y) for all α F, x V, y V 1 2 1 2 ∈ k ∈ k ∈ B(x, y1 + y2) = B(x, y1)+ B(x, y2) (This of course forces B(x, y)=0 if either input is zero.) We say B is symmetric if B(x, y)= B(y, x), for all x, y and antisymmetric if B(x, y)= B(y, x). Similarly a multilinear form (aka a k-linear form , or a tensor− of rank k) is a map B : V V F that is linear in each entry when the other entries are held fixed. ×···×(0,k) → We write V = V ∗ . V ∗ for the set of k-linear forms. The reason we use V ∗ here rather than V , and⊗ the⊗ rationale for the “tensor product” notation, will gradually become clear. The set V ∗ V ∗ of bilinear forms on V becomes a vector space over F if we define ⊗ 1. Zero element: B(x, y) = 0 for all x, y V ; ∈ 2. Scalar multiple: (αB)(x, y)= αB(x, y), for α F and x, y V ; ∈ ∈ 3. Addition: (B + B )(x, y)= B (x, y)+ B (x, y), for x, y V . -
Compactification of Classical Groups
COMMUNICATIONS IN ANALYSIS AND GEOMETRY Volume 10, Number 4, 709-740, 2002 Compactification of Classical Groups HONGYU HE 0. Introduction. 0.1. Historical Notes. Compactifications of symmetric spaces and their arithmetic quotients have been studied by many people from different perspectives. Its history went back to E. Cartan's original paper ([2]) on Hermitian symmetric spaces. Cartan proved that a Hermitian symmetric space of noncompact type can be realized as a bounded domain in a complex vector space. The com- pactification of Hermitian symmetric space was subsequently studied by Harish-Chandra and Siegel (see [16]). Thereafter various compactifications of noncompact symmetric spaces in general were studied by Satake (see [14]), Furstenberg (see [3]), Martin (see [4]) and others. These compact- ifications are more or less of the same nature as proved by C. Moore (see [11]) and by Guivarc'h-Ji-Taylor (see [4]). In the meanwhile, compacti- fication of the arithmetic quotients of symmetric spaces was explored by Satake (see [15]), Baily-Borel (see [1]), and others. It plays a significant role in the theory of automorphic forms. One of the main problems involved is the analytic properties of the boundary under the compactification. In all these compactifications that have been studied so far, the underlying compact space always has boundary. For a more detailed account of these compactifications, see [16] and [4]. In this paper, motivated by the author's work on the compactification of symplectic group (see [9]), we compactify all the classical groups. One nice feature of our compactification is that the underlying compact space is a symmetric space of compact type. -
Dual Pairs in the Pin-Group and Duality for the Corresponding Spinorial
DUAL PAIRS IN THE PIN-GROUP AND DUALITY FOR THE CORRESPONDING SPINORIAL REPRESENTATION CLÉMENT GUÉRIN, GANG LIU, AND ALLAN MERINO Abstract. In this paper, we give a complete picture of Howe correspondence for the setting (O(E, b), Pin(E, b), Π), where O(E, b) is an orthogonal group (real or complex), Pin(E, b) is the two-fold Pin-covering of O(E, b), and Π is the spinorial representation of Pin(E, b). More pre- cisely, for a dual pair (G, G′) in O(E, b), we determine explicitly the nature of its preimages (G, G′) in Pin(E, b), and prove that apart from some exceptions, (G, G′) is always a dual pair in e f e f Pin(E, b); then we establish the Howe correspondence for Π with respect to (G, G′). e f Contents 1. Introduction 1 2. Preliminaries 3 3. Dual pairs in the Pin-group 6 3.1. Pull-back of dual pairs 6 3.2. Identifyingtheisomorphismclassofpull-backs 8 4. Duality for the spinorial representation 14 References 21 1. Introduction The first duality phenomenon has been discovered by H. Weyl who pointed out a correspon- dence between some irreducible finite dimensional representations of the general linear group GL(V) and the symmetric group Sk where V is a finite dimensional vector space over C. In- d deed, considering the joint action of GL(V) and Sd on the space V⊗ , we get the following decomposition: arXiv:1907.09093v1 [math.RT] 22 Jul 2019 d λ V⊗ = M λ σ , [ ⊗ (Vλ,λ) GL(V) ∈ π [ where GL(V)π is the set of equivalence classes of irreducible finite dimensional representations d of GL(V) such that Hom (V , V⊗ ) , 0 and σ is an irreducible representation of S . -
Forms on Inner Product Spaces
FORMS ON INNER PRODUCT SPACES MARIA INFUSINO PROSEMINAR ON LINEAR ALGEBRA WS2016/2017 UNIVERSITY OF KONSTANZ Abstract. This note aims to give an introduction on forms on inner product spaces and their relation to linear operators. After briefly recalling some basic concepts from the theory of linear operators on inner product spaces, we focus on the space of forms on a real or complex finite-dimensional vector space V and show that it is isomorphic to the space of linear operators on V . We also describe the matrix representation of a form with respect to an ordered basis of the space on which it is defined, giving special attention to the case of forms on finite-dimensional complex inner product spaces and in particular to Hermitian forms. Contents Introduction 1 1. Preliminaries 1 2. Sesquilinear forms on inner product spaces 3 3. Matrix representation of a form 4 4. Hermitian forms 6 Notes 7 References 8 FORMS ON INNER PRODUCT SPACES 1 Introduction In this note we are going to introduce the concept of forms on an inner product space and describe some of their main properties (c.f. [2, Section 9.2]). The notion of inner product is a basic notion in linear algebra which allows to rigorously in- troduce on any vector space intuitive geometrical notions such as the length of an element and the orthogonality between two of them. We will just recall this notion in Section 1 together with some basic examples (for more details on this structure see e.g. [1, Chapter 3], [2, Chapter 8], [3]).