[Math.GR] 9 Jul 2003 Buildings and Classical Groups
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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. -
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. -
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. -
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). -
Automorphisms of Classical Geometries in the Sense of Klein
Automorphisms of classical geometries in the sense of Klein Navarro, A.∗ Navarro, J.† November 11, 2018 Abstract In this note, we compute the group of automorphisms of Projective, Affine and Euclidean Geometries in the sense of Klein. As an application, we give a simple construction of the outer automorphism of S6. 1 Introduction Let Pn be the set of 1-dimensional subspaces of a n + 1-dimensional vector space E over a (commutative) field k. This standard definition does not capture the ”struc- ture” of the projective space although it does point out its automorphisms: they are projectivizations of semilinear automorphisms of E, also known as Staudt projectivi- ties. A different approach (v. gr. [1]), defines the projective space as a lattice (the lattice of all linear subvarieties) satisfying certain axioms. Then, the so named Fun- damental Theorem of Projective Geometry ([1], Thm 2.26) states that, when n> 1, collineations of Pn (i.e., bijections preserving alignment, which are the automorphisms of this lattice structure) are precisely Staudt projectivities. In this note we are concerned with geometries in the sense of Klein: a geometry is a pair (X, G) where X is a set and G is a subgroup of the group Biy(X) of all bijections of X. In Klein’s view, Projective Geometry is the pair (Pn, PGln), where PGln is the group of projectivizations of k-linear automorphisms of the vector space E (see Example 2.2). The main result of this note is a computation, analogous to the arXiv:0901.3928v2 [math.GR] 4 Jul 2013 aforementioned theorem for collineations, but in the realm of Klein geometries: Theorem 3.4 The group of automorphisms of the Projective Geometry (Pn, PGln) is the group of Staudt projectivities, for any n ≥ 1. -
Arxiv:1305.5974V1 [Math-Ph]
INTRODUCTION TO SPORADIC GROUPS for physicists Luis J. Boya∗ Departamento de F´ısica Te´orica Universidad de Zaragoza E-50009 Zaragoza, SPAIN MSC: 20D08, 20D05, 11F22 PACS numbers: 02.20.a, 02.20.Bb, 11.24.Yb Key words: Finite simple groups, sporadic groups, the Monster group. Juan SANCHO GUIMERA´ In Memoriam Abstract We describe the collection of finite simple groups, with a view on physical applications. We recall first the prime cyclic groups Zp, and the alternating groups Altn>4. After a quick revision of finite fields Fq, q = pf , with p prime, we consider the 16 families of finite simple groups of Lie type. There are also 26 extra “sporadic” groups, which gather in three interconnected “generations” (with 5+7+8 groups) plus the Pariah groups (6). We point out a couple of physical applications, in- cluding constructing the biggest sporadic group, the “Monster” group, with close to 1054 elements from arguments of physics, and also the relation of some Mathieu groups with compactification in string and M-theory. ∗[email protected] arXiv:1305.5974v1 [math-ph] 25 May 2013 1 Contents 1 Introduction 3 1.1 Generaldescriptionofthework . 3 1.2 Initialmathematics............................ 7 2 Generalities about groups 14 2.1 Elementarynotions............................ 14 2.2 Theframeworkorbox .......................... 16 2.3 Subgroups................................. 18 2.4 Morphisms ................................ 22 2.5 Extensions................................. 23 2.6 Familiesoffinitegroups ......................... 24 2.7 Abeliangroups .............................. 27 2.8 Symmetricgroup ............................. 28 3 More advanced group theory 30 3.1 Groupsoperationginspaces. 30 3.2 Representations.............................. 32 3.3 Characters.Fourierseries . 35 3.4 Homological algebra and extension theory . 37 3.5 Groupsuptoorder16.......................... -
NZMRI Summer School the Classical Groups
NZMRI Summer School The classical groups Colva Roney-Dougal [email protected] School of Mathematics and Statistics, University of St Andrews Nelson, 10 January 2018 Automorphisms of quasisimple groups G is quasisimple if G = [G; G] and G=Z(G) is nonabelian simple. Lemma 43 Let G = Z :S be quasisimple, with Z central and S nonabelian simple. Then Aut(G) embeds in Aut(S). Proof. Let α 2 Aut(G). α If α induces 1 on S = G=Z, then 8g 2 G, 9 zg 2 Z s.t. g = gzg . Hence 8g; h 2 G,[g; h]α = [g; h], and α is trivial on G = [G; G]. Thus Aut(G) acts faithfully on G=Z = S. SLd (q) is (generally) quasisimple: Aut(SLd (q)) embeds in Aut(PSLd (q)). A crash course on finite fields Theorem 44 (Fundamental Theorem of Finite Fields) For each prime p and each e ≥ 1 there is exactly one field of order q = pe , up to isomorphism, and these are the only finite fields. ∗ The multiplicative gp Fq of Fq is cyclic of order q − 1. A generator ∗ λ of Fq is a primitive element of Fq. ∼ Aut(Fq) = Ce , with generator the Frobenius automorphism φ : x 7! xp. Diagonal automorphisms of PSLd (q) Defn: g 2 GLd (q). cg induces a diagonal outer aut of (P)SLd (q). ∗ PGLd (q) = GLd (q)=(FqId ). Lemma 45 Let δ = diag(λ, 1;:::; 1) 2 GLd (q). Then hSLd (q); δi = GLd (q) and jcδj = (q − 1; d) = jPGLd (q): PSLd (q)j. -
Semilinear Transformations Over Finite Fields Are Frobenius Maps U
Glasgow Math. J. 42 (2000) 289±295. # Glasgow Mathematical Journal Trust 2000. Printed in the United Kingdom SEMILINEAR TRANSFORMATIONS OVER FINITE FIELDS ARE FROBENIUS MAPS U. DEMPWOLFF FB Mathematik, UniversitaÈt Kaiserslautern, Postfach 3049, D-67653 Kaiserslautern, Germany e-mail:[email protected] J. CHRIS FISHER and ALLEN HERMAN Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada, S4S 0A2 e-mail:®[email protected], [email protected] (Received 27 October, 1998) Abstract. In its original formulation Lang's theorem referred to a semilinear map on an n-dimensional vector space over the algebraic closure of GF p: it ®xes the vectors of a copy of V n; ph. In other words, every semilinear map de®ned over a ®nite ®eld is equivalent by change of coordinates to a map induced by a ®eld automorphism. We provide an elementary proof of the theorem independent of the theory of algebraic groups and, as a by-product of our investigation, obtain a con- venient normal form for semilinear maps. We apply our theorem to classical groups and to projective geometry. In the latter application we uncover three simple yet surprising results. 1991 Mathematics Subject Classi®cation. Primary 20G40; secondary 15A21, 51A10. x1. Introduction. Let V V n; F be an n-dimensional vector space over a ®eld F, and let B fv1; ...; vng be a basis of V. Any automorphism of F induces a bijectionP fromP V to V by acting on the coordinates with respect to this basis; i.e. xivi xi vi. This bijection will also be referred to as .A-semilinear map T is the composition of this with an F-linear transformation M; i.e. -
Desarguesian Spreads and Field Reduction for Elements of The
Desarguesian spreads and field reduction for elements of the semilinear group Geertrui Van de Voorde ∗ Abstract The goal of this note is to create a sound framework for the interplay between field reduction for finite projective spaces, the general semilinear groups acting on the defining vector spaces and the projective semilinear groups. This approach makes it possible to reprove a result of Dye on the stabiliser in PGL of a Desar- guesian spread in a more elementary way, and extend it to PΓL(n,q). Moreover a result of Drudge [5] relating Singer cycles with Desarguesian spreads, as well as a result on subspreads (by Sheekey, Rottey and Van de Voorde [19]) are reproven in a similar elementary way. Finally, we try to use this approach to shed a light on Condition (A) of Csajb´ok and Zanella, introduced in the study of linear sets [4]. Keywords: Field reduction, Desarguesian spread Mathematics Subject Classfication: 51E20, 51E23, 05B25,05E20 1 Introduction In the last two decades a technique, commonly referred to as ‘field reduction’, has been used in many constructions and characterisations in finite geometry, e.g. the construction of m-systems and semi-partial geometries (SPG-reguli) [16, 21], eggs and translation generalised quadrangles [18], spreads and linear blocking sets [14], .... A survey paper of Polverino [17] collects these and other applications of linear sets in finite geometry. For projective and polar spaces, the technique itself was more elaborately discussed in [13]. The main goal of this paper is to contribute to this study by also considering the interplay between finite projective spaces and the projective (semi-)linear groups acting arXiv:1607.01549v1 [math.CO] 6 Jul 2016 on the definining spaces. -
ISOMORPHISM THEORY of CONGRUENCE GROUPS This
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 77, Number 1, January 1971 ISOMORPHISM THEORY OF CONGRUENCE GROUPS BY ROBERT E. SOLAZZI Communicated by I. N, Herstein, May 18, 1970 This note is an announcement of four theorems I proved in [5]- [9] on the isomorphisms of the linear, symplectic, and unitary con gruence groups. A sketch of the proofs is given. NOTATION. Let V be an w-dimensional vector space over the field F, n*z2. For crGGLw(F), let cr denote the contragredient of cr (inverse of the transpose). Let ~ denote the natural map of GLn( V) onto PGLn( V) ; for any subset 5 of GLn(V), S is the image of S in PGLn(V) under the "" map. A transvection r is a linear transformation of determinant one which fixes all vectors of some hyperplane, called the proper hyper- plane of r. If r 5^ 1 then (r — 1) V is a line called the proper line of r. An element f of PGLn(V) is called a (projective) transvection if r is a transvection. The proper line and proper hyperplane of f are defined as the proper line and hyperplane of r. Congruence groups. Again let V be an w-dimensional vector space over F, n è 2 ; let o be any integral domain with quotient field F. An o-module MC. V is bounded if there is an o-linear isomorphism of M into some free o-module of finite dimension. Assume If is a bounded o-module with FM = V. We define the integral linear group GLn(M) as {<rEGLn(V)\<rM=M}.