On Finite Groups Acting on Spheres and Finite Subgroups of Orthogonal
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Galois-Equivariant Mckay Bijections for Primes Dividing $ Q-1$
GALOIS-EQUIVARIANT MCKAY BIJECTIONS FOR PRIMES DIVIDING q 1 − A. A. SCHAEFFER FRY Abstract. We prove that for most groups of Lie type, the bijections used by Malle and Sp¨ath in the proof of Isaacs–Malle–Navarro’s inductive McKay conditions for the prime 2 and odd primes dividing q − 1 are also equivariant with respect to certain Galois automorphisms. In particular, this shows that these bijections are candidates for proving Navarro–Sp¨ath–Vallejo’s recently-posited inductive Galois–McKay conditions. Mathematics Classification Number: 20C15, 20C33 Keywords: local-global conjectures, characters, McKay conjecture, Galois-McKay, finite simple groups, Lie type 1. Introduction One of the main sets of questions of interest in the representation theory of finite groups falls under the umbrella of the so-called local-global conjectures, which relate the character theory of a finite group G to that of certain local subgroups. Another rich topic in the area is the problem of determining how certain other groups (for example, the group of automorphisms Aut(G) of G, or the Galois group Gal(Q/Q)) act on the set of irreducible characters of G and the related question of determining the fields of values of characters of G. This paper concerns the McKay–Navarro conjecture, which incorporates both of these main problems. For a finite group G, the field Q(e2πi/|G|), obtained by adjoining the G th roots of unity in some | | algebraic closure Q of Q, is a splitting field for G. The group G := Gal(Q(e2πi/|G|)/Q) acts naturally on the set of irreducible ordinary characters, Irr(G), of G via χσ(g) := χ(g)σ for χ Irr(G), g G, σ G . -
Expansion in Finite Simple Groups of Lie Type
Expansion in finite simple groups of Lie type Terence Tao Department of Mathematics, UCLA, Los Angeles, CA 90095 E-mail address: [email protected] In memory of Garth Gaudry, who set me on the road Contents Preface ix Notation x Acknowledgments xi Chapter 1. Expansion in Cayley graphs 1 x1.1. Expander graphs: basic theory 2 x1.2. Expansion in Cayley graphs, and Kazhdan's property (T) 20 x1.3. Quasirandom groups 54 x1.4. The Balog-Szemer´edi-Gowers lemma, and the Bourgain- Gamburd expansion machine 81 x1.5. Product theorems, pivot arguments, and the Larsen-Pink non-concentration inequality 94 x1.6. Non-concentration in subgroups 127 x1.7. Sieving and expanders 135 Chapter 2. Related articles 157 x2.1. Cayley graphs and the algebra of groups 158 x2.2. The Lang-Weil bound 177 x2.3. The spectral theorem and its converses for unbounded self-adjoint operators 191 x2.4. Notes on Lie algebras 214 x2.5. Notes on groups of Lie type 252 Bibliography 277 Index 285 vii Preface Expander graphs are a remarkable type of graph (or more precisely, a family of graphs) on finite sets of vertices that manage to simultaneously be both sparse (low-degree) and \highly connected" at the same time. They enjoy very strong mixing properties: if one starts at a fixed vertex of an (two-sided) expander graph and randomly traverses its edges, then the distribution of one's location will converge exponentially fast to the uniform distribution. For this and many other reasons, expander graphs are useful in a wide variety of areas of both pure and applied mathematics. -
CHAPTER 4 Representations of Finite Groups of Lie Type Let Fq Be a Finite
CHAPTER 4 Representations of finite groups of Lie type Let Fq be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the Fq-rational points of a connected reductive group G defined over Fq. For example, if n is a positive integer GLn(Fq) and SLn(Fq) are finite groups of Lie type. 0 In Let J = , where In is the n × n identity matrix. Let −In 0 t Sp2n(Fq) = { g ∈ GL2n(Fq) | gJg = J }. Then Sp2n(Fq) is a symplectic group of rank n and is a finite group of Lie type. For G = GLn(Fq) or SLn(Fq) (and some other examples), the standard Borel subgroup B of G is the subgroup of G consisting of the upper triangular elements in G.A standard parabolic subgroup of G is a subgroup of G which contains the standard Borel subgroup B. If P is a standard parabolic subgroup of GLn(Fq), then there exists a partition (n1, . , nr) of n (a set of positive integers nj such that n1 + ··· + nr = n) such that P = P(n1,...,nr ) = M n N, where M ' GLn1 (Fq) × · · · × GLnr (Fq) has the form A 0 ··· 0 1 0 A2 ··· 0 M = | A ∈ GL ( ), 1 ≤ j ≤ r . . .. .. j nj Fq . 0 ··· 0 Ar and In1 ∗ · · · ∗ 0 In2 · · · ∗ N = , . .. .. . 0 ··· 0 Inr where ∗ denotes arbitary entries in Fq. The subgroup M is called a (standard) Levi sub- group of P , and N is called the unipotent radical of P . Note that the partition (1, 1,..., 1) corresponds to B and (n) corresponds to G. -
The Finite Simple Groups Have Complemented Subgroup Lattices
Pacific Journal of Mathematics THE FINITE SIMPLE GROUPS HAVE COMPLEMENTED SUBGROUP LATTICES Mauro Costantini and Giovanni Zacher Volume 213 No. 2 February 2004 PACIFIC JOURNAL OF MATHEMATICS Vol. 213, No. 2, 2004 THE FINITE SIMPLE GROUPS HAVE COMPLEMENTED SUBGROUP LATTICES Mauro Costantini and Giovanni Zacher We prove that the lattice of subgroups of every finite simple group is a complemented lattice. 1. Introduction. A group G is called a K-group (a complemented group) if its subgroup lattice is a complemented lattice, i.e., for a given H ≤ G there exists a X ≤ G such that hH, Xi = G and H ∧X = 1. The main purpose of this Note is to answer a long-standing open question in finite group theory, by proving that: Every finite simple group is a K-group. In this context, it was known that the alternating groups, the projective special linear groups and the Suzuki groups are K-groups ([P]). Our proof relies on the FSGC-theorem and on structural properties of the maximal subgroups in finite simple groups. The rest of this paper is divided into four sections. In Section 2 we collect some criteria for a subgroup of a group G to have a complement and recall some useful known results. In Section 3 we deal with the classical groups, in 4 with the exceptional groups of Lie type and in Section 5 with the sporadic groups. With reference to notation and terminology, we shall follow closely those in use in [P] and [S]. All groups are meant to be finite. 2. Preliminaries. -
Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation -
Arxiv:1709.02742V3
PIN GROUPS IN GENERAL RELATIVITY BAS JANSSENS Abstract. There are eight possible Pin groups that can be used to describe the transformation behaviour of fermions under parity and time reversal. We show that only two of these are compatible with general relativity, in the sense that the configuration space of fermions coupled to gravity transforms appropriately under the space-time diffeomorphism group. 1. Introduction For bosons, the space-time transformation behaviour is governed by the Lorentz group O(3, 1), which comprises four connected components. Rotations and boosts are contained in the connected component of unity, the proper orthochronous Lorentz group SO↑(3, 1). Parity (P ) and time reversal (T ) are encoded in the other three connected components of the Lorentz group, the translates of SO↑(3, 1) by P , T and PT . For fermions, the space-time transformation behaviour is governed by a double cover of O(3, 1). Rotations and boosts are described by the unique simply connected double cover of SO↑(3, 1), the spin group Spin↑(3, 1). However, in order to account for parity and time reversal, one needs to extend this cover from SO↑(3, 1) to the full Lorentz group O(3, 1). This extension is by no means unique. There are no less than eight distinct double covers of O(3, 1) that agree with Spin↑(3, 1) over SO↑(3, 1). They are the Pin groups abc Pin , characterised by the property that the elements ΛP and ΛT covering P and 2 2 2 T satisfy ΛP = −a, ΛT = b and (ΛP ΛT ) = −c, where a, b and c are either 1 or −1 (cf. -
Algebraic Topology
Algebraic Topology Len Evens Rob Thompson Northwestern University City University of New York Contents Chapter 1. Introduction 5 1. Introduction 5 2. Point Set Topology, Brief Review 7 Chapter 2. Homotopy and the Fundamental Group 11 1. Homotopy 11 2. The Fundamental Group 12 3. Homotopy Equivalence 18 4. Categories and Functors 20 5. The fundamental group of S1 22 6. Some Applications 25 Chapter 3. Quotient Spaces and Covering Spaces 33 1. The Quotient Topology 33 2. Covering Spaces 40 3. Action of the Fundamental Group on Covering Spaces 44 4. Existence of Coverings and the Covering Group 48 5. Covering Groups 56 Chapter 4. Group Theory and the Seifert{Van Kampen Theorem 59 1. Some Group Theory 59 2. The Seifert{Van Kampen Theorem 66 Chapter 5. Manifolds and Surfaces 73 1. Manifolds and Surfaces 73 2. Outline of the Proof of the Classification Theorem 80 3. Some Remarks about Higher Dimensional Manifolds 83 4. An Introduction to Knot Theory 84 Chapter 6. Singular Homology 91 1. Homology, Introduction 91 2. Singular Homology 94 3. Properties of Singular Homology 100 4. The Exact Homology Sequence{ the Jill Clayburgh Lemma 109 5. Excision and Applications 116 6. Proof of the Excision Axiom 120 3 4 CONTENTS 7. Relation between π1 and H1 126 8. The Mayer-Vietoris Sequence 128 9. Some Important Applications 131 Chapter 7. Simplicial Complexes 137 1. Simplicial Complexes 137 2. Abstract Simplicial Complexes 141 3. Homology of Simplicial Complexes 143 4. The Relation of Simplicial to Singular Homology 147 5. Some Algebra. The Tensor Product 152 6. -
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). -
Group Theory
Group theory March 7, 2016 Nearly all of the central symmetries of modern physics are group symmetries, for simple a reason. If we imagine a transformation of our fields or coordinates, we can look at linear versions of those transformations. Such linear transformations may be represented by matrices, and therefore (as we shall see) even finite transformations may be given a matrix representation. But matrix multiplication has an important property: associativity. We get a group if we couple this property with three further simple observations: (1) we expect two transformations to combine in such a way as to give another allowed transformation, (2) the identity may always be regarded as a null transformation, and (3) any transformation that we can do we can also undo. These four properties (associativity, closure, identity, and inverses) are the defining properties of a group. 1 Finite groups Define: A group is a pair G = fS; ◦} where S is a set and ◦ is an operation mapping pairs of elements in S to elements in S (i.e., ◦ : S × S ! S: This implies closure) and satisfying the following conditions: 1. Existence of an identity: 9 e 2 S such that e ◦ a = a ◦ e = a; 8a 2 S: 2. Existence of inverses: 8 a 2 S; 9 a−1 2 S such that a ◦ a−1 = a−1 ◦ a = e: 3. Associativity: 8 a; b; c 2 S; a ◦ (b ◦ c) = (a ◦ b) ◦ c = a ◦ b ◦ c We consider several examples of groups. 1. The simplest group is the familiar boolean one with two elements S = f0; 1g where the operation ◦ is addition modulo two. -
UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title Shapes of Finite Groups through Covering Properties and Cayley Graphs Permalink https://escholarship.org/uc/item/09b4347b Author Yang, Yilong Publication Date 2017 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles Shapes of Finite Groups through Covering Properties and Cayley Graphs A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Yilong Yang 2017 c Copyright by Yilong Yang 2017 ABSTRACT OF THE DISSERTATION Shapes of Finite Groups through Covering Properties and Cayley Graphs by Yilong Yang Doctor of Philosophy in Mathematics University of California, Los Angeles, 2017 Professor Terence Chi-Shen Tao, Chair This thesis is concerned with some asymptotic and geometric properties of finite groups. We shall present two major works with some applications. We present the first major work in Chapter 3 and its application in Chapter 4. We shall explore the how the expansions of many conjugacy classes is related to the representations of a group, and then focus on using this to characterize quasirandom groups. Then in Chapter 4 we shall apply these results in ultraproducts of certain quasirandom groups and in the Bohr compactification of topological groups. This work is published in the Journal of Group Theory [Yan16]. We present the second major work in Chapter 5 and 6. We shall use tools from number theory, combinatorics and geometry over finite fields to obtain an improved diameter bounds of finite simple groups. We also record the implications on spectral gap and mixing time on the Cayley graphs of these groups. -
Lecture Notes in Physics
Lecture Notes in Physics Volume 823 Founding Editors W. Beiglböck J. Ehlers K. Hepp H. Weidenmüller Editorial Board B.-G. Englert, Singapore U. Frisch, Nice, France F. Guinea, Madrid, Spain P. Hänggi, Augsburg, Germany W. Hillebrandt, Garching, Germany M. Hjorth-Jensen, Oslo, Norway R. A. L. Jones, Sheffield, UK H. v. Löhneysen, Karlsruhe, Germany M. S. Longair, Cambridge, UK M. L. Mangano, Geneva, Switzerland J.-F. Pinton, Lyon, France J.-M. Raimond, Paris, France A. Rubio, Donostia, San Sebastian, Spain M. Salmhofer, Heidelberg, Germany D. Sornette, Zurich, Switzerland S. Theisen, Potsdam, Germany D. Vollhardt, Augsburg, Germany W. Weise, Garching, Germany For further volumes: http://www.springer.com/series/5304 The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new devel- opments in physics research and teaching—quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic • to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas • to be a source of advanced teaching material for specialized seminars, courses and schools Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com.