On the Action of the Sporadic Simple Baby Monster Group on Its Conjugacy Class 2B
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NOTES on FINITE GROUP REPRESENTATIONS in Fall 2020, I
NOTES ON FINITE GROUP REPRESENTATIONS CHARLES REZK In Fall 2020, I taught an undergraduate course on abstract algebra. I chose to spend two weeks on the theory of complex representations of finite groups. I covered the basic concepts, leading to the classification of representations by characters. I also briefly addressed a few more advanced topics, notably induced representations and Frobenius divisibility. I'm making the lectures and these associated notes for this material publicly available. The material here is standard, and is mainly based on Steinberg, Representation theory of finite groups, Ch 2-4, whose notation I will mostly follow. I also used Serre, Linear representations of finite groups, Ch 1-3.1 1. Group representations Given a vector space V over a field F , we write GL(V ) for the group of bijective linear maps T : V ! V . n n When V = F we can write GLn(F ) = GL(F ), and identify the group with the group of invertible n × n matrices. A representation of a group G is a homomorphism of groups φ: G ! GL(V ) for some representation choice of vector space V . I'll usually write φg 2 GL(V ) for the value of φ on g 2 G. n When V = F , so we have a homomorphism φ: G ! GLn(F ), we call it a matrix representation. matrix representation The choice of field F matters. For now, we will look exclusively at the case of F = C, i.e., representations in complex vector spaces. Remark. Since R ⊆ C is a subfield, GLn(R) is a subgroup of GLn(C). -
Molecular Symmetry
Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry . -
GROUP REPRESENTATIONS and CHARACTER THEORY Contents 1
GROUP REPRESENTATIONS AND CHARACTER THEORY DAVID KANG Abstract. In this paper, we provide an introduction to the representation theory of finite groups. We begin by defining representations, G-linear maps, and other essential concepts before moving quickly towards initial results on irreducibility and Schur's Lemma. We then consider characters, class func- tions, and show that the character of a representation uniquely determines it up to isomorphism. Orthogonality relations are introduced shortly afterwards. Finally, we construct the character tables for a few familiar groups. Contents 1. Introduction 1 2. Preliminaries 1 3. Group Representations 2 4. Maschke's Theorem and Complete Reducibility 4 5. Schur's Lemma and Decomposition 5 6. Character Theory 7 7. Character Tables for S4 and Z3 12 Acknowledgments 13 References 14 1. Introduction The primary motivation for the study of group representations is to simplify the study of groups. Representation theory offers a powerful approach to the study of groups because it reduces many group theoretic problems to basic linear algebra calculations. To this end, we assume that the reader is already quite familiar with linear algebra and has had some exposure to group theory. With this said, we begin with a preliminary section on group theory. 2. Preliminaries Definition 2.1. A group is a set G with a binary operation satisfying (1) 8 g; h; i 2 G; (gh)i = g(hi)(associativity) (2) 9 1 2 G such that 1g = g1 = g; 8g 2 G (identity) (3) 8 g 2 G; 9 g−1 such that gg−1 = g−1g = 1 (inverses) Definition 2.2. -
Recent Progress in the Symmetric Generation of Groups, to Appear in the Proceedings of the Conference Groups St Andrews 2009 [28] B
RECENT PROGRESS IN THE SYMMETRIC GENERATION OF GROUPS BEN FAIRBAIRN Abstract. Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial struc- ture. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for groups and prac- tically by providing succinct means of representing group elements. We give a survey of results obtained in the study of these symmet- ric generating sets. In keeping with earlier surveys on this matter, we emphasize the sporadic simple groups. ADDENDUM: This is an updated version of a survey article orig- inally accepted for inclusion in the proceedings of the 2009 ‘Groups St Andrews’ conference [27]. Since [27] was accepted the author has become aware of other recent work in the subject that we incorporate to provide an updated version here (the most notable addition being the contents of Section 3.4). 1. Introduction This article is concerned with groups that are generated by highly symmetric subsets of their elements: that is to say by subsets of el- ements whose set normalizer within the group they generate acts on them by conjugation in a highly symmetric manner. Rather than inves- tigate the behaviour of known groups we turn this procedure around and ask what groups can be generated by a set of elements that pos- sesses a certain assigned set of symmetries. This enables constructions arXiv:1004.2272v2 [math.GR] 21 Apr 2010 by hand of a number of interesting groups, including many of the spo- radic simple groups. Much of the emphasis of the research project to date has been concerned with using these techniques to construct spo- radic simple groups, and this article will emphasize this important spe- cial case. -
Representations, Character Tables, and One Application of Symmetry Chapter 4
Representations, Character Tables, and One Application of Symmetry Chapter 4 Friday, October 2, 2015 Matrices and Matrix Multiplication A matrix is an array of numbers, Aij column columns matrix row matrix -1 4 3 1 1234 rows -8 -1 7 2 2141 3 To multiply two matrices, add the products, element by element, of each row of the first matrix with each column in the second matrix: 12 12 (1×1)+(2×3) (1×2)+(2×4) 710 × 34 34 = (3×1)+(4×3) (3×2)+(4×4) = 15 22 100 1 1 0-10× 2 = -2 002 3 6 Transformation Matrices Each symmetry operation can be represented by a 3×3 matrix that shows how the operation transforms a set of x, y, and z coordinates y Let’s consider C2h {E, C2, i, σh}: x transformation matrix C2 x’ = -x -1 0 0 x’ -1 0 0 x -x y’ = -y 0-10 y’ ==0-10 y -y z’ = z 001 z’ 001 z z new transformation old new in terms coordinates ==matrix coordinates of old transformation i matrix x’ = -x -1 0 0 x’ -1 0 0 x -x y’ = -y 0-10 y’ ==0-10 y -y z’ = -z 00-1 z’ 00-1z -z Representations of Groups The set of four transformation matrices forms a matrix representation of the C2h point group. 100 -1 0 0 -1 0 0 100 E: 010 C2: 0-10 i: 0-10 σh: 010 001 001 00-1 00-1 These matrices combine in the same way as the operations, e.g., -1 0 0 -1 0 0 100 C2 × C2 = 0-10 0-10= 010= E 001001 001 The sum of the numbers along each matrix diagonal (the character) gives a shorthand version of the matrix representation, called Γ: C2h EC2 i σh Γ 3-1-31 Γ (gamma) is a reducible representation b/c it can be further simplified. -
Lecture Notes
i Applications of Group Theory to the Physics of Solids M. S. Dresselhaus 8.510J 6.734J SPRING 2002 ii Contents 1 Basic Mathematical Background 1 1.1 De¯nition of a Group . 1 1.2 Simple Example of a Group . 2 1.3 Basic De¯nitions . 4 1.4 Rearrangement Theorem . 6 1.5 Cosets . 6 1.6 Conjugation and Class . 8 1.6.1 Self-Conjugate Subgroups . 9 1.7 Factor Groups . 10 1.8 Selected Problems . 11 2 Representation Theory 15 2.1 Important De¯nitions . 15 2.2 Matrices . 17 2.3 Irreducible Representations . 18 2.4 The Unitarity of Representations . 20 2.5 Schur's Lemma (Part I) . 23 2.6 Schur's Lemma (Part 2) . 24 2.7 Wonderful Orthogonality Theorem . 27 2.8 Representations and Vector Spaces . 30 2.9 Suggested Problems . 31 3 Character of a Representation 33 3.1 De¯nition of Character . 33 3.2 Characters and Class . 34 3.3 Wonderful Orthogonality Theorem for Character . 36 3.4 Reducible Representations . 38 iii iv CONTENTS 3.5 The Number of Irreducible Representations . 40 3.6 Second Orthogonality Relation for Characters . 41 3.7 Regular Representation . 43 3.8 Setting up Character Tables . 47 3.9 Symmetry Notation . 51 3.10 Selected Problems . 73 4 Basis Functions 75 4.1 Symmetry Operations and Basis Functions . 75 4.2 Basis Functions for Irreducible Representations . 77 ^(¡n) 4.3 Projection Operators Pkl . 82 ^(¡n) 4.4 Derivation of Pk` . 83 4.5 Projection Operations on an Arbitrary Function . 84 4.6 Linear Combinations for 3 Equivalent Atoms . -
The Q-Conjugacy Character Table of Dihedral Groups
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS { N. 39{2018 (89{96) 89 THE Q-CONJUGACY CHARACTER TABLE OF DIHEDRAL GROUPS H. Shabani A. R. Ashrafi E. Haghi Department of Pure Mathematics Faculty of Mathematical Sciences University of Kashan P.O. Box 87317-53153 Kashan I.R. Iran M. Ghorbani∗ Department of Pure Mathematics Faculty of Sciences Shahid Rajaee Teacher Training University P.O. Box 16785-136 Tehran I.R. Iran [email protected] Abstract. In a seminal paper published in 1998, Shinsaku Fujita introduced the concept of Q-conjugacy character table of a finite group. He applied this notion to solve some problems in combinatorial chemistry. In this paper, the Q-conjugacy character table of dihedral groups is computed in general. As a consequence, Q- ∼ ∼ conjugacy character table of molecules with point group symmetries D = C = Dih , ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼3 3v∼ 6 D = D = C = Dih , D = C = Dih , D = D = D = C = Dih , 4 ∼ 2d ∼ 4v ∼ 8 5 ∼5v 10 ∼3d ∼3h 6 ∼6v 12 D2 = C2v = Z2 × Z2 = Dih2, D4d = Dih16, D5d = D5h = Dih20, D6d = Dih24 are computed, where Z2 denotes a cyclic group of order 2 and Dihn is the dihedral group of even order n. Keywords: Q-conjugacy character table, dihedral group, conjugacy class. 1. Introduction A representation of a group G is a homomorphism from G into the group of invertible operators of a vector space V . In this case, we can interpret each element of G as an invertible linear transformation V −! V . If n = dimV and fix a basis for V then we can construct an isomorphism between the set of all \invertible linear transformation V −! V " and the set of all \invertible n × n matrices". -
THE STRONG SYMMETRIC GENUS and GENERALIZED SYMMETRIC GROUPS of TYPE G(N, 3)
THE STRONG SYMMETRIC GENUS AND GENERALIZED SYMMETRIC GROUPS OF TYPE G(n, 3) MICHAEL A. JACKSON Abstract. The generalized symmetric groups are defined to be G(n, m) = Zm o Σn where n, m ∈ Z+. The strong symmetric genus of a finite group G is the smallest genus of a closed orientable topological surface on which G acts faithfully as a group of orientation preserving automorphisms. The present work extends work on the strong symmetric genus by Marston Conder, who studied the symmetric groups, which are G(n, 1), and by the author, who studied the hyperoctahedral groups, which are G(n, 2). We give results for the strong symmetric genus of the groups of type G(n, 3). Grove City College, 100 Campus Drive, Grove City, PA 16127 [email protected]. 1 1. Introduction The strong symmetric genus of the finite group G is the smallest genus of a compact orientable surface on which G acts as a group of orientation preserving symmetries and is denoted σ0(G). The strong symmetric genus has a long history beginning with Burnside [1]. For more general information see [10, Chapter 6]. The strong symmetric genus of many groups are known: the alternating and symmetric groups [2, 3, 4], the hyperoctahedral groups [12], the remaining finite Coxeter groups [11], the groups P SL2(q) [8, 9] and SL2(q) [15], as well as the sporadic finite simple groups [5, 16, 17, 18]. The generalized symmetric groups G(n, m) are defined for n > 1 and m ≥ 1 to be the wreath product Zm o Σn. -
Representation Theoretic Existence Proof for Fischer Group Fi 23
Cornell University Mathematics Department Senior Thesis REPRESENTATION THEORETIC EXISTENCE PROOF FOR FISCHER GROUP Fi23 HYUN KYU KIM May 2008 A thesis presented in partial fulfillment of criteria for Honors in Mathematics Bachelor of Arts, Cornell University Thesis Advisor(s) Gerhard O. Michler Department of Mathematics 1 2 HYUN KYU KIM ABSTRACT In the first section of this senior thesis the author provides some new efficient algorithms for calculating with finite permutation groups. They cannot be found in the computer algebra system Magma, but they can be implemented there. For any finite group G with a given set of generators, the algorithms calculate generators of a fixed subgroup of G as short words in terms of original generators. Another new algorithm provides such a short word for a given element of G. These algo- rithms are very useful for documentation and performing demanding experiments in computational group theory. In the later sections, the author gives a self-contained existence proof for Fis- 18 13 2 cher's sporadic simple group Fi23 of order 2 · 3 · 5 · 7 · 11 · 13 · 17 · 23 using G. Michler's Algorithm [11] constructing finite simple groups from irreducible sub- groups of GLn(2). This sporadic group was originally discovered by B. Fischer in [6] by investigating 3-transposition groups, see also [5]. This thesis gives a representa- tion theoretic and algorithmic existence proof for his group. The author constructs the three non-isomorphic extenstions Ei by the two 11-dimensional non-isomorphic simple modules of the Mathieu group M23 over F = GF(2). In two cases Michler's Algorithm fails. -
Irreducible Representations and Character Tables
MIT OpenCourseWare http://ocw.mit.edu 5.04 Principles of Inorganic Chemistry II �� Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.04, Principles of Inorganic Chemistry II Prof. Daniel G. Nocera Lecture 3: Irreducible Representations and Character Tables Similarity transformations yield irreducible representations, Γi, which lead to the useful tool in group theory – the character table. The general strategy for determining Γi is as follows: A, B and C are matrix representations of symmetry operations of an arbitrary basis set (i.e., elements on which symmetry operations are performed). There is some similarity transform operator v such that A’ = v –1 ⋅ A ⋅ v B’ = v –1 ⋅ B ⋅ v C’ = v –1 ⋅ C ⋅ v where v uniquely produces block-diagonalized matrices, which are matrices possessing square arrays along the diagonal and zeros outside the blocks ⎡ A1 ⎤ ⎡B1 ⎤ ⎡C1 ⎤ ′ = ⎢ ⎥ ′ = ⎢ ⎥ ′ = ⎢ ⎥ A ⎢ A 2 ⎥ B ⎢ B2 ⎥ C ⎢ C 2 ⎥ ⎢ A ⎥ ⎢ B ⎥ ⎢ C ⎥ ⎣ 3 ⎦ ⎣ 3 ⎦ ⎣ 3 ⎦ Matrices A, B, and C are reducible. Sub-matrices Ai, Bi and Ci obey the same multiplication properties as A, B and C. If application of the similarity transform does not further block-diagonalize A’, B’ and C’, then the blocks are irreducible representations. The character is the sum of the diagonal elements of Γi. 2 As an example, let’s continue with our exemplary group: E, C3, C3 , σv, σv’, σv” by defining an arbitrary basis … a triangle v A v'' v' C B The basis set is described by the triangles vertices, points A, B and C. -
Moonshines for L2(11) and M12
Moonshines for L2(11) and M12 Suresh Govindarajan Department of Physics Indian Institute of Technology Madras (Work done with my student Sutapa Samanta) Talk at the Workshop on Modular Forms and Black Holes @NISER, Bhubaneswar on January 13. 2017 Plan Introduction Some finite group theory Moonshine BKM Lie superalgebras Introduction Classification of Finite Simple Groups Every finite simple group is isomorphic to one of the following groups: (Source: Wikipedia) I A cyclic group with prime order; I An alternating group of degree at least 5; I A simple group of Lie type, including both I the classical Lie groups, namely the groups of projective special linear, unitary, symplectic, or orthogonal transformations over a finite field; I the exceptional and twisted groups of Lie type (including the Tits group which is not strictly a group of Lie type). I The 26 sporadic simple groups. The classification was completed in 2004 when Aschbacher and Smith filled the last gap (`the quasi-thin case') in the proof. Fun Reading: Symmetry and the Monster by Mark Ronan The sporadic simple groups I the Mathieu groups: M11, M12, M22, M23, M24; (found in 1861) I the Janko groups: J1, J2, J3, J4; (others 1965-1980) I the Conway groups; Co1, Co2, Co3; 0 I the Fischer groups; Fi22. Fi23, Fi24; I the Higman-Sims group; HS I the McLaughlin group: McL I the Held group: He; I the Rudvalis group Ru; I the Suzuki sporadic group: Suz; 0 I the O'Nan group: O N; I Harada-Norton group: HN; I the Lyons group: Ly; I the Thompson group: Th; I the baby Monster group: B and Sources: Wikipedia and Mark Ronan I the Fischer-Griess Monster group: M Monstrous Moonshine Conjectures I The j-function has the followed q-series: (q = exp(2πiτ)) j(τ)−744 = q−1+[196883+1] q+[21296876+196883+1] q2+··· I McKay observed that 196883 and 21296876 are the dimensions of the two smallest irreps of the Monster group. -
A 195,747,435 Vertex Graph Related to the Fischer Group Fi23, Part III
A 195,747,435 vertex graph related to the Fischer group Fi23, part III Rowley, Peter and Walker, Louise 2009 MIMS EPrint: 2009.18 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 A 195,747,435 VERTEX GRAPH RELATED TO THE FISCHER GROUP F i23 ,III Peter Rowley Louise Walker School of Mathematics University of Manchester Oxford Road Manchester M13 9JL UK 1 Introduction This paper, together with our earlier work in [3] and [4], ¯nishes our anatomical description of the 195,747,435 vertex graph G. This graph is the point-line collinearity graph of a certain geometry ¡ associated with the Fischer group F i23 and is described in Section 2 of [3]. We shall continue the section numbering of [3] and [4]. Our main results are stated in Section 1 { also in Section 1 and the introduction of [4] there is a discussion of various aspects of this investigation of G. We direct the reader to Section 2 for our notation as well as the all important line orbit data. Again, a denotes a ¯xed point of G. Our objective here is to complete the proofs of Theorems 9, 10, 11, 12, 13, 14 and 15. So, for the most part, we are interested in dissecting parts of ¢4(a), the fourth disc of a. In particular 3 5 6 we shall concentrate on the Ga-orbits ¢4(a), ¢4(a) and ¢4(a).