DOCSLIB.ORG

A Brief Introduction to Characters and Representation Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Brief Introduction to Characters and Representation Theory Linear Representations of Finite Groups William Hargis A Brief Introduction to Characters and Representations Theory Structures Studied Representation Theory Linear Representations Character Theory Characters Orthogonality of William Hargis Characters Character Properties Examples of Mathematics DRP Fall 2016 Characters Cyclic Groups Mentor: Xuqiang (Russell) Qin Dec. 8, 2016 Overview Linear Representations of Finite Groups William Hargis 1 Representations Theory Structures Studied Representations Theory Linear Representations Structures Studied Linear Representations Character Theory 2 Character Theory Characters Characters Orthogonality of Characters Character Properties Orthogonality of Characters Examples of Character Properties Characters Cyclic Groups 3 Examples of Characters Cyclic Groups Linear Representations of Material studied: Linear Representations of Finite Groups Finite Groups by Jean-Pierre Serre William Hargis Representations Theory Structures Studied Linear Representations Character Theory Characters Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups What is Representation Theory? Linear Representations of Finite Groups William Hargis Representations Representation theory is the study of algebraic structures Theory Structures Studied by representing the structure's elements as linear Linear Representations transformations of vector spaces. Character Theory Characters Orthogonality of Characters Character Properties This makes abstract structures more concrete by Examples of describing the structure in terms of matrices and their Characters Cyclic Groups algebraic operations as matrix operations. Structures Studied Linear Representations of Finite Groups Structures studied this way include: William Hargis Groups Representations Associative Algebras Theory Structures Studied Linear Lie Algebras Representations Character Theory Characters Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups Finite Groups Linear Representations of Finite Groups William Hargis Representations Finite Groups Theory Structures Studied Study group actions on structures. Linear Representations especially operations of groups on vector spaces; Character Theory other actions are group action on other groups or Characters Orthogonality of Characters sets. Character Properties Examples of Characters Group elements are represented by invertible matrices Cyclic Groups such that the group operation is matrix multiplication. Linear Representations Linear Representations of Finite Groups William Hargis Representations Theory Let V be a K-vector space and G a finite group. Structures Studied Linear Representations The linear representation of G is a group homomorphism Character Theory ρ : G ! GL(V ). Characters Orthogonality of Characters So, a linear representation is a map Character Properties Examples of ρ : G ! GL(V ) s:t: ρ(st) = ρ(s)ρ(t) 8s; t 2 G. Characters Cyclic Groups Importance of Linear Representations Linear Representations of Finite Groups William Hargis Representations Theory Structures Studied Linear representations allow us to state group theoretic Linear Representations problems in terms of linear algebra. Character Theory Characters Orthogonality of Linear algebra is well understood; reduces complexity of Characters Character Properties problems. Examples of Characters Cyclic Groups Applications of Linear Representations Linear Representations of Finite Groups Applications in the study of geometric structures and in William Hargis the physical sciences. Representations Space Groups Theory Structures Studied Symmetry groups of the configuration of space. Linear Representations Character Theory Characters Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups Lattice Point Groups Lattice groups define the geometric configuration of crystal structures in materials science and crystallography. Characters Linear Representations of Finite Groups William Hargis Representations Theory The character of a group representation is a function on Structures Studied Linear Representations the group that associates the trace of each group Character Theory element's matrix to the corresponding group element. Characters Orthogonality of Characters Characters contain all of the essential information of the Character Properties Examples of representation in a more condensed form. Characters Cyclic Groups Trace Review Linear Representations of Finite Groups William Hargis Note: Trace is the sum of the diagonal entries of the Representations matrix. Theory Structures Studied Linear 2 3 Representations x11 x12 ::: x1n Character Theory 6x21 x22 ::: x2n 7 Characters Tr(X ) = Tr( ) Orthogonality of 6 7 Characters 4::::::::::::::::::5 Character Properties xm1 xm2 ::: xmn Examples of Characters Cyclic Groups Pn Tr(X ) = x11 + x22 + ::: + xmn = i=1 xii Characters Linear Representations of Finite Groups William Hargis Representations Characters: Theory Structures Studied Linear For a representation ρ : G ! GL(V ) of a group G on V . Representations Character Theory the character of ρ is the function χρ : G ! F given by Characters Orthogonality of χρ(g) = Tr(ρ(g)). Characters Character Properties Examples of Where F is a field that the finite-dimensional vector Characters space V is over. Cyclic Groups Orthogonality of Characters Linear Representations of Finite Groups William Hargis The space of complex-valued class functions of a finite Representations group G has an inner product given by: Theory Structures Studied 1 Linear X ¯ Representations fα; βg := α(g)β(g) (1) G Character Theory g2G Characters Orthogonality of Characters From this inner product, the irreducible characters form Character Properties Examples of an orthonormal basis for the space of class functions and Characters Cyclic Groups an orthogonality relation for the rows of the character table. Similarly an orthogonality relation is established for the columns of the character table. Consequences of Orthogonality: Linear Representations of Finite Groups William Hargis Representations Consequences of Orthogonality: Theory Structures Studied Linear An unknown character can be decomposed as a Representations Character Theory linear combination of irreducible characters. Characters Orthogonality of The complete character table can be constructed Characters Character Properties when only a few irreducible characters are known. Examples of Characters The order of the group can be found. Cyclic Groups Character Properties Linear Representations of Finite Groups William Hargis A character χρ is irreducible if ρ is irreducible. Representations One can read the dimension of the vector space Theory Structures Studied directly from the character. Linear Representations Characters are class functions; take a constant value Character Theory Characters on a given conjugacy. Orthogonality of Characters Character Properties The number of irreducible characters of G is equal Examples of to the number of conjugacy classes of G. Characters Cyclic Groups The set of irreducible characters of a given group G into a field K form a basis of the K-vector space of all class functions G ! K. Examples of Characters Linear Representations of Finite Groups William Hargis Representations Note: the elements of any group can be partitioned in to Theory Structures Studied conjugacy classes; classes corresponding to the same Linear Representations conjugate element. Character Theory Characters Orthogonality of −1 Characters Cl(a) = b 2 Gj9g 2 G with b = gag (2) Character Properties Examples of From the definition it follows that Abelian groups have a Characters Cyclic Groups conjugacy class corresponding to each element. with conjugacy classes: f0¯g,f1¯g,f2¯g,...,fn −¯ 1g. 2πi Let !n=e n be a primitive n root of unity. (any complex number that gives 1 when raised to a positive integer power) Examples: Generalized Cyclic Group Zn Linear Representations of Note: All of Zn's irreducible characters are linear. Finite Groups ¯ ¯ ¯ ¯ William Hargis Zn is an additive group where Zn = 0; 1; 2; :::; n − 1 Representations Theory Structures Studied Linear Representations Character Theory Characters Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups 2πi Let !n=e n be a primitive n root of unity. (any complex number that gives 1 when raised to a positive integer power) Examples: Generalized Cyclic Group Zn Linear Representations of Note: All of Zn's irreducible characters are linear. Finite Groups ¯ ¯ ¯ ¯ William Hargis Zn is an additive group where Zn = 0; 1; 2; :::; n − 1 with conjugacy classes: f0¯g,f1¯g,f2¯g,...,fn −¯ 1g. Representations Theory Structures Studied Linear Representations Character Theory Characters Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups Examples: Generalized Cyclic Group Zn Linear Representations of Note: All of Zn's irreducible characters are linear. Finite Groups ¯ ¯ ¯ ¯ William Hargis Zn is an additive group where Zn = 0; 1; 2; :::; n − 1 with conjugacy classes: f0¯g,f1¯g,f2¯g,...,fn −¯ 1g. Representations Theory 2πi n Structures Studied Let !n=e be a primitive n root of unity. Linear Representations (any complex number that gives 1 when raised to a Character Theory Characters positive integer power) Orthogonality of Characters Character Properties Examples of Characters Cyclic Groups Let χ0, χ1, χ2, ..., χn−1 be the n irreducible characters ¯ jm of Zn then χm(j) = !n where 0 ≤ j ≤ n − 1 and 0 ≤ m ≤ n − 1. Examples: Generalized Cyclic Group
Load more

Recommended publications
  • An Introduction to the Trace Formula
    Clay Mathematics Proceedings Volume 4, 2005 An Introduction to the Trace Formula James Arthur Contents Foreword 3 Part I. The Unrefined Trace Formula 7 1. The Selberg trace formula for compact quotient 7 2. Algebraic groups and adeles 11 3. Simple examples 15 4. Noncompact quotient and parabolic subgroups 20 5. Roots and weights 24 6. Statement and discussion of a theorem 29 7. Eisenstein series 31 8. On the proof of the theorem 37 9. Qualitative behaviour of J T (f) 46 10. The coarse geometric expansion 53 11. Weighted orbital integrals 56 12. Cuspidal automorphic data 64 13. A truncation operator 68 14. The coarse spectral expansion 74 15. Weighted characters 81 Part II. Refinements and Applications 89 16. The first problem of refinement 89 17. (G, M)-families 93 18. Localbehaviourofweightedorbitalintegrals 102 19. The fine geometric expansion 109 20. Application of a Paley-Wiener theorem 116 21. The fine spectral expansion 126 22. The problem of invariance 139 23. The invariant trace formula 145 24. AclosedformulaforthetracesofHeckeoperators 157 25. Inner forms of GL(n) 166 Supported in part by NSERC Discovery Grant A3483. c 2005 Clay Mathematics Institute 1 2 JAMES ARTHUR 26. Functoriality and base change for GL(n) 180 27. The problem of stability 192 28. Localspectraltransferandnormalization 204 29. The stable trace formula 216 30. Representationsofclassicalgroups 234 Afterword: beyond endoscopy 251 References 258 Foreword These notes are an attempt to provide an entry into a subject that has not been very accessible. The problems of exposition are twofold. It is important to present motivation and background for the kind of problems that the trace formula is designed to solve.
    [Show full text]
  • 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).
    [Show full text]
  • 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 .
    [Show full text]
  • Symmetries on the Lattice
    Symmetries On The Lattice K.Demmouche January 8, 2006 Contents Background, character theory of finite groups The cubic group on the lattice Oh Representation of Oh on Wilson loops Double group 2O and spinor Construction of operator on the lattice MOTIVATION Spectrum of non-Abelian lattice gauge theories ? Create gauge invariant spin j states on the lattice Irreducible operators Monte Carlo calculations Extract masses from time slice correlations Character theory of point groups Groups, Axioms A set G = {a, b, c, . } A1 : Multiplication ◦ : G × G → G. A2 : Associativity a, b, c ∈ G ,(a ◦ b) ◦ c = a ◦ (b ◦ c). A3 : Identity e ∈ G , a ◦ e = e ◦ a = a for all a ∈ G. −1 −1 −1 A4 : Inverse, a ∈ G there exists a ∈ G , a ◦ a = a ◦ a = e. Groups with finite number of elements → the order of the group G : nG. The point group C3v The point group C3v (Symmetry group of molecule NH3) c ¡¡AA ¡ A ¡ A Z ¡ A Z¡ A Z O ¡ Z A ¡ Z A ¡ Z A Z ¡ Z A ¡ ZA a¡ ZA b G = {Ra(π),Rb(π),Rc(π),E(2π),R~n(2π/3),R~n(−2π/3)} noted G = {A, B, C, E, D, F } respectively. Structure of Groups Subgroups: Definition A subset H of a group G that is itself a group with the same multiplication operation as G is called a subgroup of G. Example: a subgroup of C3v is the subset E, D, F Classes: Definition An element g0 of a group G is said to be ”conjugate” to another element g of G if there exists an element h of G such that g0 = hgh−1 Example: on can check that B = DCD−1 Conjugacy Class Definition A class of a group G is a set of mutually conjugate elements of G.
    [Show full text]
  • Lie Algebras of Generalized Quaternion Groups 1 Introduction 2
    Lie Algebras of Generalized Quaternion Groups Samantha Clapp Advisor: Dr. Brandon Samples Abstract Every finite group has an associated Lie algebra. Its Lie algebra can be viewed as a subspace of the group algebra with certain bracket conditions imposed on the elements. If one calculates the character table for a finite group, the structure of its associated Lie algebra can be described. In this work, we consider the family of generalized quaternion groups and describe its associated Lie algebra structure completely. 1 Introduction The Lie algebra of a group is a useful tool because it is a vector space where linear algebra is available. It is interesting to consider the Lie algebra structure associated to a specific group or family of groups. A Lie algebra is simple if its dimension is at least two and it only has f0g and itself as ideals. Some examples of simple algebras are the classical Lie algebras: sl(n), sp(n) and o(n) as well as the five exceptional finite dimensional simple Lie algebras. A direct sum of simple lie algebras is called a semi-simple Lie algebra. Therefore, it is also interesting to consider if the Lie algebra structure associated with a particular group is simple or semi-simple. In fact, the Lie algebra structure of a finite group is well known and given by a theorem of Cohen and Taylor [1]. In this theorem, they specifically describe the Lie algebra structure using character theory. That is, the associated Lie algebra structure of a finite group can be described if one calculates the character table for the finite group.
    [Show full text]
  • Chapter 2 C -Algebras
    Chapter 2 C∗-algebras This chapter is mainly based on the first chapters of the book [Mur90]. Material bor- rowed from other references will be specified. 2.1 Banach algebras Definition 2.1.1. A Banach algebra C is a complex vector space endowed with an associative multiplication and with a norm k · k which satisfy for any A; B; C 2 C and α 2 C (i) (αA)B = α(AB) = A(αB), (ii) A(B + C) = AB + AC and (A + B)C = AC + BC, (iii) kABk ≤ kAkkBk (submultiplicativity) (iv) C is complete with the norm k · k. One says that C is abelian or commutative if AB = BA for all A; B 2 C . One also says that C is unital if 1 2 C , i.e. if there exists an element 1 2 C with k1k = 1 such that 1B = B = B1 for all B 2 C . A subalgebra J of C is a vector subspace which is stable for the multiplication. If J is norm closed, it is a Banach algebra in itself. Examples 2.1.2. (i) C, Mn(C), B(H), K (H) are Banach algebras, where Mn(C) denotes the set of n × n-matrices over C. All except K (H) are unital, and K (H) is unital if H is finite dimensional. (ii) If Ω is a locally compact topological space, C0(Ω) and Cb(Ω) are abelian Banach algebras, where Cb(Ω) denotes the set of all bounded and continuous complex func- tions from Ω to C, and C0(Ω) denotes the subset of Cb(Ω) of functions f which vanish at infinity, i.e.
    [Show full text]
  • Notes on the Chern-Character Maakestad H* Department of Mathematics, NTNU, Trondheim, Norway
    Theory an ie d L A p d p Maakestad, J Generalized Lie Theory Appl 2017, 11:1 e l z i i c l a a t Journal of Generalized Lie r DOI: 10.4172/1736-4337.1000253 i o e n n s e G ISSN: 1736-4337 Theory and Applications Research Article Open Access Notes on the Chern-Character Maakestad H* Department of Mathematics, NTNU, Trondheim, Norway Abstract Notes for some talks given at the seminar on characteristic classes at NTNU in autumn 2006. In the note a proof of the existence of a Chern-character from complex K-theory to any cohomology Lie theory with values in graded Q-algebras equipped with a theory of characteristic classes is given. It respects the Adams and Steenrod operations. Keywords: Chern-character; Chern-classes; Euler classes; Singular H* : Top→−Q algebras cohomology; De Rham-cohomology; Complex K-theory; Adams operations; Steenrod operations from the category of topological spaces to the category of graded commutative Q-algebras with respect to continuous maps of topological Introduction spaces. We say the theory satisfy the projective bundle property if the The aim of this note is to give an axiomatic and elementary following axioms are satisfied: For any rankn complex continuous treatment of Chern-characters of vectorbundles with values in a vectorbundle E over a compact space B There is an Euler class. class of cohomology-theories arising in topology and algebra. Given ∈ 2 uE H (P(E)) (1) a theory of Chern-classes for complex vectorbundles with values in → singular cohomology one gets in a natural way a Chern-character from Where π:P(E) B is the projective bundle associated to E.
    [Show full text]
  • 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.
    [Show full text]
  • Representation Theory
    M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23.
    [Show full text]
  • Lecture Notes: Basic Group and Representation Theory
    Lecture notes: Basic group and representation theory Thomas Willwacher February 27, 2014 2 Contents 1 Introduction 5 1.1 Definitions . .6 1.2 Actions and the orbit-stabilizer Theorem . .8 1.3 Generators and relations . .9 1.4 Representations . 10 1.5 Basic properties of representations, irreducibility and complete reducibility . 11 1.6 Schur’s Lemmata . 12 2 Finite groups and finite dimensional representations 15 2.1 Character theory . 15 2.2 Algebras . 17 2.3 Existence and classification of irreducible representations . 18 2.4 How to determine the character table – Burnside’s algorithm . 20 2.5 Real and complex representations . 21 2.6 Induction, restriction and characters . 23 2.7 Exercises . 25 3 Representation theory of the symmetric groups 27 3.1 Notations . 27 3.2 Conjugacy classes . 28 3.3 Irreducible representations . 29 3.4 The Frobenius character formula . 31 3.5 The hook lengths formula . 34 3.6 Induction and restriction . 34 3.7 Schur-Weyl duality . 35 4 Lie groups, Lie algebras and their representations 37 4.1 Overview . 37 4.2 General definitions and facts about Lie algebras . 39 4.3 The theorems of Lie and Engel . 40 4.4 The Killing form and Cartan’s criteria . 41 4.5 Classification of complex simple Lie algebras . 43 4.6 Classification of real simple Lie algebras . 44 4.7 Generalities on representations of Lie algebras . 44 4.8 Representation theory of sl(2; C) ................................ 45 4.9 General structure theory of semi-simple Lie algebras . 48 4.10 Representation theory of complex semi-simple Lie algebras .
    [Show full text]
  • Character Theory of Finite Groups NZ Mathematics Research Institute Summer Workshop Day 1: Essentials
    1/29 Character Theory of Finite Groups NZ Mathematics Research Institute Summer Workshop Day 1: Essentials Don Taylor The University of Sydney Nelson, 7–13 January 2018 2/29 Origins Suppose that G is a finite group and for every element g G we have 2 an indeterminate xg . What are the factors of the group determinant ¡ ¢ det x 1 ? gh¡ g,h This was a question posed by Dedekind. Frobenius discovered character theory (in 1896) when he set out to answer it. Characters of abelian groups had been used in number theory but Frobenius developed the theory for nonabelian groups. 3/29 Representations Matrices Characters ² ² A linear representation of a group G is a homomorphism ½ : G GL(V ), where GL(V ) is the group of all invertible linear ! transformations of the vector space V , which we assume to have finite dimension over the field C of complex numbers. The dimension n of V is called the degree of ½. If (ei )1 i n is a basis of V , there are functions ai j : G C such that · · ! X ½(x)e j ai j (x)ei . Æ i The matrices A(x) ¡a (x)¢ define a homomorphism A : G GL(n,C) Æ i j ! from G to the group of all invertible n n matrices over C. £ The character of ½ is the function G C that maps x G to Tr(½(x)), ! 2 the trace of ½(x), namely the sum of the diagonal elements of A(x). 4/29 Early applications Frobenius first defined characters as solutions to certain equations.
    [Show full text]
  • L-Functions and Non-Abelian Class Field Theory, from Artin to Langlands
    L-functions and non-abelian class field theory, from Artin to Langlands James W. Cogdell∗ Introduction Emil Artin spent the first 15 years of his career in Hamburg. Andr´eWeil charac- terized this period of Artin's career as a \love affair with the zeta function" [77]. Claude Chevalley, in his obituary of Artin [14], pointed out that Artin's use of zeta functions was to discover exact algebraic facts as opposed to estimates or approxi- mate evaluations. In particular, it seems clear to me that during this period Artin was quite interested in using the Artin L-functions as a tool for finding a non- abelian class field theory, expressed as the desire to extend results from relative abelian extensions to general extensions of number fields. Artin introduced his L-functions attached to characters of the Galois group in 1923 in hopes of developing a non-abelian class field theory. Instead, through them he was led to formulate and prove the Artin Reciprocity Law - the crowning achievement of abelian class field theory. But Artin never lost interest in pursuing a non-abelian class field theory. At the Princeton University Bicentennial Conference on the Problems of Mathematics held in 1946 \Artin stated that `My own belief is that we know it already, though no one will believe me { that whatever can be said about non-Abelian class field theory follows from what we know now, since it depends on the behavior of the broad field over the intermediate fields { and there are sufficiently many Abelian cases.' The critical thing is learning how to pass from a prime in an intermediate field to a prime in a large field.
    [Show full text]