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E-raamat: On Characters of Finite Groups

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On Characters of Finite GroupsSuurem pilt
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This book explores the classical and beautiful character theory of finite groups. It does it by using some rudiments of the language of categories. Originally emerging from two courses offered at Peking University (PKU), primarily for third-year students, it is now better suited for graduate courses, and provides broader coverage than books that focus almost exclusively on groups.

The book presents the basic tools, notions and theorems of character theory (including a new treatment of the control of fusion and isometries), and introduces readers to the categorical language at several levels. It includes and proves the major results on characteristic zero representations without any assumptions about the base field. The book includes a dedicated chapter on graded representations and applications of polynomial invariants of finite groups, and its closing chapter addresses the more recent notion of the Drinfeld double of a finite group and the corresponding representation of GL_2(Z).

Arvustused

The book is written by one of the best character theorists, and he has done a great job. I am sure, teachers and students interested in character theory both enjoy reading this book. (Mohammad-Reza Darafsheh, zbMATH 1391.20001, 2018)

1 Tensor Product
1(18)
1.1 Definition of the Tensor Product
1(3)
1.2 Properties of the Tensor Product
4(8)
1.2.1 Functoriality
4(1)
1.2.2 More Properties
4(3)
1.2.3 Kronecker Product of Matrices
7(2)
1.2.4 Tensor Product and Homomorphisms
9(1)
1.2.5 Extension of Scalars
10(1)
1.2.6 Trace and Restriction of Scalars
11(1)
1.3 Symmetric and Alternating Powers
12(3)
1.3.1 Symmetric and Alternating Squares
12(1)
1.3.2 Tensor, Symmetric and Exterior Algebras
13(2)
1.4 Tensor Product over an Algebra
15(4)
2 On Representations
19(22)
2.1 Generalities on Representations
19(4)
2.1.1 Introduction
19(1)
2.1.2 General Representations
20(3)
2.2 Set-Representations
23(6)
2.2.1 Union and Product
23(1)
2.2.2 Transitive Representations
24(1)
2.2.3 Classification of Transitive Representations
25(1)
2.2.4 Burnside's Marks
26(3)
2.3 Linear Representations
29(12)
2.3.1 Generalities
29(6)
2.3.2 Finite Groups: The Group Algebra
35(6)
3 Characteristic 0 Representations
41(50)
3.1 Preliminary: 1/|G| ΣgεGg
41(3)
3.1.1 Fixed Points
41(1)
3.1.2 Maschke's Theorem
42(2)
3.1.3 Spectrum
44(1)
3.2 Characters
44(16)
3.2.1 First Properties
44(3)
3.2.2 Orthogonality Relations and First Applications
47(7)
3.2.3 Splitting Fields
54(4)
3.2.4 From Permutations to Characters
58(2)
3.3 Structure of the Group Algebra
60(4)
3.3.1 A Product of Endomorphism Algebras
60(2)
3.3.2 Canonical Decomposition of a Representation
62(2)
3.4 Group Determinant
64(2)
3.5 Center and Action of Cyclotomic Galois Groups
66(9)
3.5.1 About Centers
67(1)
3.5.2 Extending and Restricting Scalars
68(4)
3.5.3 Action of (Z/eGZ)× and of Galois Groups of Cyclotomic Extensions
72(3)
3.6 More on a Splitting Field and First Applications
75(4)
3.6.1 Character Tables
75(2)
3.6.2 Products of Groups
77(1)
3.6.3 Abelian Groups
78(1)
3.7 Some Arithmetical Properties of Characters
79(12)
3.7.1 Characters and Integrality
79(3)
3.7.2 Applications: Two Theorems of Burnside
82(9)
4 PLAYING with the BASE FIELD
91(20)
4.1 Analysis of an Irreducible Module
91(9)
4.1.1 An Example: The Quaternion Group Q8
91(3)
4.1.2 Scalars of an Irreducible Module
94(3)
4.1.3 Schur Indices
97(3)
4.2 Complements on Rationality
100(11)
4.2.1 Group of Characters and Rationality
100(3)
4.2.2 Reflections and Rationality
103(2)
4.2.3 Questions of Rationality over R
105(6)
5 Induction, Restriction
111(20)
5.1 On Any Field
111(10)
5.1.1 Restriction
111(1)
5.1.2 Induction
112(7)
5.1.3 Mackey Formula
119(2)
5.2 Induction and Restriction in Characteristic Zero
121(10)
5.2.1 Induction and Normal Subgroups
121(1)
5.2.2 Induction and Restriction for Class Functions
122(4)
5.2.3 Application of Induction to an Example
126(5)
6 Brauer's Theorem and Some Applications
131(24)
6.1 Artin's Theorem
131(2)
6.2 Brauer's Characterization of Characters
133(7)
6.2.1 Statement and First Consequences
133(2)
6.2.2 Proof of Brauer's Theorem
135(5)
6.3 Fusion and Isometries
140(4)
6.3.1 π-elements and Class Functions
141(1)
6.3.2 π-Control Subgroups
141(3)
6.4 Some Fundamental Theorems about Finite Groups
144(11)
6.4.1 Existence of a Normal π-Complement
145(1)
6.4.2 π-trivial Intersection Subgroups
146(1)
6.4.3 Frobenius Groups
147(8)
7 Graded Representation and Characters
155(24)
7.1 Graded Vector Spaces, Algebras, Modules
155(12)
7.1.1 Graded Vector Spaces
155(2)
7.1.2 Graded Algebras and Modules
157(3)
7.1.3 Nakayama's Lemma
160(1)
7.1.4 Free Graded Modules
161(3)
7.1.5 Polynomial Algebras and Noether Parameters
164(3)
7.2 Graded Characters of Graded kG-modules
167(12)
7.2.1 Notation and Definitions
167(2)
7.2.2 Isotypic Components of the Symmetric Algebra
169(3)
7.2.3 Computations with Power Series
172(2)
7.2.4 A Simple Example
174(1)
7.2.5 Complement: Reflection Groups
175(4)
8 Drinfeld Double
179(38)
8.1 The Drinfeld Double of a Finite Group as an Algebra
179(10)
8.1.1 The Semidirect Product of kG and Its Dual
180(3)
8.1.2 A Description of DkGmod
183(3)
8.1.3 On the Center Z(DkG) and Central Functions Again
186(3)
8.2 Hopf Algebras: An Introduction from Scratch
189(9)
8.2.1 Notation for Multiple Tensor Products
189(2)
8.2.2 Algebras, Coalgebras, Bialgebras, Hopf Algebras
191(5)
8.2.3 On Algebra Representations of a Hopf Algebra
196(2)
8.3 The Drinfeld Double as a Ribbon Hopf Algebra
198(10)
8.3.1 Universal R-matrix for DkG
198(3)
8.3.2 The Category DtGmod is a Ribbon Category
201(1)
8.3.3 A Description of 0tcmod as a Ribbon Category
202(6)
8.4 Action of GL2(Z/eGZ)
208(9)
8.4.1 The Automorphisms S, Ω, Δn
208(3)
8.4.2 Action of GL2(Z/eGZ)
211(1)
8.4.3 The Verlinde Formula
212(5)
Appendix A Basics on Finite Groups 217(4)
Appendix B Assumed Results on Galois Theory 221(4)
Appendix C Integral Elements 225(4)
Appendix D Noetherian Rings and Modules 229(4)
Appendix E The Language of Categories and Functors 233(8)
Bibliography 241(2)
Index 243
Michel Broué is Emeritus Exceptional Professor of Mathematics at the University Paris-Diderot Paris 7. He has been the Director of the Institut Henri-Poincaré in Paris (1999-2009), and the Director of the Department of Mathematics and Computer Science (DMI) at the École normale supérieure (1986-1993) He is a Foreign Honorary Member of the American Academy of Arts and Sciences, a Fellow of the American Mathematical Society,  a Senior Member of the Institut Universitaire de France, and a frequent visitor to many institutions in the world. 

He is the Editor-in-chief of the Journal of Algebra, and a member of several editorial boards. His main research area is Algebra and Representation Theory, in particular of finite and algebraic groups.
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