Representation Theory of Finite Groups

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Representation Theory of Finite Groups
Representation Theory of Finite Groups
Benjamin Steinberg
School of Mathematics and Statistics
Carleton University
bsteinbg@math.carleton.ca
December 15, 2009
Preface
This book arose out of course notes for a fourth year undergraduate/first
year graduate course that I taught at Carleton University. The goal was to
present group representation theory at a level that is accessible to students
who have not yet studied module theory and who are unfamiliar with tensor
products. For this reason, the Wedderburn theory of semisimple algebras is
completely avoided. Instead, I have opted for a Fourier analysis approach.
This sort of approach is normally taken in books with a more analytic flavor;
such books, however, invariably contain material on representations of com-
pact groups, something that I would also consider beyond the scope of an
undergraduate text. So here I have done my best to blend the analytic and
the algebraic viewpoints in order to keep things accessible. For example,
Frobenius reciprocity is treated from a character point of view to evade use
of the tensor product.
The only background required for this book is a basic knowledge of linear
algebra and group theory, as well as familiarity with the definition of a ring.
The proof of Burnside’s theorem makes use of a small amount of Galois
theory (up to the fundamental theorem) and so should be skipped if used
in a course for which Galois theory is not a prerequisite. Many things are
proved in more detail than one would normally expect in a textbook; this
was done to make things easier on undergraduates trying to learn what is
usually considered graduate level material.
The main topics covered in this book include: character theory; the
group algebra; Burnside’s pq-theorem and the dimension theorem; permu-
tation representations; induced representations and Mackey’s theorem; and
the representation theory of the symmetric group.
It should be possible to present this material in a one semester course.
Chapters 2–5 should be read by everybody; it covers the basic character
theory of finite groups. The first two sections of Chapter 6 are also rec-
ommended for all readers; the reader who is less comfortable with Galois
theory can then skip the last section and move on to Chapter 7 on permu-
i
ii
tation representations, which is needed for Chapters 8–10. Chapter 10, on
the representation theory of the symmetric group, can be read immediately
after Chapter 7.
Although this book is envisioned as a text for an advanced undergraduate
or introductory graduate level course, it is also intended to be of use for
mathematicians who may not be algebraists, but need group representation
theory for their work.
When preparing this book I have relied on a number of classical refer-
ences on representation theory, including [2–4,6,9, 13,14]. For the represen-
tation theory of the symmetric group I have drawn from [4, 7, 8, 10–12]; the
approach is due to James [11]. Good references for applications of represen-
tation theory to computing eigenvalues of graphs and random walks are [3,4].
Discrete Fourier analysis and its applications can be found in [1, 4].
Contents
Preface i
1 Introduction 1
2 Review of Linear Algebra 3
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Complex inner product spaces . . . . . . . . . . . . . . . . . . 4
2.3 Further notions from linear algebra . . . . . . . . . . . . . . . 6
3 Group Representations 8
3.1 Basic definitions and first examples . . . . . . . . . . . . . . . 8
3.2 Maschke’s theorem and complete reducibility . . . . . . . . . 15
4 Character Theory 21
4.1 Homomorphisms of representations . . . . . . . . . . . . . . . 21
4.2 The orthogonality relations . . . . . . . . . . . . . . . . . . . 25
4.3 Characters and class functions . . . . . . . . . . . . . . . . . 30
4.4 The regular representation . . . . . . . . . . . . . . . . . . . . 37
4.5 Representations of abelian groups . . . . . . . . . . . . . . . . 43
5 Fourier Analysis on Finite Groups 47
5.1 Periodic functions on cyclic groups . . . . . . . . . . . . . . . 47
5.2 The convolution product . . . . . . . . . . . . . . . . . . . . . 48
5.3 Fourier analysis on finite abelian groups . . . . . . . . . . . . 51
5.4 An application to graph theory . . . . . . . . . . . . . . . . . 54
5.5 Fourier analysis on non-abelian groups . . . . . . . . . . . . . 59
6 Burnside’s Theorem 65
6.1 A little number theory . . . . . . . . . . . . . . . . . . . . . . 65
6.2 The dimension theorem . . . . . . . . . . . . . . . . . . . . . 68
iii
CONTENTS iv
6.3 Burnside’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 71
7 Permutation Representations 77
7.1 Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.2 Permutation representations . . . . . . . . . . . . . . . . . . . 79
8 Induced Representations 87
8.1 Induced characters and Frobenius reciprocity . . . . . . . . . 87
8.2 Induced representations . . . . . . . . . . . . . . . . . . . . . 90
8.3 Mackey’s irreducibility criterion . . . . . . . . . . . . . . . . . 94
9 Another Theorem of Burnside 101
9.1 Conjugate representations . . . . . . . . . . . . . . . . . . . . 101
10 The Symmetric Group 106
10.1 Partitions and tableaux . . . . . . . . . . . . . . . . . . . . . 106
10.2 Constructing the irreducible representations . . . . . . . . . . 111
Bibliography 119
Index 120

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