(Mathematics 4H 1998“9 )
7/11/2002
Dr A. J. Baker
Department of Mathematics, University of Glasgow, Glasgow G12 8QW,
Scotland.
Email address: a.baker@maths.gla.ac.uk
URL: http://www.maths.gla.ac.uk/∼ajb
Contents
Chapter 1. Linear and multilinear algebra 1
1. Basic linear algebra 1
2. Class functions and the CayleyHamilton Theorem 5
3. Separability 8
4. Basic notions of multilinear algebra 9
Chapter 2. Recollections and reformulations on basic group theory 13
1. The Isomorphism and Correspondence Theorems 13
2. Some de¬nitions and notation 14
3. Group actions 15
4. The Sylow theorems 17
5. Solvable groups 17
6. Product and semidirect product groups 18
7. Some useful groups 18
8. Some useful Number Theory 19
Chapter 3. Representations of ¬nite groups 21
1. Linear representations 21
2. Ghomomorphisms and irreducible representations 23
3. New representations from old 27
4. Permutation representations 28
5. Properties of permutation representations 30
6. Calculating in permutation representations 32
7. Generalized permutation representations 33
Chapter 4. Character theory 35
1. Characters and class functions on a ¬nite group 35
2. Properties of characters 37
3. Inner products of characters 38
4. Character tables 41
5. Examples of character tables 44
6. Reciprocity formul¦ 49
7. Representations of semidirect products 51
Chapter 5. Some applications to group theory 53
1. Characters and the structure of groups 53
2. A result on representations of simple groups 55
3. A Theorem of Frobenius 56
Chapter 6. Automorphisms and extensions 59
1. Automorphisms 59
2. Extensions 62
3. Classifying extensions [optional extra material] 65
3
4 CONTENTS
Chapter 7. Some further applications 69
1. Fourier series and the circle group 69
CHAPTER 1
Linear and multilinear algebra
In this chapter we will study the linear algebra required in representation theory. Some of
this will be familiar but there will also be new material, especially that on ˜multilinear™ algebra.
1. Basic linear algebra
Throughout the remainder of these notes k will denote a ¬eld, i.e., a commutative ring with
unity 1 in which every nonzero element has an inverse. Most of the time in representation
theory we will work the ¬eld of complex numbers C and occasionally the ¬eld of real numbers
R. However, a lot of what we discuss will work over more general ¬elds, including those of ¬nite
characteristic such as Z/p for a prime p. Here, the characteristic of the ¬eld k is de¬ned to be
the smallest natural number p ∈ N such that p1 = 1 + · · · + 1 = 0 if such a number exists then k
is said to have ¬nite characteristic), otherwise it has characteristic 0. In the ¬nite characteristic
case, the characteristic is always a prime.
1.1. Bases, linear transformations and matrices. Let V be a ¬nite dimensional vector
space over k, i.e., a kvector space. Recall that a basis for V is a linearly independent spanning
set for V . The dimension of V (over k) is the number of elements in any basis, and is denoted
dim V . We will often view k itself as a 1dimensional kvector space with basis {1} or indeed
any set {x} with x = 0.
Given two kvector spaces V, W , a linear transformation (or linear mapping) from V to W
is a function • : V ’’ W such that
•(v1 + v2 ) = •(v1 ) + •(v2 ),
•(tv) = t•(v),
for v1 , v2 , v ∈ V and t ∈ k. The set of all linear transformations V ’’ W will be denoted
Hom (V, W ). This is a kvector space with the operations of addition and scalar multiplication
given by
(• + θ)(u) = •(u) + θ(u),
(t•)(u) = t(•(u)) = •(tu)
for •, θ ∈ Hom (V, W ) and t ∈ k.
An important property of a basis is the following extension property.
Proposition 1.1. Let V, W be kvector spaces with V ¬nite dimensional, and {v1 , . . . , vm }
a basis for V where m = dim V . Given a function • : {v1 , . . . , vm } ’’ W , there is a unique
linear transformation ¦ : V ’’ W such that
¦(vj ) = •(vj ) (1 j m).
We can express this with the aid of the commutative diagram
inclusion /V
{v1 , . . . , vm }
LL
LL
LL
• LLL ∃! ¦
L%
W
1
2 1. LINEAR AND MULTILINEAR ALGEBRA
in which the dotted arrow is supposed to indicate a (unique) solution to the problem of ¬lling
in the diagram
inclusion /V
{v1 , . . . , vm }
LL
LL
LL
• LLL
L%
W
with a linear transformation so that composing the functions corresponding to the horizontal
and right hand sides agrees with the functions corresponding to left hand side.
Proof. The de¬nition of ¦ is
«
m m
¦ »j vj = »j •(vj ).
j=1 j=1
When using this result we will refer to ¦ as the linear extension of • and often write •.
Let V, W be ¬nite dimensional kvector spaces with bases {v1 , . . . , vm } and {w1 , . . . , wn },
where m = dim V and n = dim W . By Proposition 1.1, each function •ij : {v1 , . . . , vm } ’’ W
(1 i m, 1 j n) given by
•ij (vk ) = δik wj (1 k m)
extends uniquely to a linear transformation •ij : V ’’ W .
Proposition 1.2. The set of functions •ij : V ’’ W (1 i m, 1 j n) is a basis
for Hom (V, W ). Hence
dim Hom (V, W ) = dim V dim W = mn.
A particular and very important case of this is the dual space of V ,
V — = Hom(V, k).
Notice that dim V — = dim V . Given a basis {v1 , . . . , vm } of V , V — has as a basis the set of
— —
functions {v1 , . . . , vm } which satisfy
—
vi (vk ) = δik ,
where δij is the Kronecker δsymbol for which
1 if i = j,
δij =
0 otherwise.
We can view this giving rise to an isomorphism V ’’ V — under which
—
vj ←’ vj .
If we set V —— = (V — )— , then there is an isomorphism V — ’’ V —— under which
vj ←’ (vj )— .
— —
Here we use the fact that the vj form a basis for V — . Composing these two isomorphisms we
—
obtain a third V ’’ V —— given by
vj ←’ (vj )— .
—
In fact, this does not depend on the basis of V used, although the factors do! This is sometimes
called the canonical isomorphism V ’’ V —— .
The set of all endomorphisms of V is
End (V ) = Hom (V, V ),
1. BASIC LINEAR ALGEBRA 3
which is a ring (actually a kalgebra, and also noncommutative if dim V > 1) with addition
as above, and multiplication given by composition of functions. There is a ring monomorphism
k ’’ End (V ) given by
t ’’ t IdV
which embeds k into End (V ) as the subring of scalars. We also have
dim End (V ) = (dim V )2 .
Let GL (V ) denote the group of all invertible klinear transformations V ’’ V , i.e., the
group of units in End (V ). This is usually called the general linear group of V or the group of
linear automorphisms of V and denoted GL (V ) or Aut (V ).
Now let v = {v1 , . . . , vm } and w = {w1 , . . . , wn } be bases for V and W . Then given a linear
transformation • : V ’’ W we may de¬ne the matrix of • with respect to the bases v and w
to be the n — m matrix with coe¬cients in k,
w [•]v = [aij ],
where
n
•(vj ) = akj wk .
k=1
Now suppose we have a second pair of bases for V and W , v = {v1 , . . . , vm } and w =
{w1 , . . . , wn }. Then we can write
m n
vj = prj vr , wj = qsj ws ,
r=1 s=1
for some pij , qij ∈ k. If we form the m — m and n — n matrices P = [pij ] and Q = [qij ], then we
have the following standard result.
Proposition 1.3. The matrices w [•]v and [•]v are related by the formul¦
w
[•]v = Qw [•]v P ’1 = Q[aij ]P ’1 .
w
In particular, if W = V , w = v and w = v , then
[•]v = P v [•]v P ’1 = P [aij ]P ’1 .
v
1.2. Quotients and complements. Let W ⊆ V be a vector subspace. Then we de¬ne
the quotient space V /W to be the set of equivalence classes under the equivalence relation ∼ on
V de¬ned by
u∼v if and only if v ’ u ∈ W.
We denote the class of v by v + W . This set V /W becomes a vector space with operations
(u + W ) + (v + W ) = (u + v) + W,
»(v + W ) = (»v) + W
and zero element 0 + W . There is a linear transformation, usually called the quotient map
q : V ’’ V /W , de¬ned by
q(v) = v + W.
Then q is surjective, has kernel ker q = W and has the following universal property.
4 1. LINEAR AND MULTILINEAR ALGEBRA
Theorem 1.4. Let f : V ’’ U be a linear transformation with W ⊆ ker f . Then there is a
unique linear transformation f : V /W ’’ U for which f = f —¦ q. This can be expressed in the
diagram
q
/ V /W
V?
? ??
??
f ? ? ∃! f

U
in which all the sides represent linear transformations.
Proof. We de¬ne f by
f (v + W ) = f (v),
which makes sense since if v ∼ v, then v ’ v ∈ W , hence
f (v ) = f ((v ’ v) + v) = f (v ’ v) + f (v) = f (v).
The uniqueness follows from the fact that q is surjective.
Notice also that
dim V /W = dim V ’ dim W.
(1.1)
A linear complement (in V ) of a subspace W ⊆ V is a subspace W ⊆ V such that the
restriction qW : W ’’ V /W is a linear isomorphism. The next result sums up properties of
linear complements and we leave the proofs as exercises.
Theorem 1.5. Let W ⊆ V and W ⊆ V be vector subspaces of the kvector space V with
dim V = n. Then the following conditions are equivalent.
a) W is a linear complement of W in V .
b) Let {w1 , . . . , wr } be a basis for W , and {wr+1 , . . . , wn } a basis for W . Then
{w1 , . . . , wn } = {w1 , . . . , wr } ∪ {wr+1 , . . . , wn }
is a basis for V .
c) Every v ∈ V has a unique expression of the form
v = v1 + v2
for some elements v1 ∈ W , v2 ∈ W . In particular, W © W = {0}.
d) Every linear transformation h : W ’’ U has a unique extension to a linear transfor
mation H : V ’’ U with W ⊆ ker H.
e) W is a linear complement of W in V .
∼
=
f) There is a linear isomorphism J : V ’ W — W for which im JW = W — {0} and