1Group actions
II Representation Theory
1 Group actions
We start by reviewing some basic group theory and linear algebra.
Basic linear algebra
Notation. F always represents a field.
Usually, we take
F
=
C
, but sometimes it can also be
R
or
Q
. These fields
all have characteristic zero, and in this case, we call what we’re doing ordinary
representation theory. Sometimes, we will take
F
=
F
p
or
¯
F
p
, the algebraic
closure of F
p
. This is called modular representation theory.
Notation.
We write
V
for a vector space over
F
— this will always be finite
dimensional over
F
. We write
GL
(
V
) for the group of invertible linear maps
θ
:
V → V
. This is a group with the operation given by composition of maps,
with the identity as the identity map (and inverse by inverse).
Notation.
Let
V
be a finite-dimensional vector space over
F
. We write
End
(
V
)
for the endomorphism algebra, the set of all linear maps V → V .
We recall a couple of facts from linear algebra:
If
dim
F
V
=
n < ∞
, we can choose a basis
e
1
, ··· , e
n
of
V
over
F
. So we
can identify
V
with
F
n
. Then every endomorphism
θ ∈ GL
(
V
) corresponds to a
matrix A
θ
= (a
ij
) ∈ M
n
(F ) given by
θ(e
j
) =
X
i
a
ij
e
i
.
In fact, we have
A
θ
∈ GL
n
(
F
), the general linear group. It is easy to see the
following:
Proposition.
As groups,
GL
(
V
)
∼
=
GL
n
(
F
), with the isomorphism given by
θ 7→ A
θ
.
Of course, picking a different basis of
V
gives a different isomorphism to
GL
n
(F), but we have the following fact:
Proposition.
Matrices
A
1
, A
2
represent the same element of
GL
(
V
) with respect
to different bases if and only if they are conjugate, namely there is some
X ∈
GL
n
(F) such that
A
2
= XA
1
X
−1
.
Recall that
tr
(
A
) =
P
i
a
ii
, where
A
= (
a
ij
)
∈ M
n
(
F
), is the trace of
A
. A
nice property of the trace is that it doesn’t notice conjugacy:
Proposition.
tr(XAX
−1
) = tr(A).
Hence we can define the trace of an operator
tr
(
θ
) =
tr
(
A
θ
), which is
independent of our choice of basis. This is an important result. When we study
representations, we will have matrices flying all over the place, which are scary.
Instead, we often just look at the traces of these matrices. This reduces our
problem of studying matrices to plain arithmetic.
When we have too many matrices, we get confused. So we want to put a
matrix into a form as simple as possible. One of the simplest form a matrix
can take is being diagonal. So we want to know something about diagonalizing
matrices.
Proposition.
Let
α ∈ GL
(
V
), where
V
is a finite-dimensional vector space over
C and α
m
= id for some positive integer m. Then α is diagonalizable.
This follows from the following more general fact:
Proposition.
Let
V
be a finite-dimensional vector space over
C
, and
α ∈
End
(
V
), not necessarily invertible. Then
α
is diagonalizable if and only if there
is a polynomial f with distinct linear factors such that f(α) = 0.
Indeed, we have x
m
− 1 =
Q
(x − ω
j
), where ω = e
2πi/m
.
Instead of just one endomorphism, we can look at many endomorphisms.
Proposition.
A finite family of individually diagonalizable endomorphisms of
a vector space over
C
can be simultaneously diagonalized if and only if they
commute.
Basic group theory
We will not review the definition of a group. Instead, we look at some of our
favorite groups, since they will be handy examples later on.
Definition
(Symmetric group
S
n
)
.
The symmetric group
S
n
is the set of all
permutations of
X
=
{
1
, ··· , n}
, i.e. the set of all bijections
X → X
. We have
|S
n
| = n!.
Definition
(Alternating group
A
n
)
.
The alternating group
A
n
is the set of
products of an even number of transpositions (
i j
) in
S
n
. We know
|A
n
|
=
n!
2
.
So this is a subgroup of index 2 and hence normal.
Definition (Cyclic group C
m
). The cyclic group of order m, written C
m
is
C
m
= hx : x
m
= 1i.
This also occurs naturally, as
Z/mZ
over addition, and also the group of
n
th roots
of unity in
C
. We can view this as a subgroup of
GL
1
(
C
)
∼
=
C
×
. Alternatively,
this is the group of rotation symmetries of a regular
m
-gon in
R
2
, and can be
viewed as a subgroup of GL
2
(R).
Definition (Dihedral group D
2m
). The dihedral group D
2m
of order 2m is
D
2m
= hx, y : x
m
= y
2
= 1, yxy
−1
= x
−1
i.
This is the symmetry group of a regular
m
-gon. The
x
i
are the rotations and
x
i
y
are the reflections. For example, in
D
8
,
x
is rotation by
π
2
and
y
is any
reflection.
This group can be viewed as a subgroup of
GL
2
(
R
), but since it also acts on
the vertices, it can be viewed as a subgroup of S
m
.
Definition (Quaternion group). The quaternion group is given by
Q
8
= hx, y : x
4
= 1, y
2
= x
2
, yxy
−1
= x
−1
i.
This has order 8, and we write i = x, j = y, k = ij, −1 = i
2
, with
Q
8
= {±1, ±i, ±j, ±k}.
We can view this as a subgroup of GL
2
(C) via
1 =
1 0
0 1
, i =
i 0
0 −i
, j =
0 1
−1 0
, k =
0 i
i 0
,
−1 =
−1 0
0 −1
, −i =
−i 0
0 i
, −j =
0 −1
1 0
, −k =
0 −i
−i 0
.
Definition (Conjugacy class). The conjugacy class of g ∈ G is
C
G
(g) = {xgx
−1
: x ∈ G}.
Definition (Centralizer). The centralizer of g ∈ G is
C
G
(g) = {x ∈ G : xg = gx}.
Then by the orbit-stabilizer theorem, we have |C
G
(g)| = |G : C
G
(G)|.
Definition
(Group action)
.
Let
G
be a group and
X
a set. We say
G
acts on
X if there is a map ∗ : G × X → X, written (g, x) 7→ g ∗x = gx such that
(i) 1x = x
(ii) g(hx) = (gh)x
The group action can also be characterised in terms of a homomorphism.
Lemma.
Given an action of
G
on
X
, we obtain a homomorphism
θ
:
G →
Sym(X), where Sym(X) is the set of all permutations of X.
Proof.
For
g ∈ G
, define
θ
(
g
) =
θ
g
∈ Sym
(
X
) as the function
X → X
by
x 7→ gx
.
This is indeed a permutation of X because θ
g
−1
is an inverse.
Moreover, for any
g
1
, g
2
∈ G
, we get
θ
g
1
g
2
=
θ
g
1
θ
g
2
, since (
g
1
g
2
)
x
=
g
1
(
g
2
x
).
Definition
(Permutation representation)
.
The permutation representation of a
group action G on X is the homomorphism θ : G → Sym(X) obtained above.
In this course,
X
is often a finite-dimensional vector space over
F
(and we
write it as
V
), and we want the action to satisfy some more properties. We will
require the action to be linear, i.e. for all g ∈ G, v
1
, v
2
∈ V , and λ ∈ F.
g(v
1
+ v
2
) = gv
1
+ gv
2
, g(λv
1
) = λ(gv
1
).
Alternatively, instead of asking for a map
G → Sym
(
X
), we would require a
map G → GL(V ) instead.