2Basic definitions
II Representation Theory
2 Basic definitions
We now start doing representation theory. We boringly start by defining a
representation. In fact, we will come up with several equivalent definitions of a
representation. As always,
G
will be a finite group and
F
will be a field, usually
C.
Definition
(Representation)
.
Let
V
be a finite-dimensional vector space over
F. A (linear) representation of G on V is a group homomorphism
ρ = ρ
V
: G → GL(V ).
We sometimes write
ρ
g
for
ρ
V
(
g
), so for each
g ∈ G
,
ρ
g
∈ GL
(
V
), and
ρ
g
ρ
h
=
ρ
gh
and ρ
g
−1
= (ρ
g
)
−1
for all g, h ∈ G.
Definition
(Dimension or degree of representation)
.
The dimension (or degree)
of a representation ρ : G → GL(V ) is dim
F
(V ).
Recall that
ker ρ C G
and
G/ ker ρ
∼
=
ρ
(
G
)
≤ GL
(
V
). In the very special case
where ker ρ is trivial, we give it a name:
Definition
(Faithful representation)
.
A faithful representation is a representa-
tion ρ such that ker ρ = 1.
These are the representations where the identity is the only element that
does nothing.
An alternative (and of course equivalent) definition of a representation is to
observe that a linear representation is “the same” as a linear action of G.
Definition
(Linear action)
.
A group
G
acts linearly on a vector space
V
if it
acts on V such that
g(v
1
+ v
2
) = gv
1
+ gv
2
, g(λv
1
) = λ(gv
1
)
for all g ∈ G, v
1
, v
2
∈ V and λ ∈ F. We call this a linear action.
Now if
g
acts linearly on
V
, the map
G → GL
(
V
) defined by
g 7→ ρ
g
,
with
ρ
g
:
v 7→ gv
, is a representation in the previous sense. Conversely, given a
representation
ρ
:
G → GL
(
V
), we have a linear action of
G
on
V
via
gv
=
ρ
(
g
)
v
.
In other words, a representation is just a linear action.
Definition
(
G
-space/
G
-module)
.
If there is a linear action
G
on
V
, we say
V
is a G-space or G-module.
Alternatively, we can define a
G
-space as a module over a (not so) cleverly
picked ring.
Definition
(Group algebra)
.
The group algebra
FG
is defined to be the algebra
(i.e. a vector space with a bilinear multiplication operation) of formal sums
FG =
X
g∈G
α
g
g : α
g
∈ F
with the obvious addition and multiplication.
Then we can regard
FG
as a ring, and a
G
-space is just an
FG
-module in
the sense of IB Groups, Rings and Modules.
Definition
(Matrix representation)
. R
is a matrix representation of
G
of degree
n if R Is a homomorphism G → GL
n
(F).
We can view this as a representation that acts on
F
n
. Since all finite-
dimensional vector spaces are isomorphic to
F
n
for some
n
, every representation
is equivalent to some matrix representation. In particular, given a linear repre-
sentation
ρ
:
G → GL
(
V
) with
dim V
=
n
, we can get a matrix representation
by fixing a basis
B
, and then define the matrix representation
G → GL
n
(
F
) by
g 7→ [ρ(g)]
B
.
Conversely, given a matrix representation
R
, we get a linear representation
ρ
in the obvious way — ρ : G → GL(F
n
) by g 7→ ρ
g
via ρ
g
(v) = R
g
v.
We have defined representations in four ways — as a homomorphism to
GL
(
V
), as linear actions, as
FG
-modules and as matrix representations. Now
let’s look at some examples.
Example
(Trivial representation)
.
Given any group
G
, take
V
=
F
(the one-
dimensional space), and
ρ
:
G → GL
(
V
) by
g 7→
(
id
:
F → F
) for all
g
. This is
the trivial representation of G, and has degree 1.
Despite being called trivial, trivial representations are highly non-trivial
in representation theory. The way they interact with other representations
geometrically, topologically etc, and cannot be disregarded. This is a very
important representation, despite looking silly.
Example.
Let
G
=
C
4
=
hx
:
x
4
= 1
i
. Let
n
= 2, and work over
F
=
C
. Then
we can define a representation by picking a matrix
A
, and then define
R
:
x 7→ A
.
Then the action of other elements follows directly by
x
j
7→ A
j
. Of course, we
cannot choose
A
arbitrarily. We need to have
A
4
=
I
2
, and this is easily seen to
be the only restriction. So we have the following possibilities:
(i) A
is diagonal: the diagonal entries can be chosen freely from
{±
1
, ±i}
.
Since there are two diagonal entries, we have 16 choices.
(ii) A
is not diagonal: then it will be equivalent to a diagonal matrix since
A
4
= I
2
. So we don’t really get anything new.
What we would like to say above is that any matrix representation in which
X
is not diagonal is “equivalent” to one in which
X
is. To make this notion
precise, we need to define what it means for representations to be equivalent, or
“isomorphic”.
As usual, we will define the notion of a homomorphism of representations,
and then an isomorphism is just an invertible homomorphism.
Definition
(
G
-homomorphism/intertwine)
.
Fix a group
G
and a field
F
. Let
V, V
0
be finite-dimensional vector spaces over
F
and
ρ
:
G → GL
(
V
) and
ρ
0
:
G → GL
(
V
0
) be representations of
G
. The linear map
ϕ
:
V → V
0
is a
G-homomorphism if
ϕ ◦ ρ(g) = ρ
0
(g) ◦ ϕ (∗)
for all g ∈ G. In other words, the following diagram commutes:
V V
V
0
V
0
ρ
g
ϕ ϕ
ρ
0
g
i.e. no matter which way we go form
V
(top left) to
V
0
(bottom right), we still
get the same map.
We say
ϕ
intertwines
ρ
and
ρ
0
. We write
Hom
G
(
V, V
0
) for the
F
-space of all
these maps.
Definition
(
G
-isomorphism)
.
A
G
-homomorphism is a
G
-isomorphism if
ϕ
is
bijective.
Definition
(Equivalent/isomorphic representations)
.
Two representations
ρ, ρ
0
are equivalent or isomorphic if there is a G-isomorphism between them.
If ϕ is a G-isomorphism, then we can write (∗) as
ρ
0
= ϕρϕ
−1
. (†)
Lemma.
The relation of “being isomorphic” is an equivalence relation on the
set of all linear representations of G over F.
This is an easy exercise left for the reader.
Lemma.
If
ρ, ρ
0
are isomorphic representations, then they have the same di-
mension.
Proof.
Trivial since isomorphisms between vector spaces preserve dimension.
The converse is false.
Example. C
4
has four non-isomorphic one-dimensional representations: if
ω = e
2πi/4
, then we have the representations
ρ
j
(x
i
) = ω
ij
,
for 0 ≤ i ≤ 3, which are not equivalent for different j = 0, 1, 2, 3.
Our other formulations of representations give us other formulations of
isomorphisms.
Given a group
G
, field
F
, a vector space
V
of dimension
n
, and a representation
ρ
:
G → GL
(
V
), we fix a basis
B
of
V
. Then we get a linear
G
-isomorphism
ϕ
:
V → F
n
by
v 7→
[
v
]
B
, i.e. by writing
v
as a column vector with respect to
B
.
Then we get a representation
ρ
0
:
G → GL
(
F
n
) isomorphic to
ρ
. In other words,
every representation is isomorphic to a matrix representation:
V V
F
n
F
n
ρ
ϕ ϕ
ρ
0
Thus, in terms of matrix representations, the representations
R
:
G → GL
n
(
F
)
and
R
0
:
G → GL
n
(
F
) are
G
-isomorphic if there exists some non-singular matrix
X ∈ GL
n
(F) such that
R
0
(g) = XR(g)X
−1
for all g.
Alternatively, in terms of linear
G
-actions, the actions of
G
on
V
and
V
0
are
G-isomorphic if there is some isomorphism ϕ : V → V
0
such that
gϕ(v) = ϕ(gv).
for all
g ∈ G, v ∈ V
. It is an easy check that this is just a reformulation of our
previous definition.
Just as we have subgroups and subspaces, we have the notion of sub-
representation.
Definition
(
G
-subspace)
.
Let
ρ
:
G → GL
(
V
) be a representation of
G
. We
say W ≤ V is a G-subspace if it is a subspace that is ρ(G)-invariant, i.e.
ρ
g
(W ) ≤ W
for all g ∈ G.
Obviously, {0} and V are G-subspaces. These are the trivial G-subspaces.
Definition
(Irreducible/simple representation)
.
A representation
ρ
is irreducible
or simple if there are no proper non-zero G-subspaces.
Example.
Any 1-dimensional representation of
G
is necessarily irreducible, but
the converse does not hold, or else life would be very boring. We will later see
that D
8
has a two-dimensional irreducible complex representation.
Definition
(Subrepresentation)
.
If
W
is a
G
-subspace, then the corresponding
map
G → GL
(
W
) given by
g 7→ ρ
(
g
)
|
W
gives us a new representation of
W
.
This is a subrepresentation of ρ.
There is a nice way to characterize this in terms of matrices.
Lemma.
Let
ρ
:
G → GL
(
V
) be a representation, and
W
be a
G
-subspace of
V
. If
B
=
{v
1
, ··· , v
n
}
is a basis containing a basis
B
1
=
{v
1
, ··· , v
m
}
of
W
(with 0
< m < n
), then the matrix of
ρ
(
g
) with respect to
B
has the block upper
triangular form
∗ ∗
0 ∗
for each g ∈ G.
This follows directly from definition.
However, we do not like block triangular matrices. What we really like is
block diagonal matrices, i.e. we want the top-right block to vanish. There is
no a priori reason why this has to be true — it is possible that we cannot find
another G-invariant complement to W.
Definition
((In)decomposable representation)
.
A representation
ρ
:
G → GL
(
V
)
is decomposable if there are proper G-invariant subspaces U, W ≤ V with
V = U ⊕W.
We say ρ is a direct sum ρ
u
⊕ ρ
w
.
If no such decomposition exists, we say that ρ is indecomposable.
It is clear that irreducibility implies indecomposability. The converse is
not necessarily true. However, over a field of characteristic zero, it turns out
irreducibility is the same as indecomposability for finite groups, as we will see in
the next chapter.
Again, we can formulate this in terms of matrices.
Lemma.
Let
ρ
:
G → GL
(
V
) be a decomposable representation with
G
-invariant
decomposition
V
=
U ⊕ W
. Let
B
1
=
{u
1
, ··· , u
k
}
and
B
2
=
{w
1
, ··· , w
`
}
be
bases for
U
and
W
, and
B
=
B
1
∪ B
2
be the corresponding basis for
V
. Then
with respect to B, we have
[ρ(g)]
B
=
[ρ
u
(g)]
B
1
0
0 [ρ
u
(g)]
B
2
Example.
Let
G
=
D
6
. Then every irreducible complex representation has
dimension at most 2.
To show this, let
ρ
:
G → GL
(
V
) be an irreducible
G
-representation. Let
r ∈ G
be a (non-identity) rotation and
s ∈ G
be a reflection. These generate
D
6
.
Take an eigenvector
v
of
ρ
(
r
). So
ρ
(
r
)
v
=
λv
for some
λ 6
= 0 (since
ρ
(
r
) is
invertible, it cannot have zero eigenvalues). Let
W = hv, ρ(s)vi ≤ V
be the space spanned by the two vectors. We now check this is fixed by
ρ
. Firstly,
we have
ρ(s)ρ(s)v = ρ(e)v = v ∈ W,
and
ρ(r)ρ(s)v = ρ(s)ρ(r
−1
)v = λ
−1
ρ(s)v ∈ W.
Also,
ρ
(
r
)
v
=
λv ∈ W
and
ρ
(
s
)
v ∈ W
. So
W
is
G
-invariant. Since
V
is
irreducible, we must have W = V . So V has dimension at most 2.
The reverse operation of decomposition is taking direct sums.
Definition
(Direct sum)
.
Let
ρ
:
G → GL
(
V
) and
ρ
0
:
G → GL
(
V
0
) be
representations of G. Then the direct sum of ρ, ρ
0
is the representation
ρ ⊕ ρ
0
: G → GL(V ⊕ V
0
)
given by
(ρ ⊕ ρ
0
)(g)(v + v
0
) = ρ(g)v + ρ
0
(g)v
0
.
In terms of matrices, for matrix representations
R
:
G → GL
n
(
F
) and
R
0
: G → GL
n
0
(F), define R ⊕ R
0
: G → GL
n+n
0
(F) by
(R ⊕ R
0
)(g) =
R(g) 0
0 R
0
(g)
.
The direct sum was easy to define. It turns out we can also multiply two
representations, known as the tensor products. However, to do this, we need to
know what the tensor product of two vector spaces is. We will not do this yet.