7.2 Representations
185
morphic map of G into the set (group) of n
×
n invertible matrices.
4
Introducing a
basis
(
u
i
),
i
=
1
, . . .,
n
, on
V
n
,
D(g)
acts as an
n
×
n
matrix
D
(
g
)
≡
(
D
(
g
)
i
j
)
on
the components of a vector
V
=
(
V
i
, . . .,
V
n
)
according to the law
V
i
=
D
(
g
)
i
j
V
j
⇔
V
=
D
(
g
)
V
,
We shall denote by the bold symbol
D
the representation of a group in terms of matri-
ces. The vector space
V
n
is called the
carrier of the representation or representation
space
. In the case of the rotation group, for example, the three dimensional Euclidean
space
V
3
is the carrier of the representation studied in
Chap.4
:
g
(θ
1
, θ
1
, θ
2
)
∈
SO
(
3
)
D
−→
D
(
g
)
i
j
=
R
(θ
1
, θ
1
, θ
2
)
i
j
,
i
,
j
=
1
,
2
,
3
.
For a general representation the matrix
D
(g)
is an element of GL
(
n
,
C
)
or of
GL
(
n
,
R
)
, depending on whether the base space is a complex or real vector space.
In the active picture, for any
g
∈
G
,
D
(g)
maps vectors into vectors, all represented
with respect to a same basis
(
u
i
)
. On replacing the original basis
(
u
i
)
by a new one
(
u
i
)
, related to it through a non singular matrix
A
, as in (
7.4
), the matrix
D
(g)
gets
replaced by the matrix
D
(
g
)
=
AD
(
g
)
A
−
1
which represents the action of
D(g)
in
the new basis. This is easily shown starting from the matrix relation between the
components of a vector
V
1
and its transformed
V
2
in the old basis:
V
2
=
D
(
g
)
V
1
.
Being the components
V
1
and
V
2
of the two-vectors in the new basis given by
V
1
=
AV
1
,
V
2
=
AV
2
, we find:
V
2
=
AV
2
=
AD
(
g
)
V
1
=
AD
(
g
)
A
−
1
V
1
=
D
(
g
)
V
1
.
(7.9)
It is easily verified that the mapping
D
of a generic group element
g
into
D
(
g
)
is
still a representation, also denoted by
D
=
ADA
−
1
.
The representations
ADA
−
1
and
D
are then said
equivalent
, and we write:
D
∼
ADA
−
1
.
(7.10)
If the homomorphic mapping:
g
−→
D
(
g
),
(7.11)
is
isomorphic
, namely it is one-to-one and onto, then the representation is
faithful
,
otherwise it is
unfaithful
. A trivial, but important, representation is obtained by the
mapping:
g
−→
1
,
∀
g
∈
G
,
(7.12)
and is called the
identity, or trivial representation
, simply denoted by
1
.
4
It is obvious that the identity
g
0
=
e
element of the group is represented by the unit n-dimensional
matrix that we will denote by
1
or else by
I
.