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72
Physics Formulary by ir. J.C.A. Wevers
13.1.4
Isomorfsm and homomorfsm; representations
Two groups are
isomorphic
if they have the same multiplication table. The mapping from group
G
1
to
G
2
, so
that the multiplication table remains the same is a homomorphic mapping. It need not be isomorphic.
A
representation
is a homomorphic mapping of a group to a group of square matrices with the usual matrix
multiplication as the combining operation. This is symbolized by
Γ
. The following holds:
Γ
(
E
)=
I
I,
Γ
(
AB
)=
Γ
(
A
)
Γ
(
B
)
,
Γ
(
A

1
)=[
Γ
(
A
)]

1
For each group there are 3 possibilities for a representation:
1. A
faithful
representation: all matrices are different.
2. The representation
A
→
det
(
Γ
(
A
))
.
3. The identical representation:
A
→
1
.
An
equivalent representation
is obtained by performing an unitary base transformation:
Γ
±
(
A
)=
S

1
Γ
(
A
)
S
.
13.1.5
Reducible and irreducible representations
If
the same
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.
 Spring '10
 Ye
 Physics

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