Page 265 -

232 ALGEBRA
1
group GL(1) of all linear transformations of E , and the multiplication in this subgroup is
described by the table
A (1)
A (1) A (1)
(1)
(1)
(1)
We obtain a one-dimensional representation D (G) of the group G by letting D (I)= A ,
(1)
(1)
D (P)= A . These relations define a homomorphism of the group G to GL(1) and thus
define its representation.
2. A two-dimensional representation of the group G.
2
(2)
In E , we choose a basis e 1 , e 2 and consider the matrices A , B (2) of linear transfor-
2 2 (2)
1 0 (2)
0 1 2 2
mations A , B of this space: A = 0 1 , B = 1 0 . The transformations A , B form
2
a subgroup in the group GL(2) of linear transformations of E . The multiplication in this
subgroup is defined by the table
A (2) B (2)
A (2) A (2) B (2)
B (2) B (2) A (2)
(2)
(2)
(2)
We obtain a two-dimensional representation D (G) of the group G by letting D (I)=A ,
D (P)= B . These relations define an isomorphism of G onto the subgroup {A , B }
(2)
(2)
(2)
(2)
of GL(2), and therefore define its representation.
3. A three-dimensional representation of the group G.
3
Consider the linear transformation A (3) of E definedbythe matrix
1 0 0
( )
A (3) = 0 1 0 .
0 0 1
(3)
(3)
This transformation forms a subgroup in GL(3) with the multiplication law A A (3) = A .
(3)
(3)
One obtains a three-dimensional representation D (G) of the group G by letting D (I)=
(3)
(3)
(3)
A , D (P)= A .
4. A four-dimensional representation of the group G.
4
Consider linear transformations A (4) and B (4) of E definedbythe matrices
1 0 00 0 1 00
⎛ ⎞ ⎛ ⎞
A (4) ⎜ 0 1 00 ⎟ B (4) ⎜ 1 0 00 ⎟
⎠ ,
⎠ .
00 1 0 = ⎝ 00 0 1
= ⎝
00 0 1 00 1 0
The transformations A (4) and B (4) form a subgroup in GL(4) with the multiplication defined
by a table similar to that in the two-dimensional case. One obtains a four-dimensional
(4)
(4)
(4)
(4)
representation D (G) of the group G by letting D (I)= A , D (P)= A (B) .
A (2) 0 B (2) 0
(4)
(4)
(4)
Remark. ThematricesA andB (4) maybewrittenintheformA = (2) , B = (2) ,
0 A 0 B
(4)
(2)
(2)
(2)
(4)
and therefore the representation D (G) is sometimes denoted by D (G)= D (G)+ D (G)= 2D (G). In
(1)
(3)
a similar way, one may use the notation D (G)= 3D (G). In this way, one can construct representations of
the group G of arbitrary dimension.
◦
2 . The symmetry group G = {I, P} for the three-dimensional space is a normal subgroup of
the group O(3). The subgroup SO(3) ⊂ O(3) formed by proper orthogonal transformations
is isomorphic to the quotient group O(3)/{I, P}.