528 Chapter 18Matrices and linear transformations
represented by the number + 1 , clearly satisfies the group multiplication table,
Table 18.1, and is called the trivialor totally symmetricrepresentation of the group.
Every group has such a representation. Table 18.3, obtained from the multiplication
table 18.1 by replacing each operation by its representative± 1 inΓ
2, shows thatΓ
2is indeed a representation of the group.
In the same way, replacing each operation in Table 18.1 by its representative matrix
in the two-dimensional representationΓ
3confirms that these matrices satisfy the
multiplication table.
The matrices of the two-dimensional representation Γ
3can be derived by
considering the result of applying each symmetry operation to the coordinates of a
point in the plane. Thus, the operation Ais the anticlockwise rotation through
θ 1 = 1 120°about the origin and its representative matrix is
(18.67)
The rotation Bis the inverse of A, sinceAB 1 = 1 BA 1 = 1 E, and its representative matrix B
is the transpose matrix of A(all the matrices are orthogonal, with inverse equal to
transpose). Similarly, the operation C, rotation through 180° about the Ocaxis (the
y-axis), transforms a vectorr 1 = 1 (x, y)into the vectorr′ 1 = 1 (−x, y). Its representative
matrix is therefore
(18.68)
since
(18.69)
It is possible to construct any number of representations of all possible dimensions for
any group, but it can be shown that only a certain number of these (the ‘irreducible
−
=
−
10
01
x
y
x
y
C=
−
10
01
A=
°− °
°°
=
−
cos sin
sin cos
120 120
120 120
11
2
3
2
3
2
1
2
−
−
Table 18.3 Multiplication table of
Γ
2
1 + 1 + 1 − 1 − 1 − 1
1 + 1 + 1 + 1 − 1 − 1 − 1
1 + 1 + 1 + 1 − 1 − 1 − 1
1 + 1 + 1 + 1 − 1 − 1 − 1
− 1 − 1 − 1 − 1 + 1 + 1 + 1
− 1 − 1 − 1 − 1 + 1 + 1 + 1
− 1 − 1 − 1 − 1 + 1 + 1 + 1