Mathematical Methods for Physicists: A concise introduction - Site Map

ELEMENTS OF GROUP THEORY
Table 12.6.
~E ~A ~B
~E ~E ~A ~B
~A ~A ~B ~E
B ~B ~E ~A
axis is a threefold symmetry axis. There are three additional axes of symmetry in
the xy plane: each corner and the geometric center O de®ning an axis; each of
these is a twofold symmetry axis (Fig. 12.4). Now let us consider re¯ection operations.
The following successive operations will bring the equilateral angle onto
itself (that is, be invariant):
~E the identity; triangle is left unchanged;
~A triangle is rotated through 1208 clockwise;
~B triangle is rotated through 2408 clockwise;
~C triangle is re¯ected about axis OR (or the y-axis);
~D triangle is re¯ected about axis OQ;
~F triangle is re¯ected about axis OP.
Now the re¯ection about axis OR is just a rotation of 1808 about axis OR, thus
~C ˆ R OR …1808† ˆ 1 0
:
0 1
Next, we notice that re¯ection about axis OQ is equivalent to a rotation of 2408
about the z-axis followed by a re¯ection of the x-axis (Fig. 12.5):
~D ˆ R OQ …1808† ˆ ~C ~B ˆ 1 0 p ! p
1=2 3 =2
p
0 1
1=2
3 =2
ˆ p
!:
3 =2 1=2
3 =2 1=2
Figure 12.4.
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