5
Symmetries.
(49b)
What does it mean?
Consider a group composed of four elements (a "discrete group").
(50)
which I can write as:
(51)
The corresponding action is:
(52)
It is clear that it can reverse the x-coordinate, the y-coordinate, or both.
Schematically:
(53)
(54)
(55)
(56)
We can now construct the matrix:
(57)
We can verify that this set of matrices forms a group.
Their determinant is:
(58)
det(a) = l m (cos²a + sin²a) = l m = ±1
Let us verify that the inverse matrix is:
(59)
(60)
(61)
Thus:
(62)
whence:
(63)
SO(2) (called the special orthogonal group) is a subgroup of O(2) (called the orthogonal group), and we can form the matrices a from the matrices a via:
(64)
By the way, many are redundant. For example, if
(64b)
(65)
this means that changing (x → -x; y → -y) is equivalent to a rotation of π. See the following figure.
(66)
We know that matrices:
(67)
correspond to a simple rotation around the origin of coordinates O.
What is the meaning of more general matrices:
(68)
From:
(69)
we know that a corresponds to two combined operations:
-
A symmetry with respect to the OX axis, or the OY axis, or both.
-
A rotation by angle a around the origin of coordinates.
(70)
On the figure, the sequence of the two operations (M1 → M4) is shown.
It is clear that this is equivalent to a symmetry with respect to a straight line passing through O.
(71)
We have enriched the "special orthogonal group" SO(2), which originated from the "orthogonal group" O(2). We thus discover that this extended group contains mirror symmetries: all symmetries with respect to straight lines passing through the origin of coordinates O.
(72)
Index Dynamic Groups Theory