a4105

5

Symmetries.

(49b)

What does it mean ?

Consider a group composed by four elements ( a "discrete group" ).

(50)

that I can write :

(51)

The corresponding action is :

(52)

Clearly it may reverse the x coordinate, the y coordinate, or the two.

Schematically :

(53)

(54)

(55)

(56)

Now we may build the matrix :

(57)

We can check such set of matrixes form a group.

Their determinant is :

(58)

det (

a

) = l m ( cos 2 a + sin 2 a ) = l m = ±1

Check the inverse matrix is :

(59)

(60)

(61)

So that :

(62)

whence :

(63)

SO(2) (called special orthogonal group) is a sub-group of O(2) (called orthogonal group) and we may form the matrixes

a

from the matrixes

a

through :

(64)

By the way, many are redundant. For an example, if

(64b)

(65)

which means that changing ( x ---> - ; y ---> -y ) is equivalent to a rotation of

p

. See next figure.

(66)

We know that matrixes :

(67)

correspond to a simple rotation around the center of coordinates O.

What is the meaning of more general matrixes :

(68)

From :

(69)

we know that

a

corresponds to two combined operations :

• A symmetry with respect
  to axis OX , or OY , or both.

• A rotation a
  around the center of coordinates.

(70)

On the figure is shown the succession of the two operations

( M1 ----> M4 )

It is clear that it is equivalent to a symmetry with respect to a straight line passing by O

(71)

We have enriched the "special orthogonal group " SO(2) which began the "orthogonal group" O(2). Then we discovers that this extended group contains mirror-symmetries : all the symmetries with respect to straight lines passing by the origin of coordinates O.

(72)

Index Dynamic Groups Theory