11. The Icosahedral Groups
It seems important to outline that the tables of characters report in the first
column the name of the irreproducible representation; on the lines the symmetry
operations and the associated character of the matrix representation in a given base.
Third and fourth columns report the bases of the irreproducible representation.
Properties of Vectors
A vector in a Cartesian space can be specified by the length of its projections of the
orthogonal axes. An important operation between vectors is their product, whose
one type is called scalar product. Indicating the vectors in bold characters, it is
given by
A . B = AB cos h where A and B are the length of the two vectors and h is the
angle between the two vectors.
When we refer to a x y coordinate systems as in Fig. A.1, the angle h becomes
e − /. Thus, the direct product becomes A . B = AB (cos e − /).
Considering the projections of the vectors on the x- and y-axes,
A x ¼ A cos u B x ¼ B cos e
A y ¼ A sin u B y ¼ B sin e
A . B = AB (cos u cos e + sin u sin e) = A cos u B cos e + A sin u B sin e = A x
B x + A y B y .
The direct product of two vectors is the products of all their components along
the axes of the space.
180
Appendix
It seems important to outline that the tables of characters report in the first
column the name of the irreproducible representation; on the lines the symmetry
operations and the associated character of the matrix representation in a given base.
Third and fourth columns report the bases of the irreproducible representation.
Properties of Vectors
A vector in a Cartesian space can be specified by the length of its projections of the
orthogonal axes. An important operation between vectors is their product, whose
one type is called scalar product. Indicating the vectors in bold characters, it is
given by
A . B = AB cos h where A and B are the length of the two vectors and h is the
angle between the two vectors.
When we refer to a x y coordinate systems as in Fig. A.1, the angle h becomes
e − /. Thus, the direct product becomes A . B = AB (cos e − /).
Considering the projections of the vectors on the x- and y-axes,
A x ¼ A cos u B x ¼ B cos e
A y ¼ A sin u B y ¼ B sin e
A . B = AB (cos u cos e + sin u sin e) = A cos u B cos e + A sin u B sin e = A x
B x + A y B y .
The direct product of two vectors is the products of all their components along
the axes of the space.
180
Appendix