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5.8. SOME FACTS FROM GROUP THEORY 231

If all matrices of a representation have the form of size n × n

A 1 O
,
O A 2
then the square matrices A 1 and A 2 form representations, the sum of their dimensions
being equal to n. In this case, the representation is said to be completely reducible.The
representation induced on an invariant space by a given representation D(G) is called a part
of the representation D(G).
A representation D(G) of a group G is said to be irreducible if it has only two invariant
n
subspaces, V and O. Otherwise, it is said to be reducible. Any representation can be
expressed in terms of irreducible representations.

5.8.3-4. Characters.
Let D(G)be an n-dimensional representation of a group G,and let [D ij (g)] be the matrix
of the operator corresponding to the element g G.The character of an element g G in
the representation D(G)is defined by

n

χ(g)= D ii (g) = Tr([D ij (g)]).
i=1

Thus, the character of an element does not depend on the representation basis and is,
therefore, an invariant quantity.
An element b G is said to be conjugate to the element a G if there exists u G such
that
uau –1 = b.
Properties of conjugate elements:
1. Any element is conjugate to itself.
2. If b is conjugate to a,then a is conjugate to b.
3. If b is conjugate to a and c is conjugate to b,then c is conjugate to a.
The characters of all elements belonging to one and the same class of conjugate elements
coincide. The characters of elements for equivalent representations coincide.

5.8.3-5. Examples of group representations.

1 .Let G be a group of symmetry of three-dimensional space consisting of two elements:
◦
the identity transformation I and the reflection P with respect to the origin, G = {I, P}.
The multiplication of elements of this group is described by the table

I P
I I P
P P I

1. One-dimensional representation of the group G.
1
In the space E , we chose a basis e 1 and consider the matrix A (1) of the nondegenerate
1
1
transformation A of this space: A (1) =(1). The transformation A forms a subgroup in the