Mathematical Methods for Physicists: A concise introduction - Site Map

ELEMENTS OF GROUP THEORY
Group representations
In previous sections we have seen some examples of groups which are isomorphic
with matrix groups. Physicists have found that the representation of group
elements by matrices is a very powerful technique. It is beyond the scope of
this text to make a full study of the representation of groups; in this section we
shall make a brief study of this important subject of the matrix representations of
groups.
If to every element of a group G, g 1 ; g 2 ; g 3 ; ...; we can associate a non-singular
square matrix D…g 1 †; D…g 2 †; D…g 3 †; ...; in such a way that
g i g j ˆ g k implies D…g i †D…g j †ˆD…g k †; …12:2†
then these matrices themselves form a group G 0 , which is either isomorphic or
homomorphic to G. The set of such non-singular square matrices is called a
representation of group G. If the matrices are n n, we have an n-dimensional
representation; that is, the order of the matrix is the dimension (or order) of the
representation D n . One trivial example of such a representation is the unit matrix
associated with every element of the group. As shown in Example 12.9, the four
matrices of Problem 12.4 form a two-dimensional representation of the group G
of Problem 12.2.
If there is one-to-one correspondence between each element of G and the matrix
representation group G 0 , the two groups are isomorphic, and the representation is
said to be faithful (or true). If one matrix D represents more than one group
element of G, the group G is homomorphic to the matrix representation group
G 0 and the representation is said to be unfaithful.
Now suppose a representation of a group G has been found which consists of
matrices D ˆ D…g 1 †; D…g 2 †; D…g 3 †; ...; D…g p †, each matrix being of dimension
n. We can form another representation D 0 by a similarity transformation
D 0 …g† ˆS 1 D…g†S;
…12:3†
S being a non-singular matrix, then
D 0 …g i †D 0 …g j †ˆS 1 D…g i †SS 1 D…g j †S
ˆ S 1 D…g i †D…g j †S
ˆ S 1 D…g i g j †S
ˆ D 0 …g i g j †:
In general, representations related in this way by a similarity transformation are
regarded as being equivalent. However, the forms of the individual matrices in the
two equivalent representations will be quite di€erent. With this freedom in the
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