REPRESENTATION THEORY
column matrices (P R Q)Tand (Q P R)Trespectively. The forms of the representative
matricesMnat(X), (29.2), are now determined by equations such as, for elementE,
Q
P
R
=
010
100
001
P
Q
R
implying that
Dnat(E)=
010
100
001
T=
010
100
001
.
In this way the complete representation is obtained as
Dnat(I)=
100
010
001
, Dnat(A)=
001
100
010
, Dnat(B)=
010
001
100
,
Dnat(C)=
100
001
010
, Dnat(D)=
001
010
100
, Dnat(E)=
010
100
001
.
It should be emphasised that although the group contains six elements this representation
is three-dimensional.
We will concentrate on matrix representations offinitegroups, particularlyrotation and reflection groups (the so-called crystal point groups). The general
ideas carry over to infinite groups, such as the continuous rotation groups, but in
a book such as this, which aims to cover many areas of applicable mathematics,
some topics can only be mentioned and not explored. We now give the formal
definition of a representation.
Definition.A representationD={D(X)}of a groupGis an assignment of a non-
singular squaren×nmatrixD(X)to each elementXbelonging toG, such that
(i)D(I)=In,the unitn×nmatrix,
(ii)D(X)D(Y)=D(XY)for any two elementsXandYbelonging toG,i.e.the
matrices multiply in the same way as the group elements they represent.As mentioned previously, a representation byn×nmatrices is said to be ann-dimensional representationofG. The dimensionnis not to be confused with
g, the order of the group, which gives the number of matrices needed in the
representation, though they might not all be different.
A consequence of the two defining conditions for a representation is that thematrix associated with the inverse ofXis the inverse of the matrix associated
withX. This follows immediately from settingY=X−^1 in (ii):
D(X)D(X−^1 )=D(XX−^1 )=D(I)=In;hence
D(X−^1 )=[D(X)]−^1.