Mathematical Methods for Physics and Engineering : A Comprehensive Guide

REPRESENTATION THEORY

be non-zero the integrand must be invariant under each of these operations. In

group theoretical language,the integrand must transform as the identity, the one-

dimensional representationA 1 ofG; more accurately, some non-vanishing part of

the integrand must do so.

An alternative way of saying this is that if under the symmetry operations

ofGthe integrand transforms according to a representationDandDdoes not

contain A 1 amongst its irreps then the integralJis necessarily zero. It should be

noted that the converse is not true;JmaybezeroevenifA 1 is present, since the

integral, whilst showing the required invariance, may still have the value zero.

It is evident that we need to establish how to find the irreps that go to make

up a representation of a double or triple product when we already know the

irreps according to which the factors in the product transform. The method is

established by the following theorem.

Theorem. For each element of a group the character in a product representation is

the product of the corresponding characters in the separate representations.

Proof.Suppose that{ui}and{vj}are two sets of basis functions, that transform

under the operations of a groupGaccording to representationsD(λ)andD(μ)

respectively. Denote byuandvthe corresponding basis vectors and letXbe an

element of the group. Then the functions generated fromuiandvjby the action

ofXare calculated as follows, using (29.1) and (29.4):

u′i=Xui=

[(

D(λ)(X)

)T

u

]

i

=

[

D(λ)(X)

]

iiui+

∑

l=i

[(

D(λ)(X)

)T]

il

ul,

v′j=Xvj=

[(

D(μ)(X)

)T

v

]

j

=

[

D(μ)(X)

]

jjvj+

∑

m=j

[(

D(μ)(X)

)T]

jm

vm.

Here[D(X)]ijis just a single element of the matrixD(X)and[D(X)]kk=[DT(X)]kk

is simply a diagonal element from the matrix – the repeated subscript does not

indicate summation. Now, if we take as basis functions for a product represen-

tationDprod(X) the productswk=uivj(where thenλnμvarious possible pairs of

valuesi,jare labelled byk), we have also that

wk′=Xwk=Xuivj=(Xui)(Xvj)

=

[

D(λ)(X)

]

ii

[

D(μ)(X)

]

jjuivj+ terms not involving the productuivj.

This is to be compared with

wk′=Xwk=

[(

Dprod(X)

)T

w

]

k

=

[

Dprod(X)

]

kkwk+

∑

n=k

[(

Dprod(X)

)T]

kn

wn,

whereDprod(X) is the product representation matrix for elementXof the group.

The comparison shows that
[
Dprod(X)

]

kk=

[

D(λ)(X)

]

ii

[

D(μ)(X)

]

jj.