Mathematical Methods for Physics and Engineering : A Comprehensive Guide

REPRESENTATION THEORY

(a)

IAB

I IAB

A AB I

B BIA

(b)

IAB

I IAB

B BIA

A AB I

Table 29.3 (a) The multiplication table of the cyclic group of order 3, and

(b) its reordering used to generate the regular representation of the group.

the same multiplication structure as the groupGitself, i.e. they form a faithful

representation ofG.

Although not part of the proof, a simple example may help to make these

ideas more transparent. Consider the cyclic group of order 3. Its multiplication

table is shown in table 29.3(a) (a repeat of table 28.10(a) of the previous chapter),

whilst table 29.3(b) shows the same table reordered so that the columns are

still labelled in the orderI,A,Bbut the rows are now labelled in the order

I−^1 =I, A−^1 =B, B−^1 =A. The three matrices of the regular representation are

then

Dreg(I)=





100

010

001



, Dreg(A)=





010

001

100



, Dreg(B)=





001

100

010



.

An alternative, more mathematical, definition of the regular representation of a

group is

[

Dreg(Gk)

]

ij=

{

1ifGkGj=Gi,

0otherwise.

We now return to the proof. With the construction given, the regular representa-

tionhascharactersasfollows:

χreg(I)=g, χreg(X)=0 ifX=I.

We now apply (29.18) toDregto obtain for the numbermμof times that the irrep

Dˆ(μ)appears inDreg(see 29.11))

mμ=

1

g

∑

X

[

χ(μ)(X)

]∗

χreg(X)=

1

g

[

χ(μ)(I)

]∗

χreg(I)=

1

g

nμg=nμ.

Thus an irrepDˆ

(μ)

of dimensionnμappearsnμtimes inDreg, and so by counting

the total number of basis functions, or by consideringχreg(I), we can conclude