Mathematical Methods for Physics and Engineering : A Comprehensive Guide

REPRESENTATION THEORY

29.6.1 Orthogonality property of characters

Some of the most important properties of characters can be deduced from the

orthogonality theorem (29.13),

∑

X

[

Dˆ(λ)(X)

]∗

ij

[

Dˆ(μ)(X)

]

kl

=

g

nλ

δikδjlδλμ.

If we setj=iandl=k, so that both factors in any particular term in the

summation refer to diagonal elements of the representative matrices, and then

sum both sides overiandk, we obtain

∑

X

∑nλ

i=1

∑nμ

k=1

[

Dˆ(λ)(X)

]∗

ii

[

Dˆ(μ)(X)

]

kk

=

g

nλ

∑nλ

i=1

∑nμ

k=1

δikδikδλμ.

Expressed in term of characters, this reads

∑

X

[

χ(λ)(X)

]∗

χ(μ)(X)=

g

nλ

∑nλ

i=1

δii^2 δλμ=

g

nλ

∑nλ

i=1

1 ×δλμ=gδλμ.

(29.14)

In words, the (g-component) ‘vectors’ formed from the characters of the various

irreps of a group are mutually orthogonal, but each one has a squared magnitude

(the sum of the squares of its components) equal to the order of the group.

Since, as noted in the previous subsection, group elements in the same class

have the same characters, (29.14) can be written as a sum over classes rather than

elements. Ifcidenotes the number of elements in classCiandXiany element of

Ci,then
∑

i

ci

[

χ(λ)(Xi)

]∗

χ(μ)(Xi)=gδλμ. (29.15)

Although we do not prove it here, there also exists a ‘completeness’ relation for

characters. It makes a statement about the products of characters for a fixed pair

of group elements,X 1 andX 2 , when the products are summed over all possible

irreps of the group. This is the converse of the summation process defined by

(29.14). The completeness relation states that

∑

λ

[

χ(λ)(X 1 )

]∗

χ(λ)(X 2 )=

g

c 1

δC 1 C 2 , (29.16)

where elementX 1 belongs to conjugacy classC 1 andX 2 belongs toC 2. Thus the

sum is zero unlessX 1 andX 2 belong to the same class. For table 29.1 we can

verify that these results are valid.

(i) ForDˆ

(λ)

=Dˆ

(μ)

=A 1 or A 2 , (29.15) reads

1(1) + 2(1) + 3(1) = 6,