Mathematical Methods for Physics and Engineering : A Comprehensive Guide

29.5 THE ORTHOGONALITY THEOREM FOR IRREDUCIBLE REPRESENTATIONS

More mathematically, if we denote the entry in theith row andjth column of a

matrixD(X)by[D(X)]ij,andDˆ

(λ)

andDˆ

(μ)

are two irreps ofGhaving dimensions

nλandnμrespectively, then

∑

X

[

Dˆ(λ)(X)

]∗

ij

[

Dˆ(μ)(X)

]

kl

=

g

nλ

δikδjlδλμ. (29.13)

This rather forbidding-looking equation needs some further explanation.

Firstly, the asterisk indicates that the complex conjugate should be taken if

necessary, though all our representations so far have involved only real matrix

elements. Each Kronecker delta function on the right-hand side has the value 1

if its two subscripts are equal and has the value 0 otherwise. Thus the right-hand

side is only non-zero ifi=k,j=landλ=μ, all at the same time.

Secondly, the summation over the group elementsXmeans thatgcontributions

have to be added together, each contribution being a product of entries drawn

from the representative matrices in the two irrepsDˆ

(λ)

={Dˆ

(λ)

(X)}andDˆ

(μ)

=

{Dˆ

(μ)

(X)}.Thegcontributions arise asXruns over thegelements ofG.

Thus, putting these remarks together, the summation will produce zero if either

(i) the matrix elements are not taken from exactly the same position in every

matrix, including cases in which it is not possible to do so because the

irrepsDˆ

(λ)

andDˆ

(μ)

have different dimensions, or

(ii) even ifDˆ

(λ)

andDˆ

(μ)

do have the same dimensions and the matrix elements

are from the same positions in every matrix, they are different irreps, i.e.

λ=μ.

Some numerical illustrations based on the irreps A 1 ,A 2 and E of the group 3m

(orC 3 vorS 3 ) will probably provide the clearest explanation (see (29.12)).

(a) Takei=j=k=l= 1, withDˆ

(λ)

=A 1 andDˆ

(μ)

=A 2. Equation (29.13)

then reads

1(1) + 1(1) + 1(1) + 1(−1) + 1(−1) + 1(−1) = 0,

as expected, sinceλ=μ.

(b) Take (i, j)as(1,2) and (k, l)as(2,2), corresponding to different matrix

positions within the same irrepDˆ

(λ)

=Dˆ

(μ)

= E. Substituting in (29.13)

gives

0(1) +

(

−

√

3

2

)(

−^12

)

+

(√

3

2

)(

−^12

)

+0(1)+

(

−

√

3

2

)(

−^12

)

+

(√

3

2

)(

−^12

)

=0.

(c) Take (i, j)as(1,2), and (k, l)as(1,2), corresponding to the same matrix

positions within the same irrepDˆ

(λ)

=Dˆ

(μ)

= E. Substituting in (29.13)

gives

0(0) +

(

−

√

3

2

)(

−

√

3

2

)

+

(√

3

2

)(√

3

2

)

+0(0)+

(

−

√

3

2

)(

−

√

3

2

)

+

(√

3

2

)(√

3

2

)

=^62.