The orthogonality theorem for irreducible representations

REPRESENTATION THEORY
of simple pair interchanges to which a permutation is equivalent. That these
assignments are in accord with the group multiplication table 28.8 should be
checked.
Thus the three irreps of the group G (i.e. the group 3m or C3v or S3 ), are, using
the conventional notation A1 , A2 , E (see section 29.8), as follows:
A1
Irrep A2
E
where
MI =
,
−1 0
0
0 1
MC =
1 0
I
1
1
MI
1
MA =
MD =
Element
B
C
1
1
1
−1
MB MC
− 12
−
,
A
1
1
MA
√
3
2
1
2√
− 23
√
3
2
− 12
√
3
2
− 21
−
D
1
−1
MD
E
1
−1
ME
,
,
ME =
− 12
√
3
2
MB =
(29.12)
1
√2
3
2
√
3
2
− 12
−
√
3
2
− 12
,
.
29.5 The orthogonality theorem for irreducible representations
We come now to the central theorem of representation theory, a theorem that
justifies the relatively routine application of certain procedures to determine
the restrictions that are inherent in physical systems that have some degree of
rotational or reflection symmetry. The development of the theorem is long and
quite complex when presented in its entirety, and the reader will have to refer
elsewhere for the proof.§
The theorem states that, in a certain sense, the irreps of a group G are as
orthogonal as possible, as follows. If, for each irrep, the elements in any one
position in each of the g matrices are used to make up g-component column
matrices then
(i) any two such column matrices coming from different irreps are orthogonal;
(ii) any two such column matrices coming from different positions in the
matrices of the same irrep are orthogonal.
This orthogonality is in addition to the irreps’ being in the form of orthogonal (unitary) matrices and thus each comprising mutually orthogonal rows and
columns.
§
See, e.g., H. F. Jones, Groups, Representations and Physics (Bristol: Institute of Physics, 1998); J.
F. Cornwell, Group Theory in Physics, vol 2 (London: Academic Press, 1984); J-P. Serre, Linear
Representations of Finite Groups (New York: Springer, 1977).
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29.5 THE ORTHOGONALITY THEOREM FOR IRREDUCIBLE REPRESENTATIONS
More mathematically, if we denote the entry in the ith row and jth column of a
(λ)
(µ)
matrix D(X) by [D(X)]ij , and D̂ and D̂ are two irreps of G having dimensions
nλ and nµ respectively, then
∗ (µ)
(λ)
g
(29.13)
D̂ (X)
D̂ (X) = δik δjl δλµ .
ij
kl
n
λ
X
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 if i = k, j = l and λ = µ, all at the same time.
Secondly, the summation over the group elements X means that g contributions
have to be added together, each contribution being a product of entries drawn
(λ)
(λ)
(µ)
from the representative matrices in the two irreps D̂ = {D̂ (X)} and D̂ =
(µ)
{D̂ (X)}. The g contributions arise as X runs over the g elements of G.
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
(λ)
(µ)
irreps D̂ and D̂ have different dimensions, or
(λ)
(µ)
(ii) even if D̂ and D̂ 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 A1 , A2 and E of the group 3m
(or C3v or S3 ) will probably provide the clearest explanation (see (29.12)).
(a) Take i = j = k = l = 1, with D̂
then reads
(λ)
= A1 and D̂
(µ)
= A2 . Equation (29.13)
1(1) + 1(1) + 1(1) + 1(−1) + 1(−1) + 1(−1) = 0,
as expected, since λ = µ.
(b) Take (i, j) as (1, 2) and (k, l) as (2, 2), corresponding to different matrix
(λ)
(µ)
positions within the same irrep D̂ = D̂ = E. Substituting in (29.13)
gives
√ √ √ √ 0(1) + − 23 − 12 + 23 − 12 + 0(1) + − 23 − 12 + 23 − 12 = 0.
(c) Take (i, j) as (1, 2), and (k, l) as (1, 2), corresponding to the same matrix
(λ)
(µ)
positions within the same irrep D̂ = D̂ = E. Substituting in (29.13)
gives
√ √ √ √ √ √ √ √ 3
3
+ 0(0) + − 23 − 23 + 23
= 62 .
0(0) + − 23 − 23 + 23
2
2
1091