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 groupG(i.e. the group 3morC 3 vorS 3 ), are, usingthe conventional notation A 1 ,A 2 , E (see section 29.8), as follows:
Element
IABCDE
A 1 11 1 1 1 1
Irrep A 2 11 1− 1 − 1 − 1
E MI MA MB MC MD ME(29.12)where
MI=(
1001), MA=(
−^12√
3
2
−√
3
2 −1
2), MB=(
−^12 −√
3
√^2
3
2 −1
2),MC=(
− 1001), MD=(
1
2 −√
3
2
−√
3
2 −1
2), ME=(
1
2√
3
√^2
3
2 −1
2).29.5 The orthogonality theorem for irreducible representationsWe 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 groupGare asorthogonal as possible, as follows. If, for each irrep, the elements in any one
positionineachofthegmatricesareusedtomakeupg-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 orthogo-
nal (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).