62
4 Representations
Hence, every irrep occurs as many times as its own dimension. The sum of all these
irreducible blocks must yield the regular representation. Thus one has
k
c k dim(Γ k ) =
k
dim(Γ k )
2 =|G|
(4.43)
This result tells us that, in a group of order |G|, there are exactly |G| independent
vectors. Rewriting these vectors in the form of SALCs exhausts all possible symmetries that can be realized in this group. If a group is abelian, every class is a
singleton, and hence the number of classes is equal to the order of the group. In this
case, Eq. (4.43) can be fulfilled only if all irreps are one-dimensional. Hence, in an
abelian group all irreps are one-dimensional.
4.3 Character Tables
In Appendix A, we reproduce the character tables for the point groups and the symmetric groups, following the standard form introduced by Mulliken [1]. The top row
of the table lists the conjugacy classes. In some cases the designations of the symmetry operations can be ambiguous, and additional labels are added, such as h, v,
and d for horizontal, vertical, and dihedral, respectively (see also, e.g., Fig. 3.10).
In the final columns of the tables we list some simple functions, which transform
according to the corresponding irreps. Irreps are denoted by letters that are related
to their degeneracy. A and B stand for one-dimensional irreps, which are symmetric or antisymmetric with respect to some principal symmetry element. E and T are
used for two- and three-dimensional irreps, respectively. Sometimes, in physics textbooks, T is replaced by F . This alphabetical order is then continued for the fourfoldand fivefold-degenerate irreps in the icosahedral symmetry, which are denoted as G
and H . Further subscripts are added to distinguish symmetry characteristics with
respect to secondary symmetry elements. Best known are the g and u subscripts,
which distinguish between even (gerade) and odd (ungerade) symmetries with respect to spatial inversion. Primes or double primes are used to distinguish symmetric versus antisymmetric behavior with respect to a horizontal symmetry plane in
groups such as D (2n+1)h or C (2n+1)h . In addition, numerical indices can appear as
subscripts, such as in A 1 ,A 2 in the C 3v group. It should be clear that this labeling
is somewhat ad hoc, and one should consult the actual tables in order to find out the
precise meaning of the symbols used.
Some point groups, viz. the cyclic groups C n , C (2n+1)h , S 2n , and also T and T h ,
have irreps with complex characters. In these cases, for an irrep Γ k with complex
characters, there will always be a complementary irrep with a complex-conjugate
character string, which is denoted as ¯
Γ k . Hence, one has
χ
¯
Γ k (R) =¯ χ
Γ k (R)
(4.44)
4 Representations
Hence, every irrep occurs as many times as its own dimension. The sum of all these
irreducible blocks must yield the regular representation. Thus one has
k
c k dim(Γ k ) =
k
dim(Γ k )
2 =|G|
(4.43)
This result tells us that, in a group of order |G|, there are exactly |G| independent
vectors. Rewriting these vectors in the form of SALCs exhausts all possible symmetries that can be realized in this group. If a group is abelian, every class is a
singleton, and hence the number of classes is equal to the order of the group. In this
case, Eq. (4.43) can be fulfilled only if all irreps are one-dimensional. Hence, in an
abelian group all irreps are one-dimensional.
4.3 Character Tables
In Appendix A, we reproduce the character tables for the point groups and the symmetric groups, following the standard form introduced by Mulliken [1]. The top row
of the table lists the conjugacy classes. In some cases the designations of the symmetry operations can be ambiguous, and additional labels are added, such as h, v,
and d for horizontal, vertical, and dihedral, respectively (see also, e.g., Fig. 3.10).
In the final columns of the tables we list some simple functions, which transform
according to the corresponding irreps. Irreps are denoted by letters that are related
to their degeneracy. A and B stand for one-dimensional irreps, which are symmetric or antisymmetric with respect to some principal symmetry element. E and T are
used for two- and three-dimensional irreps, respectively. Sometimes, in physics textbooks, T is replaced by F . This alphabetical order is then continued for the fourfoldand fivefold-degenerate irreps in the icosahedral symmetry, which are denoted as G
and H . Further subscripts are added to distinguish symmetry characteristics with
respect to secondary symmetry elements. Best known are the g and u subscripts,
which distinguish between even (gerade) and odd (ungerade) symmetries with respect to spatial inversion. Primes or double primes are used to distinguish symmetric versus antisymmetric behavior with respect to a horizontal symmetry plane in
groups such as D (2n+1)h or C (2n+1)h . In addition, numerical indices can appear as
subscripts, such as in A 1 ,A 2 in the C 3v group. It should be clear that this labeling
is somewhat ad hoc, and one should consult the actual tables in order to find out the
precise meaning of the symbols used.
Some point groups, viz. the cyclic groups C n , C (2n+1)h , S 2n , and also T and T h ,
have irreps with complex characters. In these cases, for an irrep Γ k with complex
characters, there will always be a complementary irrep with a complex-conjugate
character string, which is denoted as ¯
Γ k . Hence, one has
χ
¯
Γ k (R) =¯ χ
Γ k (R)
(4.44)