Representations - Group Theory Applied to Chemistry - page 67

This also implies that the diagonal operator P kk is idempotent , i.e., applying it twice
gives exactly the same result as applying it once:

P kk P kk = P kk

(4.60)

Finally, summing over all diagonal projectors gives rise to the unit element:

P ii = E

(4.61)

Γ i

The proof is as follows:

dim (Γ )
i R D ii (R) R

1
| G |

P ii =

Γ i Γ

dim (Γ )

1

χ Γ (R) R

=

R ¯

| |

G

Γ

dim (Γ )χ Γ (R)

1
| G |

R

=

R Γ

δ R,E R =

R

= E (4.62)

Here, we have made use of the fact that the sum over all characters multiplied by the
dimension of the irrep is the character of the regular representation, and this vanishes
for all R except for the unit element, where it is equal to

| G | (see Eq. ( 4.41 )). In

Dirac terminology this reads

Φ i Φ i =

1 (4.63)

Γ i

This relation is also known as the closure relation. It is frequently applied in the
context of the crystal field theory of the lanthanides.

4.6 Subduction and Induction

Many applications are concerned with the reduction of symmetry by external or in-
ternal perturbations. Subduction corresponds to the lowering a symmetry group G
to one of its subgroups, H , and is denoted by G

↓

H . It can consist of a chain of
consecutive symmetry lowerings, following a path of descent in symmetry down
the genealogical tree of the group. In physics a typical form of external symmetry
breaking is through application of a uniform magnetic or electric field. It leads to
a subgroup that is the intersection of the molecular point group and the axial or