4.6 Subduction and Induction
73
Hence, the four sites of the square are denoted by the coset generators as
a, ˆ
C 4 , ˆ
C 2
4 , and ˆ
C 3
4 . Now we also introduce a functional basis on site a,
which is represented by the irrep γ , with component labeling m γ :
ˆ
h A |γm γ ; a=
m ′
γ
|γm
′
γ ; aD
γ
m ′
γ m γ
(h A )
(4.70)
The coset generators will once again take this functional space around in an orbit
which visits all the equivalent sites. Local basis sets are thus defined as
|γm γ ; κ= ˆ
g κ |γm γ ; a
(4.71)
The total induction space is the sum of all these basis sets on the different sites.
As we have seen, the operators of the group act transitively on the cosets. This
means that the cosets are permuted among themselves. The permutation matrix is
denoted as P(g). One has
ˆ
g( ˆ
g κ H A ) =
λ
P λκ (g) ˆ
g λ H A
(4.72)
with
P λκ (g) = 1i f ˆ
g( ˆ
g κ H A ) =ˆ g λ H A
P λκ (g) = 0i f ˆ
g( ˆ
g κ H A ) = ˆ
g λ H A
(4.73)
This permutational representation is also called the ground representation. It describes the transformation of the coset space. The dimension of this coset space is
|G|/|H |. In the case of a cluster, where each coset corresponds to a site, it represents the permutation of the positions of the sites. For this reason, it is also called
the positional representation. Indeed, Eq. (4.72) may equally well be written as
ˆ
gκ=
λ
P λκ (g)λ
(4.74)
For the λ-value, which marks the position of the nonzero element in the κth column
of the matrix P, the product ˆ
g
−1
λ ˆ
g ˆ
g κ is an element of H A . We call this the subelement
of ˆ
g in H A . As an example, for the case of the pyramidal complex, the matrices of
the positional representation are listed in Table 4.6.I f ˆ
g =ˆ g κ ˆ
h ˆ
g −1
κ , the diagonal
element will be nonzero: P κκ (g) = 1. The following sum rules will thus hold, as
can be verified from Table 4.6:
κ
P κκ
g κ hg
−1
κ
=
|G|
|H |
h∈H
P κκ
g κ hg
−1
κ
=|H |
(4.75)
73
Hence, the four sites of the square are denoted by the coset generators as
a, ˆ
C 4 , ˆ
C 2
4 , and ˆ
C 3
4 . Now we also introduce a functional basis on site a,
which is represented by the irrep γ , with component labeling m γ :
ˆ
h A |γm γ ; a=
m ′
γ
|γm
′
γ ; aD
γ
m ′
γ m γ
(h A )
(4.70)
The coset generators will once again take this functional space around in an orbit
which visits all the equivalent sites. Local basis sets are thus defined as
|γm γ ; κ= ˆ
g κ |γm γ ; a
(4.71)
The total induction space is the sum of all these basis sets on the different sites.
As we have seen, the operators of the group act transitively on the cosets. This
means that the cosets are permuted among themselves. The permutation matrix is
denoted as P(g). One has
ˆ
g( ˆ
g κ H A ) =
λ
P λκ (g) ˆ
g λ H A
(4.72)
with
P λκ (g) = 1i f ˆ
g( ˆ
g κ H A ) =ˆ g λ H A
P λκ (g) = 0i f ˆ
g( ˆ
g κ H A ) = ˆ
g λ H A
(4.73)
This permutational representation is also called the ground representation. It describes the transformation of the coset space. The dimension of this coset space is
|G|/|H |. In the case of a cluster, where each coset corresponds to a site, it represents the permutation of the positions of the sites. For this reason, it is also called
the positional representation. Indeed, Eq. (4.72) may equally well be written as
ˆ
gκ=
λ
P λκ (g)λ
(4.74)
For the λ-value, which marks the position of the nonzero element in the κth column
of the matrix P, the product ˆ
g
−1
λ ˆ
g ˆ
g κ is an element of H A . We call this the subelement
of ˆ
g in H A . As an example, for the case of the pyramidal complex, the matrices of
the positional representation are listed in Table 4.6.I f ˆ
g =ˆ g κ ˆ
h ˆ
g −1
κ , the diagonal
element will be nonzero: P κκ (g) = 1. The following sum rules will thus hold, as
can be verified from Table 4.6:
κ
P κκ
g κ hg
−1
κ
=
|G|
|H |
h∈H
P κκ
g κ hg
−1
κ
=|H |
(4.75)