In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).
The space of complex-valued class functions of a finite group G has a natural inner product:
\left\langle\alpha,\beta\right\rangle:=
1 | |
\left|G\right| |
\sumg\alpha(g)\overline{\beta(g)}
where
\overline{\beta(g)}
\beta
\left\langle\chii,\chij\right\rangle=\begin{cases}0&ifi\nej,\ 1&ifi=j.\end{cases}
For
g,h\inG
\sum | |
\chii |
\chii(g)\overline{\chii(h)}=\begin{cases}\left|CG(g)\right|&ifg,hareconjugate\ 0&otherwise.\end{cases}
where the sum is over all of the irreducible characters
\chii
G
\left|CG(g)\right|
g
The orthogonality relations can aid many computations including:
Let
\Gamma(λ)(R)mn
\Gamma(λ)
G=\{R\}
\Gamma(λ)
lλ | |
\sum | |
n=1 |
\Gamma(λ)
* \Gamma | |
(R) | |
nm |
(λ)(R)nk=\deltamk \hbox{forall} R\inG,
lλ
\Gamma(λ)
The orthogonality relations, only valid for matrix elements of irreducible representations, are:
|G| | |
\sum | |
R\inG |
\Gamma(λ)
* \Gamma | |
(R) | |
nm |
(\mu)(R)n'm'=\deltaλ\mu\deltann'\deltamm'
|G| | |
lλ |
.
Here
\Gamma(λ)
* | |
(R) | |
nm |
\Gamma(λ)(R)nm
\deltaλ\mu
\Gamma(λ)=\Gamma(\mu)
\Gamma(λ)
\Gamma(\mu)
n=n'
m=m'
Every group has an identity representation (all group elements mapped to 1).This is an irreducible representation. The great orthogonality relations immediately imply that
|G| | |
\sum | |
R\inG |
\Gamma(\mu)(R)nm=0
n,m=1,\ldots,l\mu
\Gamma(\mu)
C3v
λ=[2,1]
C3v
λ=E
\begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} \begin{pmatrix} -
1 | |
2 |
&
\sqrt{3 | |
6 | |
\sum | |
R\inG |
* \Gamma(R) | |
\Gamma(R) | |
11 |
=12+12+\left(-\tfrac{1}{2}\right)2+\left(-\tfrac{1}{2}\right)2+\left(-\tfrac{1}{2}\right)2+\left(-\tfrac{1}{2}\right)2 =3.
6 | |
\sum | |
R\inG |
* \Gamma(R) | |
\Gamma(R) | |
22 |
=12+(1)(-1)+\left(-\tfrac{1}{2}\right)\left(\tfrac{1}{2}\right) +\left(-\tfrac{1}{2}\right)\left(\tfrac{1}{2}\right) +\left(-\tfrac{1}{2}\right)2+\left(-\tfrac{1}{2}\right)2 =0.
The trace of a matrix is a sum of diagonal matrix elements,
\operatorname{Tr}(\Gamma(R))=
l | |
\sum | |
m=1 |
\Gamma(R)mm.
\chi\equiv\{\operatorname{Tr}(\Gamma(R)) | R\inG\}
\chi(λ)
\chi(λ)(R)\equiv\operatorname{Tr}\left(\Gamma(λ)(R)\right).
In this notation we can write several character formulas:
|G| | |
\sum | |
R\inG |
\chi(λ)(R)*\chi(\mu)(R)=\deltaλ\mu|G|,
which allows us to check whether or not a representation is irreducible. (The formula means that the lines in any character table have to be orthogonal vectors.)And
|G| | |
\sum | |
R\inG |
\chi(λ)(R)*\chi(R)=n(λ)|G|,
which helps us to determine how often the irreducible representation
\Gamma(λ)
\Gamma
\chi(R)
For instance, if
n(λ)|G|=96
and the order of the group is
|G|=24
then the number of times that
\Gamma(λ)
\Gamma
n(λ)=4.
See Character theory for more about group characters.
The generalization of the orthogonality relations from finite groups to compact groups (which include compact Lie groups such as SO(3)) is basically simple: Replace the summation over the group by an integration over the group.
Every compact group
G
dg
(\pi\alpha)
G
\alpha | |
\phi | |
v,w |
(g)=\langlev,\pi\alpha(g)w\rangle
\pi\alpha
1) If
\pi\alpha\ncong\pi\beta
\intG
\alpha | |
\phi | |
v,w |
\beta | |
(g)\phi | |
v',w' |
(g)dg=0
2) If
\{ei\}
\pi\alpha
\intG
\alpha | |
\phi | |
ei,ej |
\alpha | |
(g)\overline{\phi | |
em,en |
(g)}dg=\deltai,m\deltaj,n
1 | |
d\alpha |
d\alpha
\pi\alpha
An example of an r = 3 parameter group is the matrix group SO(3) consisting of all 3 × 3 orthogonal matrices with unit determinant. A possible parametrization of this group is in terms of Euler angles:
x=(\alpha,\beta,\gamma)
0\le\alpha,\gamma\le2\pi
0\le\beta\le\pi
Not only the recipe for the computation of the volume element
\omega(x)dx1dx2 … dxr
\omega(x)
For instance, the Euler angle parametrization of SO(3) gives the weight
\omega(\alpha,\beta,\gamma)=\sin\beta,
\omega(\psi,\theta,\phi)=2(1-\cos\psi)\sin\theta
0\le\psi\le\pi, 0\le\phi\le2\pi, 0\le\theta\le\pi.
It can be shown that the irreducible matrix representations of compact Lie groups are finite-dimensional and can be chosen to be unitary:
\Gamma(λ)(R-1)=\Gamma(λ)(R)-1=\Gamma(λ)(R)\dagger \hbox{with} \Gamma(λ)
\dagger | |
(R) | |
mn |
\equiv\Gamma(λ)
* | |
(R) | |
nm |
.
\Gamma(λ)(x)=\Gamma(λ)(R(x))
| |||||||
\int | |||||||
|
…
| |||||||
\int | |||||||
|
\Gamma(λ)
* | |
(x) | |
nm |
\Gamma(\mu)(x)n'm' \omega(x)dx1 … dxr =\deltaλ\deltan\deltam
|G| | |
lλ |
,
|G|=
| |||||||
\int | |||||||
|
…
| |||||||
\int | |||||||
|
\omega(x)dx1 … dxr.
D\ell(\alpha\beta\gamma)
2\ell+1
|SO(3)|=
2\pi | |
\int | |
0 |
d\alpha
\pi | |
\int | |
0 |
\sin\betad\beta
2\pi | |
\int | |
0 |
d\gamma=8\pi2,
2\pi | |
\int | |
0 |
\pi | |
\int | |
0 |
2\pi | |
\int | |
0 |
D\ell(\alpha
* | |
\beta\gamma) | |
nm |
D\ell'(\alpha\beta\gamma)n'm' \sin\betad\alphad\betad\gamma=\delta\ell\ell'\deltann'\deltamm'
8\pi2 | |
2\ell+1 |
.
Any physically or chemically oriented book on group theory mentions the orthogonality relations. The following more advanced books give the proofs:
The following books give more mathematically inclined treatments:
lλ