Great orthogonality theorem

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The great orthogonality theorem (GOT) defines properties of irreducible matrix representations of compact Lie groups. It is the central theorem of the representation theory of such groups. Examples of compact Lie groups are: finite groups, such as point groups and permutation groups, and continuous groups with a bounded parameter space, such as the rotation group SO(3). These groups have in common that continuous maps of the group into C or R—functions on the group—may be summed/integrated over the group (give finite results).

1 Finite groups
2 Continuous groups
- 2.1 References

[edit] Finite groups

Let $Γ (λ) (R) m n$ be a matrix element of an irreducible matrix representation (irrep) $Γ (λ)$ of the finite group $G = {R}$ , which is of order |G|, i.e., G has |G| elements. Since it can be proved that the matrix representation of any finite group may be chosen to be unitary, we assume this to be the case,

$\sum_{n=1}^{l_\lambda} \; \Gamma^{(\lambda)} (R)_{nm}^*\;\Gamma^{(\lambda)} (R)_{nk} = \delta_{mk} \quad \hbox{for all}\quad R \in G,$

where $l λ$ is the dimension of irrep $Γ (λ)$ —a finite number.^[1]

The great orthogonality relations (only valid for matrix elements in irreducible representations) are:

$\sum_{R\in G}^{|G|} \; \Gamma^{(\lambda)} (R)_{nm}^*\;\Gamma^{(\mu)} (R)_{n'm'} = \delta_{\lambda\mu} \delta_{nn'}\delta_{mm'} \frac{|G|}{l_\lambda}.$

Here $\Gamma^{(\lambda)} (R)_{nm}^*$ is the complex conjugate of $\Gamma^{(\lambda)} (R)_{nm}\,$ and the sum is over all elements of G. The Kronecker delta $δ λμ$ is unity if the matrices are in the same irreducible representation $Γ (λ) = Γ (μ)$ , otherwise it is zero. The other two Kronecker delta's state that the row and column indices must be equal ( $n = n'$ and $m = m'$ ) in order to obtain a non-vanishing result.

Every group has an identity representation (all group elements mapped onto the real number 1). This obviously is an irreducible representation. The great orthogonality relations immediately imply that

$\sum_{R\in G}^{|G|} \; \Gamma^{(\mu)} (R)_{nm} = 0$

for $n,m=1,\ldots,l_\mu$ and any irrep $\Gamma^{(\mu)}\,$ not equal to the identity irrep.

[edit] Example

The 3! permutations of three objects form a group of order 6, commonly denoted by $S 3$ (symmetric group). This group is isomorphic to the point group $C 3 v$ , consisting of a three-fold rotation axis and three vertical mirror planes. The groups have a 2-dimensional irrep (l = 2). In the case of $S 3$ one usually labels this irrep by the Young tableau $λ = [2,1]$ and in the case of $C 3 v$ one usually writes $λ = E$ . In both cases the irrep consists of the following six real matrices, each representing a single group element:^[2]

$\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} \quad \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix} \quad \begin{pmatrix} -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2}& \frac{1}{2} \\ \end{pmatrix} \quad \begin{pmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2}& \frac{1}{2} \\ \end{pmatrix} \quad \begin{pmatrix} -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2}& -\frac{1}{2} \\ \end{pmatrix} \quad \begin{pmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2}& -\frac{1}{2} \\ \end{pmatrix}$

The normalization of the (1,1) element:

$\sum_{R\in G}^{6} \; \Gamma(R)_{11}^*\;\Gamma(R)_{11} = 1^2+1^2+\left(-\tfrac{1}{2}\right)^2+\left(-\tfrac{1}{2}\right)^2 +\left(-\tfrac{1}{2}\right)^2 +\left(-\tfrac{1}{2}\right)^2 = 3 .$

In the same manner one can show the normalization of the other matrix elements: (2,2), (1,2), and (2,1). The orthogonality of the (1,1) and (2,2) elements:

$\sum_{R\in G}^{6} \; \Gamma(R)_{11}^*\;\Gamma(R)_{22} = 1^2+(1)(-1)+\left(-\tfrac{1}{2}\right)\left(\tfrac{1}{2}\right) +\left(-\tfrac{1}{2}\right)\left(\tfrac{1}{2}\right) +\left(-\tfrac{1}{2}\right)^2 +\left(-\tfrac{1}{2}\right)^2 = 0 .$

Similar relations hold for the orthogonality of the elements (1,1) and (1,2), etc. One verifies easily in the example that all sums of corresponding matrix elements vanish because of the orthogonality of the given irrep to the identity irrep.

[edit] Footnotes

^ The finiteness of $l λ$ follows from the fact that the regular representation of a finite group G is a reducible representation of dimension |G|. Further, it can be shown that any irrep of G is contained in the regular representation of G and so it follows that finite groups have finite-dimensional irreps.
^ This choice is not unique, any orthogonal similarity transformation applied to the matrices gives a valid irrep.

[edit] Direct implications

The trace of a matrix is a sum of diagonal matrix elements,

$\operatorname{Tr}\big(\Gamma(R)\big) = \sum_{m=1}^{l} \Gamma(R)_{mm}$ .

The collection of traces is the character $\chi \equiv \{\operatorname{Tr}\big(\Gamma(R)\big)\;|\; R \in G\}$ of a representation. Often one writes for the trace of a matrix in an irrep with character $χ (λ)$

$\chi^{(\lambda)} (R)\equiv \operatorname{Tr}\left(\Gamma^{(\lambda)}(R)\right)$ .

In this notation we can write several character formulas:

$\sum_{R\in G}^{|G|} \chi^{(\lambda)}(R)^* \, \chi^{(\mu)}(R)= \delta_{\lambda\mu} |G|,$

which allows us to check the whether or not a representation is irreducible. And

$\sum_{R\in G}^{|G|} \chi^{(\lambda)}(R)^* \, \chi(R) = n^{(\lambda)} |G|,$

which helps us to determine how often the irrep $Γ (λ)$ is contained within the reducible representation $\Gamma \,$ with character $χ(R)$ .

For instance, if

$n^{(\lambda)}\, |G| = 96$

and the order of the group is

$|G| = 24\,$

then the number of times that $\Gamma^{(\lambda)}\,$ is contained within the given reducible representation $\Gamma \,$ is

$n^{(\lambda)} = 4\, .$

See Character theory for more about group characters.

[edit] Continuous groups

The generalization of the great orthogonality relations from finite groups to continuous compact Lie groups is basically simple: Replace the summation over the group by an integration over the group. This generalization requires (i) variables to integrate over and (ii) a volume element expressed in these variables.

An r parameter continuous Lie group has the general element $R(\mathbf{x})$ , where the vector of parameters x is in R^r. In the case of a compact group the r parameters $x_i\,$ are bounded: $x^0_i \le x_i \le x_i^1$ , where the bounds $x^0_i$ and $x^1_i$ are real finite numbers.

An example of an r = 3 parameter group is the matrix group SO(3) consisting of all 3 x 3 orthogonal matrices with unit determinant. A possible parametrization of this group is in terms of Euler angles: $\mathbf{x} = (\alpha, \beta, \gamma)$ (see e.g., this article for the explicit from of an element of SO(3) in terms of Euler angles). The bounds are $0 \le\alpha, \gamma \le 2\pi$ and $0 \le \beta \le\pi$ .

Not only the recipe for the computation of the volume element $\omega(\mathbf{x})\, dx_1 dx_2\cdots dx_r$ depends on the chosen parameters, but also the final result, i.e., the analytic form of the weight function (measure) $\omega(\mathbf{x})$ .

For instance, the Euler angle parametrization of SO(3) gives the weight $\omega(\alpha,\beta,\gamma) = \sin \beta \,$ , while the n, ψ parametrization gives the weight $\omega(\psi,\theta,\phi) = 2(1-\cos\psi)\sin\theta \,$ with $0\le \psi \le \pi, \;\; 0 \le\phi\le 2\pi,\;\; 0 \le \theta \le \pi$ .

It can be shown that the irreducible matrix representations of compact Lie groups are finite-dimensional and can be chosen to be unitary:

$\Gamma^{(\lambda)}(R^{-1}) =\Gamma^{(\lambda)}(R)^{-1}=\Gamma^{(\lambda)}(R)^\dagger\quad \hbox{with}\quad \Gamma^{(\lambda)}(R)^\dagger_{mn} \equiv \Gamma^{(\lambda)}(R)^*_{nm}.$

With the short-hand notation

$\Gamma^{(\lambda)}(\mathbf{x})= \Gamma^{(\lambda)}\Big(R(\mathbf{x})\Big)$

the great orthogonality relations take the form

$\int_{x_1^0}^{x_1^1} \cdots \int_{x_r^0}^{x_r^1}\; \Gamma^{(\lambda)}(\mathbf{x})^*_{nm} \Gamma^{(\mu)}(\mathbf{x})_{n'm'}\; \omega(\mathbf{x}) dx_1\cdots dx_r \; = \delta_{\lambda \mu} \delta_{n n'} \delta_{m m'} \frac{|G|}{l_\lambda},$

with the volume of the group:

$|G| = \int_{x_1^0}^{x_1^1} \cdots \int_{x_r^0}^{x_r^1} \omega(\mathbf{x}) dx_1\cdots dx_r .$

As an example we note that the irreps of SO(3) are Wigner D-matrices $D^\ell(\alpha \beta \gamma)$ , which are of dimension $2\ell+1$ . Since

$|SO(3)| = \int_{0}^{2\pi} d\alpha \int_{0}^{\pi} \sin\beta d\beta \int_{0}^{2\pi} d\gamma = 8\pi^2,$

they satisfy

$\int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{2\pi} D^{\ell}(\alpha \beta\gamma)^*_{nm} \; D^{\ell'}(\alpha \beta\gamma)_{n'm'}\; \sin\beta d\alpha d\beta d\gamma = \delta_{\ell\ell'}\delta_{nn'}\delta_{mm'} \frac{8\pi^2}{2\ell+1}.$

[edit] References

Any physically or chemically oriented book on group theory mentions the great orthogonality relations. The following more advanced books give the proofs:

M. Hamermesh, Group Theory and its Applications to Physical Problems, Addison-Wesley, Reading (1962).
W. Miller, Jr., Symmetry Groups and their Applications, Academic Press, New York (1972).
J. F. Cornwell, Group Theory in Physics, (Three volumes), Volume 1, Academic Press, New York (1997).

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Category: Representation theory of groups