|
|
| Line 1: |
Line 1: |
| In mathematics, the '''Schur orthogonality relations''' express a central fact about [[group representation|representations]] of finite [[group (mathematics)|groups]].
| | Hello, my name is Felicidad but I don't like when individuals use my complete name. My occupation is a messenger. Kansas is exactly where her house is but she requirements to transfer simply because of her family. To play badminton is some thing he really enjoys performing.<br><br>Also visit my homepage extended auto warranty, [http://Www.gettingtherefromhere.info/User-Profile/userId/13851 resource for this article], |
| They admit a generalization to the case of [[compact group]]s in general, and in particular [[compact group|compact Lie groups]], such as the
| |
| [[Rotation group SO(3)|rotation group ''SO''(3)]].
| |
| | |
| ==Finite groups==
| |
| | |
| ===Intrinsic statement===
| |
| The space of complex-valued [[class function]]s of a finite group G has a natural [[inner product space|inner product]]:
| |
| | |
| :<math>\left \langle \alpha, \beta\right \rangle := \frac{1}{ \left | G \right | }\sum_{g \in G} \alpha(g) \overline{\beta(g)}</math>
| |
| | |
| where <math>\overline{\beta(g)}</math> means the complex conjugate of the value of <math>\beta</math> on ''g''. With respect to this inner product, the irreducible [[character theory|characters]] form an orthonormal basis
| |
| for the space of class functions, and this yields the orthogonality relation for the rows of the character
| |
| table:
| |
| | |
| :<math>\left \langle \chi_i, \chi_j \right \rangle = \begin{cases} 0 & \mbox{ if } i \ne j, \\ 1 & \mbox{ if } i = j. \end{cases}</math>
| |
| | |
| For <math>g, h \in G</math> the orthogonality relation for columns is as follows:
| |
| | |
| :<math>\sum_{\chi_i} \chi_i(g) \overline{\chi_i(h)} = \begin{cases} \left | C_G(g) \right |, & \mbox{ if } g, h \mbox{ are conjugate } \\ 0 & \mbox{ otherwise.}\end{cases}</math>
| |
| | |
| where the sum is over all of the irreducible characters <math>\chi_i</math> of ''G'' and the symbol <math>\left | C_G(g) \right |</math> denotes the order of the [[centralizer]] of <math>g</math>.
| |
| | |
| The orthogonality relations can aid many computations including:
| |
| * decomposing an unknown character as a linear combination of irreducible characters;
| |
| * constructing the complete character table when only some of the irreducible characters are known;
| |
| * finding the orders of the centralizers of representatives of the conjugacy classes of a group; and
| |
| * finding the order of the group.
| |
| | |
| ===Coordinates statement===
| |
| Let <math>\Gamma^{(\lambda)} (R)_{mn}</math> be a [[Matrix (mathematics)|matrix]] element of an [[irreducible representation|irreducible]] [[Group representation|matrix representation]]
| |
| <math>\Gamma^{(\lambda)}</math> of a finite group <math>G=\{R\}</math> of order |''G''|, i.e., ''G'' has |''G''| elements. Since it can be proven that any matrix representation of any finite group is equivalent to a [[unitary representation]], we assume <math>\Gamma^{(\lambda)}</math> is unitary:
| |
| :<math>
| |
| \sum_{n=1}^{l_\lambda} \; \Gamma^{(\lambda)} (R)_{nm}^*\;\Gamma^{(\lambda)} (R)_{nk} = \delta_{mk} \quad \hbox{for all}\quad R \in G,
| |
| </math>
| |
| where <math>l_\lambda</math> is the (finite) dimension of the irreducible representation <math>\Gamma^{(\lambda)}</math>.<ref>The finiteness of <math>l_\lambda</math> follows from the fact that any irreducible representation of a finite group ''G'' is contained in the [[regular representation]].</ref> | |
| | |
| The '''orthogonality relations''', only valid for matrix elements of ''irreducible'' representations, are:
| |
| | |
| :<math>
| |
| \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}.
| |
| </math>
| |
| | |
| Here <math>\Gamma^{(\lambda)} (R)_{nm}^*</math> is the complex conjugate of <math>\Gamma^{(\lambda)} (R)_{nm}\,</math> and the sum is over all elements of ''G''.
| |
| The [[Kronecker delta]] <math>\delta_{\lambda\mu}</math> is unity if the matrices are in the same irreducible representation <math>\Gamma^{(\lambda)}= \Gamma^{(\mu)}</math>. If <math>\Gamma^{(\lambda)}</math> and <math>\Gamma^{(\mu)}</math> are non-equivalent
| |
| it is zero. The other two Kronecker delta's state that
| |
| the row and column indices must be equal (<math>n=n'</math> and <math>m=m'</math>) in order to obtain a non-vanishing result. This theorem is also known as the Great (or Grand) Orthogonality Theorem.
| |
| | |
| Every group has an identity representation (all group elements mapped onto the real number 1).
| |
| This is an irreducible representation. The great orthogonality relations immediately imply that
| |
| :<math>
| |
| \sum_{R\in G}^{|G|} \; \Gamma^{(\mu)} (R)_{nm} = 0
| |
| </math>
| |
| for <math>n,m=1,\ldots,l_\mu</math> and any irreducible representation <math>\Gamma^{(\mu)}\,</math> not equal to the identity representation.
| |
| | |
| ===Example of the permutation group on 3 objects===
| |
| | |
| The 3! permutations of three objects form a group of order 6, commonly denoted by <math>S_3</math> ([[symmetric group]]). This group is isomorphic to the [[Point groups in three dimensions#The seven infinite series|point group]] <math>C_{3v}</math>, consisting of a threefold rotation axis and three vertical mirror planes. The groups have a 2-dimensional irreducible representation (''l'' = 2). In the case of <math>S_3</math> one usually labels this representation
| |
| by the [[Young tableau]] <math> \lambda = [2,1]</math> and in the case of <math>C_{3v}</math>
| |
| one usually writes <math> \lambda = E</math>. In both cases the representation consists of the following six real matrices, each representing a single group element:<ref>This choice is not unique, any orthogonal similarity transformation
| |
| applied to the matrices gives a valid irreducible representation.</ref>
| |
| :<math>
| |
| \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}
| |
| </math>
| |
| The normalization of the (1,1) element:
| |
| :<math> \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 .
| |
| </math>
| |
| 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:
| |
| :<math> \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 .
| |
| </math>
| |
| 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 irreducible representation to the identity representation.
| |
| | |
| ===Direct implications===
| |
| The [[trace (linear algebra)|trace]] of a matrix is a sum of diagonal matrix elements,
| |
| | |
| :<math>\operatorname{Tr}\big(\Gamma(R)\big) = \sum_{m=1}^{l} \Gamma(R)_{mm}.</math>
| |
| The collection of traces is the ''character'' <math>\chi \equiv \{\operatorname{Tr}\big(\Gamma(R)\big)\;|\; R \in G\}</math> of a representation. Often one writes for
| |
| the trace of a matrix in an irreducible representation with character <math>\chi^{(\lambda)}</math>
| |
| | |
| :<math>\chi^{(\lambda)} (R)\equiv \operatorname{Tr}\left(\Gamma^{(\lambda)}(R)\right).</math>
| |
| | |
| In this notation we can write several character formulas:
| |
| | |
| :<math>\sum_{R\in G}^{|G|} \chi^{(\lambda)}(R)^* \, \chi^{(\mu)}(R)= \delta_{\lambda\mu} |G|,</math>
| |
| | |
| 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
| |
| | |
| :<math>\sum_{R\in G}^{|G|} \chi^{(\lambda)}(R)^* \, \chi(R) = n^{(\lambda)} |G|,</math>
| |
| | |
| which helps us to determine how often the irreducible representation <math>\Gamma^{(\lambda)}</math> is contained within the reducible representation <math>\Gamma \,</math> with character <math>\chi(R)</math>.
| |
| | |
| For instance, if
| |
|
| |
| :<math>n^{(\lambda)}\, |G| = 96</math>
| |
| | |
| and the order of the group is
| |
| | |
| :<math>|G| = 24\,</math>
| |
| | |
| then the number of times that <math>\Gamma^{(\lambda)}\,</math> is contained within the given
| |
| ''reducible'' representation <math>\Gamma \,</math> is
| |
| | |
| :<math>n^{(\lambda)} = 4\, .</math>
| |
| | |
| ''See [[Character theory]] for more about group characters.''
| |
| <!---
| |
| | |
| I don't understand the meaning of the following lines [P.wormer]
| |
| or
| |
| | |
| :<math>n_j = 1^2 1^2 1^2 1^2. \,</math> | |
| | |
| And lastly,
| |
| | |
| :<math>\sum_{\hat R}^N {\begin{vmatrix}{\Chi_i(\hat R)} \end{vmatrix}}^2 = h \sum_{j}^N n_j^2</math>
| |
| -->
| |
| | |
| ==Compact Groups==
| |
| 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 <math>G</math> has unique bi-invariant [[Haar measure]], so that the volume of the group is 1. Denote this measure by <math>dg</math>. Let <math>( \pi^\alpha )</math> be a complete set of irreducible representations of <math>G</math>, and let <math>\phi^\alpha_{v,w}(g)=\langle v,\pi^\alpha(g)w\rangle </math> be a [[matrix coefficient]] of the representation <math>\pi^\alpha</math>. The orthogonality relations can the be stated in two parts:
| |
| | |
| 1) If <math>\pi^\alpha \ncong \pi^\beta </math> then
| |
| :<math>
| |
| \int_G \phi^\alpha_{v,w}(g)\phi^\beta_{v',w'}(g)dg=0
| |
| </math>
| |
| | |
| 2) If <math>\{e_i\}</math> is an [[orthonormal basis]] of the representation space <math>\pi^\alpha</math> then
| |
| :<math>
| |
| d^\alpha\int_G \phi^\alpha_{e_i,e_j}(g)\overline{\phi^\alpha_{e_m,e_n}(g)}dg=\delta_{i,m}\delta_{j,n}
| |
| </math>
| |
| where <math>d^\alpha</math> is the dimension of <math>\pi^\alpha</math>. These orthogonality relations and the fact that all of the representations have finite dimensions are consequences of the [[Peter-Weyl theorem]]
| |
| | |
| ===An Example SO(3)===
| |
| | |
| 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: <math>\mathbf{x} = (\alpha, \beta, \gamma)</math> (see e.g., this article for the explicit form of an element of SO(3) in terms of Euler angles). The bounds are <math>0 \le\alpha, \gamma \le 2\pi</math> and <math>0 \le \beta \le\pi</math>.
| |
| | |
| Not only the recipe for the computation of the volume element <math> \omega(\mathbf{x})\, dx_1 dx_2\cdots dx_r </math> depends on the chosen parameters, but also the final result, i.e., the analytic form of the weight function (measure) <math>\omega(\mathbf{x})</math>.
| |
| | |
| For instance, the Euler angle parametrization of SO(3) gives the weight <math>\omega(\alpha,\beta,\gamma) = \sin\! \beta \,,</math> while the n, ψ parametrization gives the weight <math>\omega(\psi,\theta,\phi) = 2(1-\cos\psi)\sin\!\theta\, </math>
| |
| with <math>0\le \psi \le \pi, \;\; 0 \le\phi\le 2\pi,\;\; 0 \le \theta \le \pi.</math>
| |
| | |
| It can be shown that the irreducible matrix representations of compact Lie groups are finite-dimensional and can be chosen to be unitary:
| |
| :<math>
| |
| \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}.
| |
| </math>
| |
| With the shorthand notation
| |
| :<math>
| |
| \Gamma^{(\lambda)}(\mathbf{x})= \Gamma^{(\lambda)}\Big(R(\mathbf{x})\Big)
| |
| </math>
| |
| the orthogonality relations take the form
| |
| :<math>
| |
| \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},
| |
| </math>
| |
| with the volume of the group:
| |
| :<math>
| |
| |G| = \int_{x_1^0}^{x_1^1} \cdots \int_{x_r^0}^{x_r^1} \omega(\mathbf{x}) dx_1\cdots dx_r .
| |
| </math>
| |
| As an example we note that the irreducible representations of SO(3) are Wigner D-matrices <math>D^\ell(\alpha \beta \gamma)</math>, which are of dimension <math>2\ell+1 </math>. Since
| |
| :<math>
| |
| |SO(3)| = \int_{0}^{2\pi} d\alpha \int_{0}^{\pi} \sin\!\beta\, d\beta \int_{0}^{2\pi} d\gamma = 8\pi^2,
| |
| </math>
| |
| they satisfy
| |
| :<math>
| |
| \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}.
| |
| </math>
| |
| | |
| ==Notes==
| |
| | |
| {{reflist}}
| |
| | |
| ==References ==
| |
| Any physically or chemically oriented book on group theory mentions the orthogonality relations. The following more advanced books give the proofs:
| |
| * M. Hamermesh, ''Group Theory and its Applications to Physical Problems'', Addison-Wesley, Reading (1962). (Reprinted by Dover).
| |
| * 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).
| |
| | |
| [[Category:Representation theory of groups]]
| |