# 舒尔正交关系

## 有限群

${\displaystyle \Gamma ^{(\lambda )}(R)_{mn}}$ 是一个 |G| 阶（即 G 有 |G| 个元素）有限群 ${\displaystyle G=\{R\}}$ 的一个不可约矩阵表示 ${\displaystyle \Gamma ^{(\lambda )}}$ 的矩阵元素。因为可以证明任何有限群的不可约矩阵表示等价于一个酉表示，我们假设 ${\displaystyle \Gamma ^{(\lambda )}}$ 是酉的：

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

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

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

${\displaystyle n,m=1,\ldots ,l_{\mu }}$ ，此式对任何不等于单位表示的不可约表示 ${\displaystyle \Gamma ^{(\mu )}\,}$成立。

### 例子

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

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

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

### 直接推论

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

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

${\displaystyle \sum _{R\in G}^{|G|}\chi ^{(\lambda )}(R)^{*}\,\chi ^{(\mu )}(R)=\delta _{\lambda \mu }|G|,}$

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

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

${\displaystyle |G|=24\,}$

${\displaystyle \Gamma ^{(\lambda )}\,}$ 在给定“可约”表示 ${\displaystyle \Gamma \,}$ 中包含的次数是

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

## 紧群

${\displaystyle \int _{G}\phi _{v,w}^{\alpha }(g)\phi _{v',w'}^{\beta }(g)dg=0}$

2)如果 ${\displaystyle \{e_{i}\}}$ 是表示空间 ${\displaystyle \pi ^{\alpha }}$ 的一个正交规范基，则：

${\displaystyle d^{\alpha }\int _{G}\phi _{e_{i},e_{j}}^{\alpha }(g)\phi _{e_{m},e_{n}}^{\alpha }(g)dg=\delta _{i,m}\delta _{j,n}}$

### 例 ${\displaystyle SO(3)}$

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

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

${\displaystyle \int _{x_{1}^{0}}^{x_{1}^{1}}\cdots \int _{x_{r}^{0}}^{x_{r}^{1}}\;\Gamma ^{(\lambda )}(\mathbf {x} )_{nm}^{*}\Gamma ^{(\mu )}(\mathbf {x} )_{n'm'}\;\omega (\mathbf {x} )dx_{1}\cdots dx_{r}\;=\delta _{\lambda \mu }\delta _{nn'}\delta _{mm'}{\frac {|G|}{l_{\lambda }}},}$

${\displaystyle |G|=\int _{x_{1}^{0}}^{x_{1}^{1}}\cdots \int _{x_{r}^{0}}^{x_{r}^{1}}\omega (\mathbf {x} )dx_{1}\cdots dx_{r}.}$

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

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

### 脚注

1. ^ ${\displaystyle l_{\lambda }}$ 的有限性是由于一个有限群 G 的不可约表示包含于正则表示
2. ^ 这种选择不是惟一的，这个矩阵的任意正交相似变换给出一个等价的不可约表示。

## 参考文献

• 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).