years student of mathematics and write my script for my bachelor. The topic is "Representations of groups and applications in physics". I understand the representations very good but now i want to make some applications with name Molecular vibrations. For finite groups (for example: the symmetry-group of $H_2O$ is $D_2$ and such a symmetrygroup of a molecul $M$ in $\Bbb{R}^3$ with $n$ atoms is given by: $$G=\{A\in O(\Bbb{R^3}):\ \forall i\ \exists j: m_i=m_j, Aq_i=q_j\}$$ with $m_i$ the massa and $q=(q_1,\cdots,q_n)\in\Bbb{R^{3n}}$ the equilibrium position of $M$. Notice that $G$ is a subgroup of $O(\Bbb{R^3})$ and that there exists a natural representation $\Pi$ of $G$ on $\Bbb{R^{3n}}$ given by: $$\Pi(A)(x_1,\cdots,x_n)=(Ax_{\sigma^{-1}_A}(1),\cdots,Ax_{\sigma^{-1}_A}(n))$$ with $x=(x_1,\cdots,x_n)\in\Bbb{R^{3n}}$ small displacements from $q$, and $\sigma_A\in S_n$ a permutation "induced" by $A\in O(\Bbb{R^3})$. Suppose $\chi$ is the charakter from we standartrepresentation of $G$ (as subgroup of $O(\Bbb{R^3})$) with name $\pi:G\rightarrow GL(\Bbb{C^3})$. Then there is Wigner's rule given by:
The character $X$ from the representation $\Pi$ of $G$ on $\Bbb{R^{3n}}$ is given by: $X(A)$=#$\{i:\sigma_A(i)=i\}\cdot\chi(A)$
The proof of this fact is very simple. But now my question: Suppose $G$ is a compact group (also suppose $G$ is non-finite!) (for example the circle group $S^1$), can we generalize Wigner's rule???
Form my point of view is this a difficult (research)-question because i can't find any notes of this and we can't make the same formule because we can't count such fixed points of permutaions because the groups is not finite. Someone an idea??
Thank you very much.