Representations of U(1), part 1

0.00 We now state the classification theorem for representations of $U(1)$ and illustrate it with an example. We will prove the theorem next time.

Theorem:

If $R\colon U(1)\to GL(n,\mathbf{C})$ is a smooth representation then there exists a basis of $\mathbf{C}^{n}$ with respect to which $R(e^{i\theta})=\begin{pmatrix}e^{im_{1}\theta}&&{\large 0}\\ &\ddots&\\ {\large 0}&&e^{im_{n}\theta}\end{pmatrix}$ where $m_{1},\ldots,m_{n}$ are integers called the weights of the representation. A fancier way of saying this is that $\mathbf{C}^{n}=\bigoplus_{i=1}^{n}V_{i}$ where each $V_{i}$ is a 1-dimensional subrepresentation and $R=R_{1}\oplus\cdots\oplus R_{n}$ with $R_{i}=R|_{V_{i}}$ .

2.50 This means that the basis with respect to which $R$ has this form is a basis of eigenvectors $v_{1},\ldots,v_{n}$ . Moreover, $v_{k}$ is simultaneously an eigenvector of all the matrices $R(e^{i\theta})$ with eigenvalue $e^{im_{k}\theta}$ .

Example:

3.50 Take $R(e^{i\theta})=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}\in GL(2,\mathbf{C})$ . The characteristic polynomial of this matrix is

$\det\begin{pmatrix}\cos\theta-\lambda&-\sin\theta\\ \sin\theta&\cos\theta-\lambda\end{pmatrix}=\lambda^{2}-2\lambda\cos\theta+1,$ 4.55 so the eigenvalues are

$\lambda=\frac{2\cos\theta\pm\sqrt{4\cos^{2}\theta-4}}{2}=\cos\theta\pm i\sin% \theta=e^{\pm i\theta}$ . Therefore the weights of this representation are

$\pm 1$ .

5.35 The eigenvectors are $(i,1)$ and $(-i,1)$ . These are therefore our vectors $v_{1}\in V_{1}$ and $v_{2}\in V_{2}$ . With respect to this basis of eigenvectors, $R(e^{i\theta})=\begin{pmatrix}e^{i\theta}&0\\ 0&e^{-i\theta}\end{pmatrix}$ .

Pre-class exercise

Exercise:

Check that $(\pm i,1)$ is an eigenvector of $\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}$ with eigenvalue $e^{\pm i\theta}$ .