Week 5

Session 1

Pre-class videos 42.42

SU(2): Overview, accessible version 17.51
New representations from old, accessible version 24.51

Pre-class exercises

Exercise:

Check that $\frac{1}{2i}\begin{pmatrix}i&-i\\ 1&1\end{pmatrix}\begin{pmatrix}e^{i\theta}&0\\ 0&e^{-i\theta}\end{pmatrix}\begin{pmatrix}1&i\\ -1&i\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}$

Exercise:

Let $R\colon SU(2)\to GL(2,\mathbf{C})$ and $S\colon SU(2)\to GL(2,\mathbf{C})$ be two copies of the standard representation. Figure out the full 4-by-4 matrix $(R\otimes S)\begin{pmatrix}a&b\\ -\bar{b}&\bar{a}\end{pmatrix}$ .

Session 2

Pre-class videos 29.26

Weight space decomposition, accessible version 13.12
X and Y, accessible version 16.14

Pre-class exercises

Exercise:

Figure out the weight diagram for $\mathrm{Sym}^{n}(\mathbf{C}^{2})$ (where $\mathbf{C}^{2}$ is the standard representation of $SU(2)$ ).

Exercise:

Suppose that $R\colon SU(2)\to GL(V)$ is a representation with weight decomposition $V=\bigoplus W_{m}$ . Show that $R_{*}^{\mathbf{C}}(Y)$ sends $W_{m}$ to $W_{m-2}$ .