# Week 5

## Session 1

### Pre-class exercises

Exercise:

Check that 1 over 2 i times the 2-by-2 matrix i, -i; 1, 1, times the 2-by-2 matrix e to the i theta, 0; 0, e to the minus i theta, times the 2-by-2 matrix 1, i; minus 1, i, equals cos theta, minus sine theta; sine theta, cos theta

Exercise:

Let R\colon SU(2)\to GL(2,\CC) and S\colon SU(2)\to GL(2,\CC) 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 exercises

Exercise:

Figure out the weight diagram for Sym n C 2 (where C 2 is the standard representation of SU(2)).

Exercise:

Suppose that R from SU(2) to G L V is a representation with weight decomposition V equals the direct sum of subspaces W_m. Show that R star superscript C of Y sends W_m to W m minus 2.