# Week 7

## Session 1

### Pre-class exercises

Exercise:

If $\mathbf{C}^{3}$ denotes the standard representation of $SU(3)$ , what is the weight diagram of $\mathrm{Sym}^{3}(\mathbf{C}^{3})$ ? Can you guess the weight diagram of $\mathrm{Sym}^{n}(\mathbf{C}^{3})$ ?

Exercise:

Verify that the matrices $E_{ij}$ live in the following weight spaces: $E_{12}\in W_{1,-1},\ E_{21}\in W_{-1,1},\ E_{13}\in W_{2,1},\ E_{31}\in W_{-2,% -1},\ E_{23}\in W_{1,2},\ E_{32}\in W_{-1,-2}$ Plot these weights and verify that the hexagonal picture from the lecture is correct.

## Session 2

### Pre-class exercises

Exercise:

What do you think the weight diagram of the standard 4-dimensional representation of $SU(4)$ would look like? How do you think the matrices $E_{ij}$ act?

Exercise:

How does $\mathrm{Sym}^{3}(\mathbf{C}^{3})$ decompose into irreps of the $\mathfrak{sl}(2,\mathbf{C})$ -subalgebra spanned by $E_{12}$ , $E_{21}$ and $H_{12}$ ?