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 ${\mathrm{Sym}}^{n}({\mathbf{C}}^{3})$ ?

# Week 7

## Session 1

### Pre-class videos

### Pre-class exercises

Exercise:

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 videos

### 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 $\U0001d530\U0001d529(2,\mathbf{C})$ -subalgebra spanned by ${E}_{12}$ , ${E}_{21}$ and ${H}_{12}$ ?