# Week 7

## Session 1

### Pre-class exercises

Exercise:

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

Exercise:

Verify that the matrices E_{i j} live in the following weight spaces: E_(1 2) in W_(1, minus 1); E_(2 1) in W_(minus 1, 1); E_(1 3) in W_(2 1); E_(3 1) in W_(minus 2, minus 1); E_(2 3) in W_(1 2); and E_(3 1) in W_(minus 1, minus 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_{i j} act?

Exercise:

How does \mathrm{Sym}^3(\CC^3) decompose into irreps of the \mathfrak{sl}(2,\CC)-subalgebra spanned by E_{1 2}, E_{2 1} and H_{1 2}?