Compute the action of ${\mathrm{Sym}}^{2}(Y)$ on the basis ${e}_{1}^{2}$ , ${e}_{1}{e}_{2}$ , ${e}_{2}^{2}$ .

# Week 6

## Session 1

### Pre-class videos

### Pre-class exercises

Exercise:

Exercise:

Check that $$X{Y}^{k}v=(m+1-k+1)(k-1){Y}^{k-1}v+(m-2k+2){Y}^{k-1}v,$$ simplifies to give $$X{Y}^{k}v=(m+1-k)k{Y}^{k-1}v.$$

## Session 2

### Pre-class videos

### Pre-class exercises

Exercise:

Compute the weight diagrams and decomposition of ${\mathrm{Sym}}^{k}({\mathbf{C}}^{2})\otimes {\mathrm{Sym}}^{\mathrm{\ell}}({\mathbf{C}}^{2})$ for a few different values of $k$ and $\mathrm{\ell}$ and see if you can formulate a conjecture as to what the answer should be in general.

## Optional extras