# Week 6

## Session 1

### Pre-class exercises

Exercise:

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

Exercise:

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

## Session 2

### Pre-class exercises

Exercise:

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