Week 6

Session 1

Pre-class videos 37.17

X and Y: example, accessible version 16.33
Classification of SU(2) representations, accessible version 20.44

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 videos 16.57

Decomposing into irreducibles, accessible version 16.57

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.

Optional extras

Invariant theory, accessible version 19.00