Week 6

Compute the action of ${\mathrm{Sym}}^2(Y)$ on the basis $e_1^2$ , $e_1{e}_2$ , $e_2^2$ .

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 $$X{Y}^{k}v=(m+1-k)k{Y}^{k-1}v.$$

Compute the weight diagrams and decomposition of ${\mathrm{Sym}}^k({𝐂}^2)\otimes {\mathrm{Sym}}^{\mathrm{\ell }}({𝐂}^2)$ 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.