Exercise:
Compute the action of \mathrm{Sym}^2(Y) on the basis e_1^2, e_1e_2, e_2^2.
Week 6
Session 1
Pre-class videos
Pre-class exercises
Exercise:
Check that X Y to the k v equals (m plus 1 minus k plus 1) times (k minus 1) Y to the k minus 1, plus (m minus 2 k plus 2) Y to the k minus 1 v, simplifies to give X Y to the k v equals (m plus 1 minus k), times k, times Y to the k minus 1 v.
Session 2
Pre-class videos
Pre-class exercises
Exercise:
Compute the weight diagrams and decomposition of \mathrm{Sym}^k(\CC^2)\otimes\mathrm{Sym}^\ell(\CC^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