Tuesday April 18

Define V_lambda as the irrep of high weight lambda we already constructed.
Lemma. There are finitely many dominant weights dominated by a given weight.
Corollary. Every S_n-invariant multiplicity diagram is a Z-linear combination of characters of V_lambdas.
Theorem. The V_lambdas are all the irreps of U(n).
Lemma. If a polynomial on M_n vanishes on U(n), it vanishes everywhere.
Def. Rational rep of GL(n).
Thm. Every rep of U(n) extends uniquely to a rational rep of GL(n).

Then we described (without proof) the multiplicity diagrams of irreps of U(3), and computed the decomposition of the tensor square of V_(2,1,0).