We won't prove that compact Lie groups have finite, positive, left-invariant measures -- leave that for a differential topology class. With that, our character theorems apply to any continuous representations.
Weight lattices of tori. Weight multiplicity diagrams. Ex: Symn C2.
Theorem. Two reps of U(n) are isomorphic iff when restricted to T = diagonals, they're isomorphic. (This is why we did character theory.) So we can study reps by their multiplicity diagrams.
Ex: Symn C2 @ Symm C2 ~ \sum_{k satisfying triangle inequalities, same parity as m+n} Symk C2 @ det@(m+n-k).
Highest weight of a rep. Dominant weight. Dominance order.
Stated the classification of irreps of U(n) in terms of their highest weights; will prove it next time.
Lemma (a): if V is a U(n)-rep, its multiplicity diagram is Sn-invariant.
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Sym m ℂ ⊗ Sym n ℂ ~ ∑_{k satisfying triangle inequalities, same parity as m+n} Symk ℂ ⊗ det⊗(m+n-k).
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