1. Figure out how to compute the Sym2 and Alt2 of a weight multiplicity diagram. Do so for the diagram for Sym3 C3 = V(3,0,0). In particular, show that Sym2 Sym3 C3 contains Sym6 C3 plus one other irrep, and Alt2 Sym3 C3 contains Sym3 Alt2 C3 plus one other irrep. Write down all the high weights in this example.
2. Give an example of a group homomorphism in which the center doesn't go to the center.
3. Using our construction of the irreps of GL(n), show that in an irrep GL(n) -> GL(m) the center does go to the center. More concretely, how does the center of GL(n) act on Vlambda?
4. Using Schur's lemma, show that in any irrep G -> GL(m) the center goes to the center.
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if G-> Gl(n,C) irrep then
C(G)-> M(n,C) surjective (Burnside) so the center maps to the center
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