Tuesday April 4

I outlined the main topics I hope to cover (as on the course web page), and for each one a key result that I hope to get to. Those results were

1. irreps of compact groups are determined by their characters
2. classification and construction of irreps of U(n) & GL_n(C)
3. Weyl character formula and Kostant multiplicity formula for GL_n, Gel'fand-Cetlin patterns, Young tableaux
4. hives compute tensor product multiplicities
5. the Borel-Weil theorem for GL(n)
6. the link between tensor products and sums of Hermitian matrices.

Then we discussed finite-dimensional complex representations. Isomorphisms, intertwiners, subreps. Decomposable vs. reducible. Unitary reps are direct sums of irreps. Reps of finite groups (or groups with left-invariant measures) are unitarizable.

Next time: characters and their orthonormality. Representation rings.