Jan 27

We proved:
Irreps of GxH are each the tensor product of an irrep of G with one of H.
C[G] is the sum over irreps V of V^* @ V, as a GxG-representation.
Character tables are square, i.e. the number of irreps is the number of conjugacy classes.
If G > H, two finite groups, then H misses some conjugacy class of G.
If G = U(n), H = T^n, then H hits every conjugacy class of G.

We didn't prove, but it's true:
As an algebra, C[G] is the direct sum of matrix algebras End(V), V running over irreps.

Jan 25

Overview:
1. Representation theory of finite groups.
2. ... of SL(2).
3. ... of U(n) and GL(n).
4. The adjoint representation of a Lie group; root system and Weyl group.
5. Classification of nice Lie groups.
6. Rep theory of general Lie groups.

Defs. Reps of finite groups on finite-dim complex vector spaces.
Irreducible, indecomposable.
Hom(V,W) as a rep. Equivariant maps.
Schur's lemma.
\pi_G|_V = 1/|G| \sum_G g|_V is a projection whose image is the G-invariants,
and whose trace is the dimension of the G-invariants.

Thm. Orthonormality of characters.
Cor. # irreps is at most # conjugacy classes.
Ex. S_3, S_4.

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