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.