# Math 6500: Lie groups, Spring 2016

Due to the large number of undergrads in the course (welcome!)
we will be having homework, due Thursdays at the beginning of class.
Graduates' homework is unlikely to get graded.

Initial plan:

If there's time, we'll touch on the 21st century approach to
representation theory through topology of loop spaces.

Homework (here @ denotes tensor product):

Lectures so far:
1/28: (smooth) Lie groups. G-rep, subrep, irrep, Schur's lemma.
2/2: reps are unitarizable hence completely reducible. Characters
are an orthonormal basis of class functions. S_4 character table.
2/4: The ring Rep(G), and its map Chi -> class functions.
If H < G finite, then Rep(G) -> Rep(H) is not injective.
Compact Lie groups have left-invariant volume forms w/volume 1,
so we again have complete reducibility and orthonormality.
Reps of S^1, reps of T = (S^1)^n.
2/9: Weight multiplicity diagrams, as Fourier transform. Convolution.
Dominance order on U(2)-dominant weights.
A rep V is **strongly dominated** by a weight lambda if lambda's
multiplicity is 1, and all other weights of V are dominated by lambda.
Theorems: exists a unique irrep strongly dominated by each dominant weight,
these give all the irreps each exactly once,
and (x,y)'s is Sym^{x-y}(C^2) @ det^{@y}.
The map Rep(U(2)) -> Z_2-symmetric multiplicity diagrams is an isomorphism.
2/11: Lemma: the set of strongly dominated weights is closed under +.
Fundamental weights for U(n). They're strongly dominated. Hence all
irreps are, and they're classified by their highest weights.
2/18: Weyl character formula for U(n).
2/23: Steinberg tensor product rule. Branching to U(n-1).
Gel'fand-Cetlin patterns.
2/25-3/13: Lie algebras, enveloping algebras, (g,B)-reps,
Verma modules, statement of the BGG resolution.
3/15: The Harish-Chandra isomorphism for GL(n).
Real, complex, and quaternionic representations are distinguished
by the Schur indicator.
3/17: Starting general compact Lie groups...
Closed subgroups are Lie (Cartan's theorem).
Maximal tori are self-centralizing.
Side topic: Conjugacy classes in SU(n) correspond to points
in the Weyl alcove.
Next time: each compact group has a root system.
4/28: Each simply-laced Dynkin
diagram has an associated Lie algebra (over ZZ).