# 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:

• Linear reps of finite groups, following my notes and ch1-2 of Serre.
• Irreps of tori and of U(n).
• The Borel-Weil theorem for GL(n).
• Irreps of SU(2) (very easy after U(n)).
• Root systems, abstractly and from compact groups. My notes, and also ch21 of Fulton and Harris.
• Classification of compact simple Lie groups.
• 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):

• HW#1 due Thursday 2/4:
• Show directly that the action of S_n on {n-vectors with sum=0} is irreducible.
• Let C[G] be the C-vector space with basis G, whose multiplication extends the group multiplication; this makes it an "algebra". Find the elements in the center of C[S_3].
• In class Tuesday 2/2 we'll compute the character table of S_4. Compute the character table of S_5.
• Answer: The obvious reps are the trivial T, sign S, and defining rep (which is the trivial + an irrep R). Then S@R is another. Another handy rep comes from looking at the action of S_5 on 2-element subsets, with character (10,4,1,0,0,2,1). This turns out to contain T and R (if you dot with them you get 1), and what's left over is a 5-d irrep W (its norm square is 1). Then W@S is the other 5-d rep, and you can determine the 6-d irrep from column orthogonality. (Plenty of other approaches work too.)

• HW#2 due Thursday 2/11:
• Show that the automorphism group of T^n is GL(n,ZZ).
• Describe the character table of G x H in terms of those of G and H. Show that every irrep of GxH is a tensor product V@W of irreps V,W of G,H.
• If G ->> H is onto, how do G and H's character tables relate? Which leads to...
• How many rows of G's character table start with 1? and
• How can you determine from G's character table whether it's a simple group?
• HW#3 due Thursday 2/18:
• Use weight multiplicity diagrams to compute the U(2)-decomposition of Sym^n(C^2) @ Sym^m(C^2). Your answer should be multiplicity-free.
• Find the center of U(n), a copy of U(1). Call a representation of level k if the center acts with weight k. Show that U(n)'s irreps have levels, and that the level is additive under tensor product. Calculate the level for the irreps of U(2).
• HW#4 due Thursday 2/25:
• Use the Weyl character formula to prove (a special case of) the Pieri rule: V_lambda @ C^n breaks as the sum of V_lambda' where lambda' - lambda = (0..010..0).
• Consider the action of U(m) x S_n on (C^m)^@n, where S_n permutes the factors and U(m) acts diagonally. Which U(m) reps V_lambda occur in here, and with what multiplicity? Note that Hom_{U(m)}(V_lambda, (C^m)^@n) is naturally a representation of S_n.
• Consider the maps SU(n) -> SU(n) x U(1) -> U(n). Every irrep of SU(n) extends to many irreps of SU(n) x U(1); show that some of those descend to irreps of U(n). Use this to classify the irreps of SU(n).
• Let SU(2) act by conjugation on the 3-dimensional real space of traceless 2x2 Hermitian matrices. Since it must preserve some positive definite form (by averaging -- but it's easy to find one), this gives a map SU(2) -> O(3), whose image is in fact SO(3). Determine the kernel and classify the representations of SO(3).
• HW#5 due Thursday 3/10:
• Classify the nonnegative Gel'fand-Cetlin patterns that aren't sums of two others. Hint: you should get 2^n-1 answers.
• HW#6 due Thursday 3/17: Consider the action of GL(a)xGL(b) on axb matrices by (X,Y).M = X M Y^T, and from there on the ring of polynomials Sym(M_{a,b}) in the matrix entries.
• Let d_i be the determinant of the NW ixi block (i \leq min(a,b)). Show that d_i is invariant under the unipotent group N_a x N_b.
• If a=2, show that these two determinants generate the ring of N_a x N_b invariants in Sym(M_{a,b}), and that they are algebraically independent. (It's true for a>2 too.)
• Use this to decompose Sym(M_{a,b}) into irreps of GL(a)xGL(b).
• We therefore have two weight bases for this infinite-dimensional representation: monomials, and certain pairs of Gel'fand-Cetlin patterns. For each fixed weight, can you find a bijection between these two indexing sets? (Again a=2 is a reasonable start. And likely, end.)
• HW#7 due Thursday 3/24:
• Let M \in M_n(Ugl_n) be the matrix whose (i,j) entry is e_ij \in gl_n. Show that C_k := Trace(M^k) lies in the center of Ugl_n.
• Which irreps of U(n) are self-dual?
• Let SD be the cone of dominant weights whose irreps are self-dual. Show that the map Schur indicator: SD -> {+/-1} takes addition to multiplication.
• For U(2), which of the self-dual irreps are real, and which quaternionic?
• 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).