Thursday April 27 - no class next Tuesday

Side note: to compute the character of a GL(n) rep on an arbitrary element of GL(n), take its eigenvalues and feed that diagonal matrix into the WCF.

Given the multiplicity diagram of a G-rep V, apply difference operators in the direction of the negative roots, and the result (in the dominant Weyl chamber) tells you the multiplicities of the irreps in V.
The branching rule from GL(n) to GL(n-1).
The weight of a Gel'fand-Cetlin pattern is the list of differences in the row sums.
Theorem: there is a basis of V_lambda indexed by Gel'fand-Cetlin patterns, where each basis vector is a T-weight vector whose weight is that of the GC pattern.

I am away next Tuesday.

Tuesday April 25

Statement of the Weyl character formula (WCF) with a single denominator, Den.
The Kostant partition function.
The Kostant multiplicity formula, as the Fourier transform of WCF.
Pictures of the weight multiplicity diagrams as given by the Kostant multiplicity formula, for the GL(3) case.
Proof that the WCF is actually a Laurent polynomial with integer coefficients (given by the Kostant multiplicity formula).
Weyl's proof that the WCF is the character of an irrep, indeed of V_lambda.

Gel'fand-Cetlin patterns, the integer points in the (compact, convex) Gel'fand-Cetlin polytope.
Example: Gel'fand-Cetlin polytopes for GL(3).

No office hour next Tuesday

There's a special combinatorics seminar next Tuesday at 2:30, so no office hour after class that day.

Thursday April 20

Lemma. Let D be an S_n-invarant weight multiplicity diagram with some D^mu > 0. Then D defines a class function Chi_D:U(n) -> C. If the integral over U(n) of |Chi_D|² = 1, then D is the weight diagram of an irrep.
Weyl integration formula, in terms of volumes of conjugacy classes.
Volumes of conjugacy classes of U(n). (statement, no proof; look in the notes on the class web page for full details)
The Weyl denominator Den, as a sort of square root of the volume of the conjugacy class.
Statement of the Weyl character formula (WCF), manifestly S_n-invariant version.
Theorem. GL(2) acting on Sym^k C² is an irrep.
The WCF for GL(2).
The rho-shifting in Den * WCF.

Exercises #4

1. Figure out how to compute the Sym² and Alt² of a weight multiplicity diagram. Do so for the diagram for Sym³ C³ = V_(3,0,0). In particular, show that Sym² Sym³ C³ contains Sym⁶ C³ plus one other irrep, and Alt² Sym³ C³ contains Sym³ Alt² C³ plus one other irrep. Write down all the high weights in this example.

2. Give an example of a group homomorphism in which the center doesn't go to the center.

3. Using our construction of the irreps of GL(n), show that in an irrep GL(n) -> GL(m) the center does go to the center. More concretely, how does the center of GL(n) act on V_lambda?

4. Using Schur's lemma, show that in any irrep G -> GL(m) the center goes to the center.

Tuesday April 18

Define V_lambda as the irrep of high weight lambda we already constructed.
Lemma. There are finitely many dominant weights dominated by a given weight.
Corollary. Every S_n-invariant multiplicity diagram is a Z-linear combination of characters of V_lambdas.
Theorem. The V_lambdas are all the irreps of U(n).
Lemma. If a polynomial on M_n vanishes on U(n), it vanishes everywhere.
Def. Rational rep of GL(n).
Thm. Every rep of U(n) extends uniquely to a rational rep of GL(n).

Then we described (without proof) the multiplicity diagrams of irreps of U(3), and computed the decomposition of the tensor square of V_(2,1,0).

Exercises #3

1. Show that dominance order is indeed a partial order.

2. Let lambda be dominant, and mu a convex combination of the permutations of lambda.
(This means: let x_pi be a nonnegative real number for each permutation pi in S_n, with \sum_pi x_pi = 1. Let mu = \sum_pi x_pi pi.lambda . Then mu is a "convex combination".)
Show that lambda is more than mu in dominance order.

3. Call a weight multiplicity diagram D "saturated" if the conditions
a) mu is integral
b) mu is a convex combination of weights in D
together imply that mu is itself a weight of D.
Show that Symⁿ C² is the smallest S₂-symmetric saturated multiplicity diagram with high weight (n,0), for some appropriate definition of "smallest".

4. Show that the reducible representations we made by tensoring fundamental representations together are saturated.

Thursday April 13

Gah! I'm already slipping on keeping this blog up-to-date.

Lemma (b). The "fundamental representations" Alt^k Cⁿ have weights all of multiplicity 1. The high weight is (1,...,1,0,...,0) with k 1s and n-k 0s. The other weights are permutations thereof, and all other weights are less than the high weight in dominance order.

Lemma (c). Call a rep "good" if the high weight is multiplicity one, and all other weights are smaller in dominance order. (So the high weight is unique.) Then the tensor product of two such reps is again "good", and the new high weight is the sum of the two individual high weights.

Theorem 1. Every irrep has a dominant high weight.

Theorem 2. For every dominant weight, there is an irrep with that high weight, and it's "good".

Tuesday April 11

We won't prove that compact Lie groups have finite, positive, left-invariant measures -- leave that for a differential topology class. With that, our character theorems apply to any continuous representations.

Weight lattices of tori. Weight multiplicity diagrams. Ex: Symⁿ C².

Theorem. Two reps of U(n) are isomorphic iff when restricted to T = diagonals, they're isomorphic. (This is why we did character theory.) So we can study reps by their multiplicity diagrams.

Ex: Symⁿ C² @ Sym^m C² ~ \sum_{k satisfying triangle inequalities, same parity as m+n} Sym^k C² @ det^@(m+n-k).

Highest weight of a rep. Dominant weight. Dominance order.

Stated the classification of irreps of U(n) in terms of their highest weights; will prove it next time.

Lemma (a): if V is a U(n)-rep, its multiplicity diagram is S_n-invariant.

Exercises #2 (corrected)

I'll write tensor product as @.

1. If V,W are two reps of G, show that Chi_{V @ W} = Chi_V Chi_W.

2. If V,W are two reps of G, and |Chi_V(g)| < dim V except for g=1, show that W is isomorphic to a subrepresentation of V@V@....@V for some number of factors.

3. Show that the condition in #2 holds for G simple, V not zeroa trivial representation.

4. If V is a rep, compute the character of G on the symmetric tensors Sym²V and the alternating tensors Alt²V, in terms of the character of V. For each irrep W of S₄, confirm that the characters of Sym²W and Alt²W are nonnegative integer combinations of the irreducible characters.

Thursday April 6

We defined the averaging operator pi_V on a representation, and proved it to be a projection onto V's invariant subspace. Schur's lemma. Orthonormality of irreducible characters. Class functions. Reps are isomorphic iff they have the same character. Some character table stuff, with S₃, S₄ as examples.

We didn't prove that irreducible characters give an orthonormal basis of class functions; you can find that in the notes on the course web page.

Main results of the survey

Surprisingly, noone circled "Combinatorics is a scourge" without also circling "is fun, interesting, and instructive". I suppose such people were scared off by the course description, as is probably best.

Almost everybody circled "Pretty solid" or "Please, don't bore me" for tensor products. As such, I'm not going to do them in class. You can find them in any book entitled "Algebra", e.g. Hungerford, Artin, Lang.

People liked having office hours directly after class.

Exercises #1 (corrected)

I'm putting these here so you can check that you're keeping up. If you find something weird, or want a (nother) hint, post a comment. (If you've got a hint, you can post that too.) Or you can just come and ask me about them after class.

1. Let V be a finite-dimensional irrep of an abelian group A (possibly infinite, even infinitely generated!). Show that V is 1-dimensional.
Hint: either every group element acts a multiple of the identity, or some element doesn't.

2. Let G be a finite abelian group. Show that the number of distinct irreps of G is |G|. (Hint: start with the cyclic case.)

3. Let G be a group, and Rep(G) resp. Irr(G) the set of {reps resp. irreps of G up to isomorphism}. Let Pic(G) be the set of {1-d reps of G up to isomorphism}. Show that Pic(G) is a group under tensor product (which includes checking some well-definedness), and that Pic(G) acts on Rep(G) by tensor product, preserving the subset Irr(G).

4. Calculate Pic(S₃), and find an element of Rep(S₃) that is fixed under the action from #3.

5. Generalizing #2 & #3, show that Irr(G)Pic(G) is isomorphic to G/[G,G]. Warning: there is no natural isomorphism!

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.

Spring quarter 2006: Representations of GL(n)

This quarter I'm teaching a topics-in-algebra graduate class, Representations of GL(n). The course web site is here.