Group Representation Theory

Lecture 1: Review Group Theory and Linear Algebra.

Topics covered: 

§1. Review relevant group theory: define a group, examples: cyclic group, symmetric group, alternating group, quaternions, dihedral group, define conjugacy class, define group action, define permutation representation. 

§2. Review relevant linear algebra: define GL(V) for V finite dim over F, GLnF, state: conjugate matrices describe the same element of GL(V), define trace, define End(V), fact: a finite set of pairwise commutative diagonalizable automorphisms of V/C is simultaneously diagonalizable. 

§3. (If time permits): define linear representation.

Class notes are available here.

Exercise session: 

•  Two elements in Sn are conjugate if and only if they have the same cycle type.

• Commuting sets of diagonalizable endomorphisms of V are simultaneously diagonalizable. 

Lecture 2: Linear Representations.

In this class session, we will define the representation of a finite group G on a finite-dimensional vector space V, and look at 4 different ways of describing them:

• as a homomorphism from G to GL(V)

• as a matrix representation

• as a representation of F[G], the group algebra

• as an action of G on V. 

Class notes are available here.

Also, since some students wanted suggested readings and exercises, I highly recommend looking at Serre's book "Linear representations of finite groups" Sections 1.1-1.3. You can also read pages 1-3 of Isaac's (up til Definition 1.4).

For examples, I recommend trying to find m-dimensional representations of C_n, for n and m integers. Can you find a pattern/classification?

Exercise session: Find all irreducible representations of S3 (or D6) of dimensions 1 and 2. In particular, find an irrep of S3 of dimension 2 by looking at the permutation rep, and show that this is the same as an irrep of D6 obtained by looking at a triangle embedded in \R^2. 

Lecture 3: Maschke's Theorem.

In this class session, we will state and prove Maschke's theorem in two different ways. 

We will also talk about regular representation, and show that every irreducible representation of a finite group is isomorphic to a subrepresentation of the regular representation.

Maschke's theorem, in some sense, reduces the study of representations of a finite group over a field of characteristic 0, to the study of its irreps. The regular representation, on the other hand, gives us a way to explicitly construct these irreps. 

Class notes are available here.

Here you can find HW 1: https://www.dropbox.com/s/rushtavffoyau4j/2022-Feb-3-HW1-RepresentationTheory-Spring22.pdf?dl=0

It is due by midnight on Thursday, February 24th, 2022. Please submit it directly to Kurser. 

Exercise session: 

• Decompose the permutation representation of S3 as a sum of irreducible representations. 

• Show that there are no irreducible representations of S3 of dimension >2.

Lecture 4: Schur's Lemma.

Class notes are available here.

Exercise session: 

• Show that the decomposition into irreps is unique up to a permutation of factors. In particular, if X is an irrep and V is a rep of G, it makes sense to define the multiplicity of X in V. 

• If G has a faithful irrep, then the center of G has a 1 dimensional faithful representation. For finite G, this means that Z(G) is cyclic. For G=GL(n), Z(G) is scalar matrices.

Lecture 5: Character Theory. 

In this week, we defined the character of a representation and proved that the character is a complete invariant, i.e. complex representations of finite groups are completely characterized by their character. 

In order to prove this, we have to show that irreducible characters of G form an orthonormal basis of the space of class functions of G. This theorem is very important and has a lot of very useful consequences. We will give a proof and several corollaries next week. 

Class notes are available here.

Exercise session: Compute character tables of finite groups of order <8.

Lecture 6: Orthogonality Relations. 

In this class session, we proved that the complex irreducible characters of a finite group G are orthonormal. Next time, we will prove that they span the space of class functions, and thus form a basis.

This theorem is very important in character theory and has a bunch of consequences, e.g.

1) A complex rep is irreducible iff its character chi satisfies <chi, chi>=1.

2) The sum of squares of the dimensions of irreps of G is |G|. In particular, this gives an upper bound on the dimension of any non-trivial irrep: \sqrt{|G|-1}. 

For S3, this shows that no irrep can have dimension >2. You might remember proving this in an exercise session using more complicated arguments.

3) The number of G irreps equals the class number of G. 

4) Two elements g and h are conjugate in G if and only if for all irreducible characters chi of G, chi(g)=chi(h).

Next time, we will finish proving the main theorem and consider even more consequences. In particular, we will compute character tables for small groups.

Class notes are available here.

Exercise session: Compute character tables of finite groups of order <12.

Lecture 7: Orthogonality relations+Permutation Characters.

In this class session, we finish proving that the irreducible characters form a basis of the class functions of G. We then define a character table and look at several examples of such tables. 

In TA sessions, you've seen character tables of small groups (up to order 11). 

Another family of groups we can study on their own are the symmetric groups, S_n. 

One of the main features of Sn is that they have a natural n-dimensional representation, the permutation representation. In the next class, we will study the properties of the permutation character and prove that you can always decompose it as a direct sum of the trivial character and another irreducible character called the standard character. This gives two more rows of the character table of Sn for free.

Class notes are available here.

HW 2 is also available now.

Exercise session: We will prove the following facts that are useful when computing character tables of Sn and An.

• If σ in An, then the elements of the conjugacy class of σ in Sn (which is just all elements of the same cycle type as σ) are conjugate in An if and only if σ commutes with some odd permutation.

• The number of elements in each of the conjugacy classes in Sn is given by ...?

Lecture 8: Character Table for S_n 

In this class session, we finished talking about permutation representations. We proved that for a 2-transitive group such as S_n, the permutation representation decomposes into the trivial representation and an irreducible representation. We call this irrep the standard rep of Sn. We wrote down a character table for S4 and started writing down one for S5. Next time, we will see how to get lift characters from quotients.

Lecture 9: Dual representations and lifting of Characters.

§ HW2 comment: 

Question 5 from HW 2 asks about self-dual representations, so we will spend some time at the start of class today talking about the dual of a representation.

§ Lifting characters:

Given a group G and subgroup H e.g. S5 and S3, we want to be able to use the information we have about the character table of H to get characters of G (and vice versa.) It turns out that this is much easier to do for quotients instead of subgroups.

In this class, we learn how to lift characters of quotients to characters of the parent group. We investigate whether this preserves irreducibility. 

Class Notes are available here.

Some problems for you to try after class:

- Prove that the commutator subgroup of A4 is the Klien 4 group. 

- Lift characters from A4/V4 and write down a character table for A4. 

- Consider V4 as a subgroup of S4. Find the quotient, S4/V4, and lift characters to write down a character table for S4.

Exercise Session: Discussion of Tensor Products of vector spaces and representations via universal property. Some discussion of character ring.

Lecture 10: Induction and Restriction.

Let G be a finite group and H be a subgroup of G. In this class session, we will see what happens when we restrict a character (or representation) of G to a character (or representation) of H. Furthermore, we will learn how to induce a character of G from a character of H. We will learn that these ideas are connected via Frobenius reciprocity. 

Class notes are available here.

Exercise session: More on Tensor Products, Symmetric Square, Exterior Square of representations. 

Lecture 11: More on induced modules.

In this class session, we finished talking about induced characters and induced modules. Class notes are available here.

HW 3 is now available.

Exercise session: Induce a character of S4 from a linear character of C4 in detail.

Lecture 12: Characters of GxH, Symmetric and Exterior Powers of Characters, Character Ring.

Exercise Session: Background for algebraic integers needed to prove Burnside's p^a q^b theorem. Reference: James and Liebeck, Section 22 (22.1-22.6). 

Lecture 13: Frobenius Groups.

Exercise Session: Prove that the degree of any irreducible character of G divides |G|.

Lecture 14: Burnside's p^a q^b Theorem.