## Weekly outline

• ### General

• The final written exam will be worth 30 points.

Due to Coronavirus, there will be no in-class exam as planned. Instead we aim for a combination of (i) An at home exam (ii) An oral exam via skype or zoom. We will discuss this during class via zoom.

#### Teaching

• Teacher: Wushi Goldring
• Email: wgoldring@math.su.se
• Office: House 6, room 107
• Office Hours: Posted each week during the course before the beginning of the week.
• Next Office Hours: Posted on my website https://sites.google.com/site/wushijig/ No more in-office office hours due to virus. Email me if you have questions, we can discuss them via email or skype/zoom as you prefer.

• Teaching Assistant: Marcel Rubio
• Email: marcel.rubio@math.su.se
• Course Time and Room
• Course literature: I plan to follow the book "Character theory of finite groups" by I. M. Isaacs (available for free through the online SU library https://www.su.se/english/library/ using a university login). The book contains a lot of material, but the course will only cover a few chapters. A few other relevant texts are suggested below, but they are not required (sometimes it is helpful to get different perspectives on the same theorem/proof/topic).

#### Examination

• Examiner: Wushi Goldring

• Examination Form: The grading scheme for the Course will comprise of two parts: (1) Final Written exam, (2) homework bonus. (No oral exam.) Revised due to coronavirus.

Examination Rules at the Department of Mathematics.

• Old Exams: There are no old exams as this course was previously given without written exam.
• The final written exam will be worth 30 points.
• You need to score 12.5 points or higher on the final exam to pass.
• Bonus system: You can raise your grade by up to 3 extra credit points by successfully Completing the homework assignments. For example, if you get 11 on the exam and 2 on the homework, your overall score for the exam is 13 and you pass.

#### Resources

• "Abstract Algebra" by Dummit & Foote: This book, which is the textbook for the "Abstract Algebra" MM5020 class, is helpful in many ways: (1) To review groups and rings, (2) Review linear algebra, (3) Some new topics from linear algebra, such as tensor products and "multilinear algbera", (4) Representation theory of finite groups is discussed in Part VI (Chapters 18-19).
• "Representation theory of finite groups" by J.-P. Serre. I used this book the last time I taught the course.

If you do look at all three books (Isaacs, Dummit & Foote, Serre), it will be interesting to see which of them you find best or most helpful.

• ### 1. Introduction

The first goal of the course will be to understand the basic structure of representations of finite groups over the field of complex numbers. As we advance, I will give examples, remarks and homework about how things are much more complicated if the group is not finite or if the field is either of characteristic p>0 or not algebraically closed (the theory is basically the same if we replace the complex numbers with an algebraically closed field of characteristic zero, such as the field of algebraic numbers -- the subfield of the complex numbers consisting of those complex numbers which satisfy a polynomial equation in one variable with rational coefficients).

#### A. Basic definitions and questions

1. Definition: Let G be a finite group and F a field. A (linear) representation of G on an F-vector space V is a group action of G on V which is linear: g(cu+dv)=cgu+dgv for all scalars c,d in F, all vectors u,v in V and all g in G. By the dictionary between group actions and homomorphisms, a representation of G on V is also a group homomorphism \rho:G-->GL(V). If V has finite dimension n and e_1,..,e_n is a choice of basis of V, then GL(V) is isomorphic to GL(n) (write an invertible linear transformation in our basis (e_i)) and we get a group homomorphism G-->GL(n).
2. Notation:We write (V, \rho) to stress both the vector space and the homomorphism \rho: G -->GL(V).
3. Definition: The character \chi of a representation (V, \rho) is the function \chi: G -->F which gives the trace of \rho, i.e. \chi(g)=tr \rho(g). A character is a class function: It is constant on conjugacy classes: \chi(xgx^{-1})=\chi(g).
4. Question: To what extent does the character \chi capture the representation \rho?
5. Question: Given a class function f: G --> F, how can we tell if f is the character of a representation?
6. Definition: If (V, \rho) is a representation of G and W is a subspace of V, we say W is G-stable if gw is in W for all g in G and all w in W. The trivial subspaces (0) and V are always G-stable. We say that (V, \rho) is irreducible if it has no nontrivial G-stable subspaces.
7. Question: What can we say about the irreducible representations of G? More precisely:
8. Question: Given G, can we classify the irreducible representations of G?
9. Question: Is every representation of G a direct sum of irreducible representations?

#### B. Summary of basic results for complex representations of finite groups

Assume G is finite and restrict attention to finite-dimensional representations (V, \rho) over the complex numbers (i.e. F=C).=complex numbers and that V is finite-dimensional over C. Then:

1. Theorem (Maschke): Every representation of G is a direct sum of irreducible representations; if W is a G-stable subspace of a representation V, then there exists a G-stable complement W' in V, meaning that V is the direct sum of W and W'.
2. Theorem (Frobenius, Schur): Setting <\chi, \psi>=1/|G| \sum_{g \in G} <\chi(g) \overline{\psi(g)}> yields an inner product on the complex vector space of class functions f:G-->C. The irreducible characters of G form an orthonormal basis of the space of class functions relative the inner product <,>. In particular, the number of irreducible representations of G (up to isomorphism) equals the number of conjugacy classes in G.
3. Theorem (Frobenius, Schur): The regular representation of G decomposes as a direct sum where every irreducible representation of G occurs precisely as many times as its dimension. Consequently, the sums of the squares of the dimensions of the irreducible representations of G equals the order of G.
4. Theorem: A representation (V, \rho) of G is uniquely determined (up to isomorphism) by its character: If the character of V_1 equals the character of V_2, then V_1 and V_2 are isomorphic.
5. Theorem: The group ring C[G] is a semisimple Artin ring; it is isomorphic to a direct sum of matrix rings M_{n_1}(C)+...+M_{n_r}(C) where the n_i are the dimensions of the irreducible representations of G and r is the number of conjugacy classes in G.

• ### Lecture 3: February 6

#### Part I: Schur's Lemma

• Statement
• Proof
• Applications:
1. The center of GL(n) consists of the scalar matrices because the standard representation of GL(n) is irreducible.
2. Given an irreducible representation of a group G over an algebraically closed field, the center Z(G) acts via a character, the central character.
3. Every irreducible representation of an abelian group over an algebraically closed field is one-dimensional.

#### Part II: Maschke's Theorem

• Statement
• Proof
• Discussion of how each assumption is used: Finiteness of the group, characteristic doesn't divide the order.
• Examples of how the theorem fails when an assumption is dropped.

• ### Lecture 4: February 13

#### Part I

1.  There are no finite-dimensional division rings over an algebraically closed field.
2. Wedderburn's Theorem: A finite division ring is a field
3. A finite-dimensional division ring over the real numbers R is either R, C or the Hamilton Quaternions H.

• Definition of what it means for a representation of a group G over a field K to be realizable over a subfield F of K.
• Discussion from 13.2 of Serre about when a representation over C is realizable over R and when the character values lie in R:
1. The character values are real if and only if the representation admits a nondegenerate invariant bilinear form.
2. The representation is realizable over R if and only if there exists a non-degenerate, invariant bilinear form which is moreover symmetric.
3. We proved (1), modulo knowing that equality of characters implies isomorphism of representations, for representations of finite groups over C.

#### Part II:

1. Definition of modules and algebras.
2. Examples of modules: Over Z, a field F, F[x] and F[G].
3. Simple modules, simple rings, simple algebras, semisimple algebras.

• ### Lecture 7 : March 5

##### 1. Set-up & Review
• We now specialize to A=C[G] the group ring of a finite group G over C=complex numbers. Everything will work equally well if C is replaced by an algebraically closed field k of characteristic 0. Most things will still work if char(k) doesn't divide the order of G, but different arguments are sometimes necessary (see Chapter 9 of Isaacs).
• We know there are finitely many irreducible representations M_1,...,M_s of A=C[G] and that s=dim Z(C[G]): the e_i= ith component of 1 relative the isotypic decomposition C[G]=sum of M_i(C[G]) give a first basis of C[G], we're about to see another one.
• If n_i:=dim M_i, we know that (n_1)^2+...+(n_s)^2=dim A in general, and dim A=|G| when A=C[G].

##### 2. Theorem (The center of C[G] via conjugacy classes)

• Let K_1,...,K_r be the conjugacy classes of G and put k_i=(sum of elements of K_i) in C[G].
• Then k_1,..,k_r is another basis of Z(C[G]).
• If we write k_i k_j in the basis (k_l), then the structure constants are all nonnegative integers.
##### Corollary
• The number of irreducible characters equals the number of conjugacy classes (r=s).
##### Proof of theorem:

• Conjugation leaves stable a conjugacy class (it is an orbit for the action of G on itself by conjugation), so conjugation of k_i permutes the summands and leaves the sum fixed. So the k_i lie in the center. Any two vectors v,w which have disjoint support in some basis (support of v= basis vectors used to v in the given basis) are linearly independent. The k_i have pairwise disjoint support in the basis G of C[G], so the k_i are linearly independent.
• If z= sum of a_g g lies in the center, then hzh^{-1}=z. The coefficient of g in z is a_g, the coefficient of g in hzh^{-1} a_{h^{-1}gh}; since G is a basis of C[G], we conclude that a_g=a_{h^{-1}gh} for all h in G i.e. the coefficients are constant on conjugacy classes. So z is in the span of the k_i. So the k_i are linearly independent and span, hence they form a basis.
• For the claim about the structure constants, observe that the coefficient of g in K_l in a product k_ik_k is the size of the set {(x,y) in K_i x K_j | xy=g}; the size of a finite set is a non-negative integer.
##### (Non-)Example: The Hamilton quaternions H

• Here {1,i,j,k} is a basis of H over R and we know that the structure constants can be negative: If ij=k, then ji=-k.
##### Remark:

• It is a recurring theme in representation theory that structure constants miraculously turn out to be integral and non-negative. For example, this arises in the theory of Schubert varieties inside flag varieties and in the closely related "Littlewood-Richardson rules" on how to decompose a tensor product of two irreducible representations (say of GL(n)).
##### 3. Lemma

• Let \chi_i be the character of M_i. Then the \chi_1,..,\ch_s are pairwise distinct functions C[G] (and hence also on G, as a character on C[G] is determined by its values on G). In other words an irreducible representation is determined (up to isomorphism) by its character (this holds for an arbitrary character, see below).
##### Proof:

• Let \rho_i:C[G]--.End(M_i) be the representation of C[G] on M_i (hom of algebras); when we restrict to G we get \rho_i: G-->GL(M_i) (hom of groups)
• We look at \rho_i(e_j), where 1=e_1+...+e_s as before. For i different from j, \rho_i(e_j)=0 since e_i is in M_j(C[G]) and the ideal M_j(C[G]) annihilates M_i.
• Since \rho_i(1)=I=identity matrix, we get that \rho_i(e_i)=1.
• Taking traces, we get that \chi_i(e_j)=0 if i not j, and \chi_i(e_i)=\chi_i(1) (=dim(M_i) not zero in characteristic 0, but potentially \chi_i(1) =0 in characteristic p. Ludvig points out that \chi_i(1)=dim(M_i) divides |G| (proved later in Isaacs), so in fact \char(k) doesn't divide |G| implies \char(k) does not divide \chi_i(1).
• In sum, at least in char(k)=0, the \chi_i are distinguished by their values on the e_j; the (\chi_i) is almost the dual basis to the basis (e_j).
##### Definition:

• Irr(G) is the set of irreducible characters of G.
##### Corollary:

• The sum of \chi(1)^2 over \chi in Irr(G) is the order of G.
##### 4. Theorem:

• Irr(G) is a basis of the space Class(G, C) of complex-valued class functions on G.
##### Proof:

• The dimension of Class(G,C) is the number of conjugacy classes=s=|Irr(G)|. So it's enough to show that Irr(G) is linearly independent. If a linear combination sum of a_i\chi_i is 0, then evaluation at e_j gives a_j=0, so all the coefficients a_j are zero.
##### Corollary

• Two representations of G over C are isomorphic if and only if their characters are equal.
##### Proof:

• We saw at the beginning of the course that isomorphic representations have equal characters. If two representations V,W both have character \chi, since the irreducible characters form a basis, the coefficient of each irreducible character in \chi is uniquely defined and it is both n_{M_i}(V) and n_{M_i}(W) i.e. M_i are appears the same number of times in V as it does in W. Since V, W are semisimple we conclude that both are isomorphic to the sum of M_i taken n_{M_i}(V)=n_{M_i}(W) times.

• ### Lecture 8: March 19 (Moving to Zoom)

See attached notes and audio file.

Video is shared here https://drive.google.com/file/d/1t4XHkkGobD8AKyI8YPcBnDAHOLNyvZ8u/view?usp=sharing from google drive (the video file is >2.5 GB, so I can't upload it here).

• ### Lecture 9: March 26

Join Zoom Meeting
https://stockholmuniversity.zoom.us/j/133296604