Weekly outline

  • General

    • The final written exam will be worth 30 points.


    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/

    • 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:TBA

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

    Examination Rules at the Department of Mathematics.

    • Old Exams: There are no old exams as this course was previously given without written exam.
    • Grading criteria: 
      • 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

    • Optional additional texts: 
      • "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 2: January 30

      • Completely reducible / semisimple representation: The following are equivalent for a representation V of G; we then say V is completely reducible or synonymously that V is semisimple.
      1. Every G-stable subspace W of V has a G-stable complement.
      2. V is a sum of irreducible representations of G.
      3. V is a direct sum of irreducible representations of G.

      • Representations of Z correspond to invertible matrices. Examples of non-semisimple representations.

      • 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.

        Exercise Session: Discussion about tensor products of vector spaces

        • Lecture 4: February 13

          Part I

          • Discussion about division rings: 
          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.



          • This week

            Lecture 6: March 1

            Not available