-1+1862 
Introduction to Group Representations

Group representations are where group theory meets linear algebra, and important applications arise in various math subjects (number theory, analysis, algebraic geometry), physics, and chemistry.

We consider the basic representation theory of finite groups. Goals include a look at Fourier series/analysis using groups and elementary character theory.

Lectures
  1. Week 1 - Representation Theory Basics

    Representations are defined, as are subrepresentations and irreducibility. Links include Linear Algebra Review, video RT1, Problem Set 1, and optional background videos.

  2. Week 2 - Unitary Representations

    Unitary representations are those that preserve an inner product. We show that any representation can be made unitary and, in turn, is fully reducible. Links include Linear Algebra Review 2, Solution Set 1, Problems Set 2 (with solutions), video RT2, and optional background videos.

  3. Week 3 - Equivalence and Examples

    We introduce the notion of equivalence of representations and present some low-dimensional examples, including a look at one-dimensional representations in general. Links include Solution Set 2, video RT3, Problem Set 3, and an option background video.

  4. Week 4 - Constructions from Linear Algebra

    We present some methods for constructing new representations from old ones. Techniques include direct sums, dual spaces, and tensors. Links include Solution Set 3, Linear Algebra Review 3, Problem Set 4, and videos RT4.1, RT4.1.1.

  5. Week 5 - Schur's Lemma

    Schur's lemma is a powerful tool for working with irreducible representations. Consequences include uniqueness results for equivalences and invariant forms. Links include Solution Set 4, Problem Set 5, and videos RT4.2. and RT5.

  6. Week 6 - Representations on Function Spaces

    For the rest of the course, we explore the connections between representations and functions. Here we explore basic properties, noting Fourier series as the prototype for groups (or vice versa if new). Links include Solution Set 5, Problem Set 6, and video RT6.

  7. Week 7 - Finite Abelian Groups: Characters

    We consider the vector space of functions on finite abelian G as a representation space for G. An orthonormal basis is given by the characters of G. Links include Solution Set 6, Problem Set 7, videos RT7 and GT18.1. (Dihedral Group review), an updated glossary, and a list of main results for analysis on groups.

  8. Week 8 - Finite Abelian Groups: Fourier Analysis

    Using the orthonormal basis of characters fof L^2 (G), we apply Fourier's Trick and Parseval's Identity. We also define convolution, and we apply convolution to obtain projection operators onto irreducible types in representations. Links include videos RT7.2 and RT7.3, Solution Set 7, and Problem Set 8.

  9. Week 9 - Finite Groups: Matrix Coefficients and L^2 (G)

    We repeat the story for finite abelian groups for general finite groups. We give orthonormal bases for L^2 (G) and Class(G), and, in turn, obtain important numeric information about G. The main result is the Schur Orthogonality Relations. Links include RT8.1 and RT8.2, Solution Set 8, and Problem Set 9.

  10. Week 10 - Finite Groups 2: Projection to Irreducibles

    Now that we can identify multiplicities of irreducibles in a given representation using characters, we give a formula for the orthogonal projections to the spaces for each irreducible type. Links include video RT8.3, Solution Set 9, and Problem Set 10.

  11. Week 11 - Basic Tensor Analysis

    Now that we have a basic theory of irreducibles using characters, we turn to the question of decomposing tensor product representations. In particular, we focus on 2-tensors, and the special cases of alternating and symmetric 2-tensors. we give character formulas for each case. Links include video RT9, Solution Set 10, and Problem Set 11.

  12. Week 12 - Application: Normal Modes (last class)

    We finish with an application of tensor analysis - finding normal modes of oscillating systems. In particular, we consider mass-spring systems in dimensions one and two. Links include RT9.1 and Solution Set 11.

Prerequisites

One semester of group theory - group actions and conjugacy classes are needed. No need for Sylow Theory that I can see.

One semester of linear algebra covering up to inner products, orthonormal bases, and orthogonal projections. A second course will help greatly, but further required topics will be addressed in the class.

I'll try to pitch the course towards advanced undergraduates, which means a heavy focus on teaching through examples.

Syllabus

The course will be run out of the subreddit /r/grouptheory2012 and material will be hosted at mathdoctorbob.org. Class will start the week of June 10 and should run from 9-12 weeks.

Tentative list of topics:

  1. Basic Definitions and Concepts

  2. Unitary Representations

  3. Equivalence and Examples

  4. Constructions from Linear Algebra

  5. Schur's Lemma

  6. Fourier series and the circle group

  7. Analysis on finite abelian groups

  8. Analysis on finite groups

  9. Symmetric and alternating tensors

The format will be the same as for the Group Theory class: each week links to materials are provided, and comments and questions can be sent directly to me or the subreddit. Of course, since students can add in at any time, questions are welcome on any covered sections.

Additional information

As with the group theory class, I don't have a book in mind. Fulton and Harris' Representation Theory is great (lots of exercises with hints, and Lie theory in the second half) if you have the background. Most of what I want to do is in the first 30 pages, but the analysis results are often pushed off to the exercises.

Terras' Fourier Analysis on Finite Groups and Applications is a good source for the specific direction of the course and provides a great deal of background and real-world applications.

Teacher qualifications

SchurThing - Over a decade teaching experience. Ph.D. from Stony Brook on Representations of Semi-simple Lie Groups. Post-docs include IAS and MIT.

See the U.Reddit class on Group Theory for previous work.

Latest Update
Group Representations - Last Lecture (2012-08-21 19:03:29)

"Hi All,

I just put up the last lecture and problem set for this course. Next week will be the last solution set.

If you are just signing up, no hurry. The course is designed to be taken at your own pace, and I'll keep an eye on the subreddit for questions once the course is archived.

Any comments or questions are appreciated.

Best, Bob (SchurThing)"

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