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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
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.
Tentative list of topics:
Basic Definitions and Concepts
Equivalence and Examples
Constructions from Linear Algebra
Fourier series and the circle group
Analysis on finite abelian groups
Analysis on finite groups
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.
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.
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.