This is an introduction to the representation theory of finite and compact groups aimed primarily at advanced undergraduate and beginning graduate students. I assume that you have a strong basis in linear algebra and multivariate calculus, but no previous exposure to abstract algebra. The subject has applications in a wide range of fields in mathematics and science. I will emphasize, in particular, connections with harmonic analysis and, to a small degree, quantum theory, without assuming any knowledge of these subjects.
I plan to rely on the two texts given below. Important! Do not buy them. They are available in electronic form through the Olin Library web site. (Paper copies may also be purchased under Springer MyCopy plan, and can be had for $15.00 plus taxes, also through the Library’s web site.)
Representation Theory of Finite Groups: An Introductory Approach by Benjamin Steinberg. Universitext, Springer, 2012. Library link here. I plan to use this text in the first half or so of the course, covering representation of finite groups.
Groups and Symmetries: From Finite Groups to Lie Groups by Yvette Kosmann-Schwarzbach. Universitext, Springer, 2010. Library link here. I will rely on this text for the material on compact Lie groups and a few physical applications.
BSt \(=\) B.Steinberg and YKS \(=\) Kosmann-Schwarzbach.
Review of linear algebra (BSt Chapter 1.)
Group representations, general definitions and basic results (BSt Chapters 3, 4.)
Fourier analysis on finite groups (BSt Chapter 5.)
Additional topics on finite groups, as time permits, from chapters BSt 8, 10, 11. (Induced representations, the symmetric group, probability and random walks on groups.)
Matrix groups (classical Lie groups) and their Lie algebras (YKS Chapter 2 and 4, plus some supplementary material from elsewhere as needed, if what is in YKS turns out to be too sketchy.)
Representations of compact groups (YKS Chapter 3.)
The unitary groups (in dimensions 2 and 3) and their representations (YKS Chapters 5 and 6.)
Spherical harmonics and a few physics applications (YKS Chapters 7 and 8.)
There will be weekly homework assignments (I plan approximately 8 of them; we won’t drop the lowest grade.) They will be due Friday in my office. Important! We expect that you’ll write solutions neatly and cleanly; as a general advice, never turn in your first draft! Points will be taken for messy copies (for example, torn from your spiral notebook with that scraggly stuff on the edge, crossed out lines, etc.) Letter grades will be given according to the following scale: A= (93,100], A-=[90,93]. Similar for B, C, D, and F=[0,60). (A+ will be given sparingly.)
Following the suggestion of one of your classmates, I’m posting also the .tex files. This may be useful as a template if you would like to type your assignments in latex.