Math 362: Representation Theory
Course notes
Preface
These lectures notes are intended as the primary reference for an introductory course in Representation Theory, with a preliminary emphasis on finite groups and eventual broadening into the theory of compact groups. The material is intended, perhaps non-exhaustively and with room for extended forays into topics of interest, for a trimester-long course at the advanced undergraduate or early graduate level. Any errors are due to Claudio, who welcomes feedback and corrections.
Potential additional references include:
- Representation Theory: A First Course (Fulton and Harris 1991). Often regarded as the definitive introductory text, though it relies on familiarity with abstract frameworks.
- Linear Representations of Finite Groups (Scott and Serre 2012). This is affectionately known as “Serre’s little book,” a concise and elegant translation that hits a variety of topics. This book was written for Josiane Heulot-Serre (the author, Jean-Pierre, was her husband) to teach students in quantum chemistry, though it is decidely a mathematical text.
- Linear Algebra and Group Representations (Shaw 1982). Features many examples, where much of the needed linear algebra and group theory is developed as the book progresses.
- Character Theory of Finite Groups (Isaacs 1994). A great reference on the relevant techniques in character theory, including an introduction to the theory of Schur covers.
- Abstract Algebra (Dummit and Foote 2003). This textbook was used recently for Math 342. Its chapters 18 and 19 cover some of the material we will discuss using distinct frameworks and with different motivations.
In addition, Diaconis’ Group Representations in Probability and Statistics (Diaconis 1988) gives a succinct introduction geared towards developing Fourier theory, which is a primary focus of this course.