Math 362

Representation Theory

Author

Claudio Gómez-Gonzáles

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 for a trimester-long course at the advanced undergraduate or early graduate level, with additional topics of interest included to supplement potential forays beyond the usual scope of such a course. 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 (Serre 1977). 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.
  • Group Representations in Probability and Statistics (Diaconis 1988). A succinct introduction, geared towards developing Fourier theory for finite groups.
  • Lie Groups: An Introduction through Linear Groups (Rossmann 2002). A concrete introduction to Lie theory, emphasizing structure and representation with an eye toward applications in physics and geometry.
  • Abstract Algebra (Dummit and Foote 2003). This textbook was used recently for Math 342 and is a generally useful reference. Its chapters 18 and 19 cover some of the material we will discuss using distinct frameworks and with different motivations.