Representation theory #

Representation theory is a beautiful field of mathematics. My PhD was in representation theory, in an area related to symmetric groups.

It’s a beautiful field. It has all the things I love about algebra, computations, classifications, beautiful emergent structure and clever tricks. It’s something I like to return to every now and then, even these days when I don’t do very much mathematics, because it reminds me well of the potent, austere beauty of mathematics.

Syllabus #

Here’s a rough syllabus of interesting topics in representation theory, say for a first graduate course like what is covered in Fulton and Harris (see below) or Linear Representations of Finite Groups by Serre.

  • Groups, rings, fields
  • Basic definitions and theorems for group representations
  • Character theory of finite groups
  • Symmetric and alternating groups (and lots of computations)
    • classification of representations
    • dimension formula (Frobenius)
  • Lie groups and Lie algebras
  • Fulton and Harris’ (graduate-level) introductory textbook on the subject. Lots of great exercises, examples and computations.
  • The Representation Theory of the Symmetric Groups by Gordon James. A classic.