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.
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
Links and resources #
- 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.