Representation theory
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Representation Theory — scope drawn from open representation theory notes and typical US graduate representation theory syllabi.
Finite groups
- Definitions — Definitions
- Irreducibles — Irreducibles
- Maschke's theorem — Maschke's theorem
- Complete reducibility — Complete reducibility
- Characters and orthogonality relations — Characters and orthogonality relations
- Induced and restricted representations — Induced and restricted representations
- Burnside's theorem and applications — Burnside's theorem and applications
Lie groups and Lie algebras
- Lie groups: examples and exponential map — Lie groups: examples and exponential map
- Lie algebras — Lie algebras
- Semisimplicity — Semisimplicity
- Representations of sl(2, C) — Representations of sl(2, C)
- Weight spaces — Weight spaces
- Root systems and Weyl group (introduction) — Root systems and Weyl group (introduction)
- Highest weight theory for semisimple Lie algebras (overview) — Highest weight theory for semisimple Lie algebras (overview)
Advanced topics
- Peter–Weyl theorem (compact groups) — Peter–Weyl theorem (compact groups)
- Tensor products — Tensor products
- Clebsch–Gordan decomposition (introduction) — Clebsch–Gordan decomposition (introduction)
- Category of representations — Category of representations
- Schur's lemma — Schur's lemma
- Introduction to geometric representation theory — Introduction to geometric representation theory
- Applications to symmetric functions — Applications to symmetric functions
- Combinatorics — Combinatorics
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Finite groups
Definitions
Coming soon - Lie groups and Lie algebras
Lie groups: examples and exponential map
Coming soon - Advanced topics
Peter–Weyl theorem (compact groups)
Coming soon
Notes
Graduate Representation Theory — scope drawn from open representation theory notes and typical US graduate representation theory syllabi. Topic outline: `content/topics/graduate/representation_theory.json`.