Theoretical / proof-based
Representation theory · Graduate · Math
Learning objectives from the Representation theory syllabus, grouped by unit. Click an objective for study materials.
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
Learning objectives
Click an objective for study materials.
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
Definitions and structure
- Definitions — Definitions
- Irreducibles — Irreducibles
- Maschke's theorem — Maschke's theorem
- Complete reducibility — Complete reducibility
- Characters and orthogonality relations — Characters and orthogonality relations
Proofs and reasoning
- Induced and restricted representations — Induced and restricted representations
- Burnside's theorem and applications — Burnside's theorem and applications
- Lie groups: examples and exponential map — Lie groups: examples and exponential map
- Lie algebras — Lie algebras
- Semisimplicity — Semisimplicity
Abstraction and generalization
- 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)
- Peter–Weyl theorem (compact groups) — Peter–Weyl theorem (compact groups)
Modeling and computation
- Apply definitions in engineering contexts — Apply definitions in engineering contexts
- Apply irreducibles in engineering contexts — Apply irreducibles in engineering contexts
- Apply maschke's theorem in engineering contexts — Apply maschke's theorem in engineering contexts
- Apply complete reducibility in engineering contexts — Apply complete reducibility in engineering contexts
- Apply characters and orthogonality relations in engineering contexts — Apply characters and orthogonality relations in engineering contexts
Data and technology
- Use software to explore representation theory problems numerically — Use software to explore representation theory problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Induced and restricted representations — Induced and restricted representations
- Burnside's theorem and applications — Burnside's theorem and applications
- Lie groups: examples and exponential map — Lie groups: examples and exponential map
- Lie algebras — Lie algebras
- Semisimplicity — Semisimplicity
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
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
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