Representation theory
Graduate · Math
Syllabus focus
Theoretical / proof-based
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Topics typically covered
Theoretical / proof-based
Finite groups
- Definitions: representations, subrepresentations, and irreducibles
- Maschke's theorem and complete reducibility
- Characters and orthogonality relations
- Induced and restricted representations
- Burnside's theorem and applications
Lie groups and Lie algebras
- Lie groups: examples and exponential map
- Lie algebras: derivations, ideals, and semisimplicity
- Representations of sl(2, C) and weight spaces
- Root systems and Weyl group (introduction)
- Highest weight theory for semisimple Lie algebras (overview)
Advanced topics
- Peter–Weyl theorem (compact groups)
- Tensor products and Clebsch–Gordan decomposition (introduction)
- Category of representations and Schur's lemma
- Introduction to geometric representation theory
- Applications to symmetric functions and combinatorics
Notes
Topics reflect common graduate representation theory syllabi at US universities. Courses may emphasize finite groups, Lie theory, or algebraic aspects depending on department strengths.