Representations of finite groups

Seventeen lessons with complete arguments, worked examples and solved exercises: character theory and Fourier analysis, induction and Clifford theory, integer Brauer induction, rationality, symmetric groups, Schur–Weyl duality and general linear groups over finite fields.

Start with finite-dimensional linear algebra and elementary group theory. Each lesson states its additional imports and supplies links to the earlier course proofs.

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1. Representations and complete reducibilitySource
2. Characters and the orthogonality relationsSource
3. The group algebra and Fourier analysis on a finite groupSource
4. Tensor products, duals and real representationsSource
5. Integrality of characters and Burnside's \(p^a q^b\) theoremSource
6. Induced representations and Frobenius reciprocitySource
7. Mackey theory and Clifford's theoremSource
8. Groups with an abelian normal subgroup: the little-group methodSource
9. Frobenius groups and Frobenius's theoremSource
10. Artin's induction theorem and rationalitySource
11. Brauer's induction theoremSource
12. Consequences of Brauer's theorem: characterization of characters and splitting fieldsSource
13. The symmetric groups I: Young tableaux and Young symmetrizersSource
14. The symmetric groups II: branching, Jucys–Murphy elements and Young's seminormal formSource
15. The symmetric groups III: characters and symmetric functionsSource
16. Schur–Weyl dualitySource
17. Representations of GL₂ over finite fieldsSource

The proof index includes exact source and reader hashes. All lessons are self-checked by the writing AI; it records no blanket independent review of this edition.