Representation Theory IV Jack Shotton (August 19, 2026) Contents 1 Introduction 1.1 Outline 1.2 References 1.3 Background review 1.4 Linear algebra 1.4.1 Fields 1.4.2 Linear maps 1.4.3 Subspaces, quotients, sums 1.4.4 Eigenspaces 1.5 Group theory 1.5.1 Subgroups, cosets, quotients 1.5.2 Homomorphisms 1.5.3 Symmetric groups 1.5.4 Actions 1.5.5 Zoo 2 Representation theory of finite groups 2.1 Definitions and first examples 2.2 Formalities 2.2.1 Subrepresentations and irreducibility 2.2.2 Homomorphisms and isomorphisms 2.2.3 Change of group 2.3 Example: dihedral groups 2.4 Schur’s Lemma 2.5 Maschke’s theorem and complete reducibility 2.5.1 Maschke’s theorem 2.5.2 Complete reducibility 2.5.3 Unitarizability 2.6 The group ring and the regular representation 3 Character theory 3.1 Characters 3.2 Orthogonality of characters 3.3 Example: S4 3.3.1 Characters of lifts 3.3.2 Decomposing a representation 3.4 Inner products and homomorphisms 3.5 Universal projections 3.6 Linear algebra constructions 3.6.1 The dual representation 3.6.2 Tensor products 3.6.3 Symmetric and alternating powers 3.6.4 Matrices in dimension 2 3.7 The character table of S5 4 Induced representations 4.1 Definition 4.2 Frobenius reciprocity 4.2.1 Example: S3 to S4 4.3 Characters 4.4 Example: D4 to S4 4.5 Mackey’s formula — nonexaminable 4.5.1 Double cosets 4.5.2 Mackey’s formula 4.5.3 Irreducibility criterion 4.5.4 Mackey examples 4.5.5 Index two 4.6 Example: GL2(Fp) — nonexaminable 4.6.1 The group GL2(Fp) 4.6.2 Characters of B and their inductions. 4.6.3 Computing the induced character 5 Linear Lie groups and their Lie algebras 5.1 Linear Lie groups 5.2 The exponential map 5.3 One-parameter subgroups 5.4 Lie algebras 5.5 Lie group and Lie algebra homomorphisms 5.6 Complex Lie groups and holomorphic homomorphisms 5.7 Topological properties 5.8 The example of SU(2) and SO(3) 6 Representations of Lie groups and Lie algebras - generalities 6.1 Basics 6.2 Standard constructions for representations 6.2.1 Functional constructions 6.3 The adjoint representation 6.4 The homomorphism SU(2)→SO(3) 6.5 Representations of U(1) and Maschke’s theorem 7 SL2 7.1 Weights 7.1.1 Highest weights 7.2 Classification of representations of sl2,C. 7.3 Decomposing representations 7.4 Real forms and complete reducibility 7.5 SO(3) 7.5.1 Classification of irreducible representations 7.5.2 Harmonic functions 8 SL3 8.1 The Lie algebra sl3,C 8.2 Weights 8.3 Visualising weights 8.4 Representations and weights 8.5 Tensor constructions 8.6 Highest weights 8.7 Dominant weights and Weyl symmetry 8.8 Irreducible representations of sl3,C. 8.9 Proof of theorem 8.28 — nonexaminable 9 Problems for Michaelmas 9.1 Problems for section 2 9.2 Problems for section 3 9.3 Problems for section 4 10 Problems for Epiphany 10.1 Problems for section 5 10.2 Problems for section 6 10.3 Problems for section 7 10.4 Problems for section 8