Representation Theory
Year 3 · 8 sections · 35 concepts · 203 practice questions
What's covered
free1 · Representations4 concepts
- Definition 1.1 · Representation
- Definition 1.2 · Subrepresentations, irreducibility and equivalence
- Example 1.3 · The dihedral representation
- Lemma 1.4 · G-homomorphisms
2 · Maschke's theorem4 concepts
- Theorem 2.1 · Maschke's theorem
- Proposition 2.2 · Invariant inner products and diagonalisability
- Corollary 2.3 · Orthogonal complements are invariant
- Example 2.4 · When complete reducibility fails
3 · Schur's lemma4 concepts
- Theorem 3.1 · Schur's lemma
- Corollary 3.2 · Irreducible representations of abelian groups
- Corollary 3.3 · Central elements act by scalars
- Corollary 3.4 · Multiplicities are well defined
4 · Characters and orthogonality6 concepts
- Definition 4.1 · Characters
- Theorem 4.2 · Orthogonality of irreducible characters
- Corollary 4.3 · Characters determine representations
- Corollary 4.4 · The irreducibility test
- Proposition 4.5 · What a character says about a single element
- Check 4.6 · Check: decomposing a representation
5 · Character tables7 concepts
- Theorem 5.1 · The regular representation and Σ dᵢ² = |G|
- Theorem 5.2 · Number of irreducibles = number of conjugacy classes
- Example 5.3 · Character tables of C₃ and S₃
- Corollary 5.4 · Column orthogonality
- Example 5.5 · Character tables of abelian groups
- Proposition 5.6 · Permutation characters
- Check 5.7 · Check: building a character table
6 · Constructing representations4 concepts
- Proposition 6.1 · Tensor products, duals, symmetric and exterior squares
- Proposition 6.2 · Lifting from quotients and linear characters
- Definition 6.3 · Induced representations
- Theorem 6.4 · Frobenius reciprocity
7 · More character tables3 concepts
- Example 7.1 · Character tables of D₈ and Q₈
- Example 7.2 · The character table of S₄
- Example 7.3 · The character table of A₄
8 · Arithmetic of characters and Burnside's theorem3 concepts
- Proposition 8.1 · Kernels, centres and normal subgroups from the table
- Theorem 8.2 · Degrees divide the group order
- Theorem 8.3 · Burnside's pᵃqᵇ theorem
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Lemma 1.4 · G-homomorphisms
Equivariant maps have invariant kernels and images; invertible ones are equivalences.
Proof
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