ρ:G→GL(V) homomorphism. Subrepresentation, irreducible. Maschke: char∤|G| ⇒ complete reducibility. Decomposition into irreducibles.
graph TD
D1["Def: Representation\nρ:G→GL(V)"]
D2["Def: Subrepresentation\nW ≤ V, GW⊆W"]
D3["Def: Irreducible\nno proper subrep"]
D4["Def: Direct sum\nV=⊕Vᵢ"]
T1["Thm: Maschke\nchar∤|G| ⇒ semisimple"]
T2["Thm: Complete reducibility\nV=⊕ irreducibles"]
T3["Thm: Schur's lemma\nHom irreps 1-dim"]
T4["Thm: Rep = kG-mod\nkG module"]
T5["Thm: Regular rep\ndim⊕Vᵢ²=|G|"]
D1 --> D2
D2 --> D3
D3 --> T2
D4 --> T2
T1 --> T2
T3 --> T5
T4 --> D1
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
class D1,D2,D3,D4 definition
class T1,T2,T3,T4,T5 theorem
Process Statistics
- Nodes: 14
- Edges: 12
- Definitions: 4
- Theorems: 5
Frontier: Modular reps, blocks, derived categories. math.RT