Theorems · Theorem · group theory

Representation.char_orthonormal

∀ {G : Type u_1} {k : Type u_2} {V : Type u_3} {W : Type u_4} [inst : Group G] [inst_1 : Field k]
  [inst_2 : AddCommGroup V] [inst_3 : Module k V] [FiniteDimensional k V] [inst_5 : AddCommGroup W]
  [inst_6 : Module k W] [FiniteDimensional k W] (ρ : Representation k G V) (σ : Representation k G W)
  [inst_8 : Fintype G] [Invertible ↑(Nat.card G)] [IsAlgClosed k] [ρ.IsIrreducible] [σ.IsIrreducible],
  (↑(Nat.card G))⁻¹ * ∑ g, ρ.character g * σ.character g⁻¹ = if Nonempty (σ.Equiv ρ) then 1 else 0

Orthogonality of characters for irreducible representations of finite group over an algebraically closed field whose characteristic doesn't divide the order of the group.

Defined in

Mathlib.RepresentationTheory.Character

Cited by

0 results in Mathlib

Foundations

Depth 136 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

GroupFieldAddCommGroupModuleFiniteDimensionalAddCommGroupModuleFiniteDimensionalFintypeInvertibleIsAlgClosedRepresentation.IsIrreducibleRepresentation.IsIrreducible