| import Mathlib | |
| import OAI.RepresentationTheory.PartialPermutation.MatrixCoefficients | |
| namespace OAI | |
| section | |
| open scoped Classical | |
| open scoped BigOperators ComplexConjugate MonoidAlgebra | |
| open scoped BigOperators ComplexConjugate | |
| namespace PartialPermutation | |
| open scoped BigOperators ComplexConjugate MonoidAlgebra | |
| noncomputable section | |
| variable {G V : Type*} [Group G] [Fintype G] | |
| [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] | |
| (ρ : Representation ℂ G V) (hρ : IsUnitary ρ) | |
| include hρ | |
| omit [Fintype G] in | |
| lemma unitary_adjoint (g : G) : LinearMap.adjoint (ρ g) = ρ g⁻¹ := by | |
| ext v | |
| apply ext_inner_left ℂ | |
| intro u | |
| rw [LinearMap.adjoint_inner_right] | |
| have h := hρ g u (ρ g⁻¹ v) | |
| simpa using h | |
| omit [Fintype G] in | |
| lemma unitary_character_inv (g : G) : ρ.character g⁻¹ = conj (ρ.character g) := by | |
| let e := stdOrthonormalBasis ℂ V | |
| rw [Representation.character, Representation.character, ← unitary_adjoint ρ hρ, | |
| LinearMap.trace_eq_sum_inner _ e, LinearMap.trace_eq_sum_inner _ e, map_sum] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [LinearMap.adjoint_inner_right, inner_conj_symm] | |
| variable [Representation.IsIrreducible ρ] | |
| lemma character_norm_sq_sum : ∑ g, ‖ρ.character g‖ ^ 2 = (Fintype.card G : ℝ) := by | |
| let := invertibleOfNonzero (show (Nat.card G : ℂ) ≠ 0 by simp [Nat.card_eq_fintype_card]) | |
| have h := Representation.char_orthonormal ρ ρ | |
| have he : Nonempty (Representation.Equiv ρ ρ) := ⟨Representation.Equiv.refl ρ⟩ | |
| simp only [ite_eq_left he, unitary_character_inv ρ hρ] at h | |
| have hN : (Nat.card G : ℂ) ≠ 0 := by simp [Nat.card_eq_fintype_card] | |
| have hsum : (∑ g, ρ.character g * conj (ρ.character g)) = (Nat.card G : ℂ) := by | |
| simpa [hN, mul_assoc] using congrArg (fun z : ℂ => (Nat.card G : ℂ) * z) h | |
| have ht := congrArg Complex.re hsum | |
| simpa only [Complex.re_sum, Complex.mul_conj, Complex.ofReal_re, | |
| Complex.normSq_eq_norm_sq, Complex.natCast_re, Nat.card_eq_fintype_card] using ht | |
| lemma character_kernel_identity : | |
| complexFourier ρ (fun g => ((Module.finrank ℂ V : ℂ) / Fintype.card G) * | |
| conj (ρ.character g)) = 1 := by | |
| classical | |
| let e := stdOrthonormalBasis ℂ V | |
| have : Nontrivial V := IsSimpleModule.nontrivial ℂ[G] ρ.asModule | |
| have hD : (Module.finrank ℂ V : ℂ) ≠ 0 := by | |
| exact_mod_cast (Module.finrank_pos (R := ℂ) (M := V)).ne' | |
| have hN : (Fintype.card G : ℂ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero | |
| ext y | |
| apply ext_inner_left ℂ | |
| intro x | |
| simp only [complexFourier, LinearMap.sum_apply, LinearMap.smul_apply, Module.End.one_apply] | |
| simp only [inner_sum, inner_smul_right, mul_assoc, ← Finset.mul_sum] | |
| have hs : (Fintype.card G : ℂ)⁻¹ * ∑ g, conj (ρ.character g) * inner ℂ x (ρ g y) = | |
| (Module.finrank ℂ V : ℂ)⁻¹ * inner ℂ x y := by | |
| simp_rw [Representation.character, LinearMap.trace_eq_sum_inner _ e, map_sum, | |
| Finset.sum_mul] | |
| rw [Finset.sum_comm, Finset.mul_sum] | |
| have hc (i : Fin (Module.finrank ℂ V)) := | |
| matrixCoefficient_orthogonality ρ hρ x (e i) y (e i) | |
| simp_rw [mul_comm (conj _), hc, div_eq_mul_inv] | |
| simpa [Finset.mul_sum, mul_assoc, mul_left_comm, mul_comm] using | |
| congrArg (fun z : ℂ => (Module.finrank ℂ V : ℂ)⁻¹ * z) (e.sum_inner_mul_inner x y) | |
| rw [div_eq_mul_inv, mul_assoc, hs, ← mul_assoc, mul_inv_cancel₀ hD, one_mul] | |
| end | |
| end PartialPermutation | |
| end | |
| end OAI | |