| import Mathlib |
| import OAI.RepresentationTheory.PartialPermutation.Characters |
| import OAI.RepresentationTheory.PartialPermutation.IsotypicMatrices |
|
|
| namespace OAI |
|
|
| section |
| open scoped Classical |
| open scoped BigOperators ComplexConjugate MonoidAlgebra |
| open scoped BigOperators ComplexConjugate |
| open scoped MonoidAlgebra BigOperators |
| open scoped BigOperators MonoidAlgebra Classical |
|
|
| attribute [local instance] Classical.propDecidable |
| open scoped BigOperators |
| open scoped MonoidAlgebra BigOperators |
|
|
| namespace PartialPermutation |
| open scoped MonoidAlgebra BigOperators ComplexConjugate |
|
|
| noncomputable section |
| variable {G M : Type*} [Group G] |
| [NormedAddCommGroup M] [InnerProductSpace ℂ M] |
| [Module ℂ[G] M] [IsScalarTower ℂ ℂ[G] M] |
|
|
| lemma ofModule'_algebra : (Representation.ofModule' (k := ℂ) (G := G) M).asAlgebraHom = |
| Algebra.lsmul ℂ ℂ M := by |
| exact (MonoidAlgebra.lift ℂ (Module.End ℂ M) G).apply_symm_apply _ |
|
|
| lemma ofModule'_apply (g : G) (x : M) : |
| Representation.ofModule' (k := ℂ) (G := G) M g x = |
| (MonoidAlgebra.single g (1 : ℂ)) • x := by |
| rw [← Representation.asAlgebraHom_single_one, ofModule'_algebra] |
| rfl |
| |
| noncomputable def ofModule'ModuleEquiv : |
| (Representation.ofModule' (k := ℂ) (G := G) M).asModule ≃ₗ[ℂ[G]] M where |
| toFun x := x |
| invFun x := x |
| left_inv _ := rfl |
| right_inv _ := rfl |
| map_add' _ _ := rfl |
| map_smul' a x := by |
| change (Representation.ofModule' (k := ℂ) (G := G) M).asAlgebraHom a x = a • (show M from x) |
| rw [ofModule'_algebra] |
| rfl |
| |
| lemma ofModule'_irreducible [IsSimpleModule ℂ[G] M] : |
| Representation.IsIrreducible (Representation.ofModule' (k := ℂ) (G := G) M) := by |
| rw [Representation.irreducible_iff_isSimpleModule_asModule] |
| exact IsSimpleModule.congr (ofModule'ModuleEquiv (G := G) (M := M)) |
|
|
| structure IsotypicKernelModel (G M : Type*) [Group G] [Fintype G] |
| [NormedAddCommGroup M] [InnerProductSpace ℂ M] |
| [Module ℂ[G] M] [IsScalarTower ℂ ℂ[G] M] extends IsotypicMatrixModel ℂ[G] M where |
| kernel : G → ℂ |
| kernel_norm : ∑ g, ‖kernel g‖^2 = (degree : ℝ)^2 / Fintype.card G |
| kernel_inv : ∀ g, kernel g⁻¹ = conj (kernel g) |
| kernel_action : ∀ x : M, ∑ g, kernel g • (MonoidAlgebra.single g (1 : ℂ)) • x = x |
|
|
| variable [Fintype G] |
|
|
| lemma exists_isotypicKernelModel [FiniteDimensional ℂ M] [Nontrivial M] |
| (h : IsIsotypic ℂ[G] M) |
| (hu : ∀ (g : G) (x y : M), inner ℂ ((MonoidAlgebra.single g (1 : ℂ)) • x) |
| ((MonoidAlgebra.single g (1 : ℂ)) • y) = inner ℂ x y) : |
| Nonempty (IsotypicKernelModel G M) := by |
| have : Module.Finite ℂ[G] M := Module.Finite.of_restrictScalars_finite ℂ ℂ[G] M |
| obtain ⟨m, hm, S, hS, ⟨e⟩⟩ := h.linearEquiv_fun |
| let : IsSimpleModule ℂ[G] S := hS |
| have : Nontrivial S := IsSimpleModule.nontrivial ℂ[G] S |
| have : FiniteDimensional ℂ S := |
| Module.Finite.of_injective (S.subtype.restrictScalars ℂ) S.subtype_injective |
| let : NormedAddCommGroup S := inferInstanceAs (NormedAddCommGroup (S.restrictScalars ℂ)) |
| let : InnerProductSpace ℂ S := inferInstanceAs (InnerProductSpace ℂ (S.restrictScalars ℂ)) |
| let σ := Representation.ofModule' (k := ℂ) (G := G) S |
| have : Representation.IsIrreducible σ := ofModule'_irreducible |
| have hσ : IsUnitary σ := by |
| intro g x y |
| change inner ℂ (Representation.ofModule' (k := ℂ) (G := G) S g x) |
| (Representation.ofModule' (k := ℂ) (G := G) S g y) = _ |
| rw [ofModule'_apply, ofModule'_apply] |
| exact hu g x.val y.val |
| let k : G → ℂ := fun g => ((Module.finrank ℂ S : ℂ) / Fintype.card G) * conj (σ.character g) |
| let F := (isotypicLift e).comp (Matrix.toLinAlgEquiv (Module.finBasis ℂ S)).toAlgHom |
| refine ⟨{ degree := Module.finrank ℂ S |
| degree_pos := Module.finrank_pos |
| lift := F |
| lift_image := fun A => isotypicLift_range e _ |
| orbit := fun v => ⟨LinearMap.range (isotypicOrbitMap e v), |
| mem_isotypicOrbitMap_range e v, isotypicOrbitMap_range_invariant e v, |
| finrank_isotypicOrbitMap_range e v⟩ |
| kernel := k |
| kernel_norm := ?_ |
| kernel_inv := ?_ |
| kernel_action := ?_ }⟩ |
| · have hN : (Fintype.card G : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero |
| simp only [k, norm_mul, norm_div, Complex.norm_natCast, Complex.norm_conj, |
| mul_pow, ← Finset.mul_sum, character_norm_sq_sum σ hσ] |
| field_simp |
| · intro g |
| simp [k, unitary_character_inv σ hσ] |
| · have he' := congrArg (isotypicLift e) (character_kernel_identity σ hσ) |
| have ha (g : G) : isotypicLift e (σ g) = |
| Algebra.lsmul ℂ ℂ M (MonoidAlgebra.single g (1 : ℂ)) := by |
| change isotypicLift e (Representation.ofModule' (k := ℂ) (G := G) S g) = _ |
| rw [← Representation.asAlgebraHom_single_one, ofModule'_algebra, isotypicLift_action] |
| change isotypicLift e (complexFourier σ k) = isotypicLift e 1 at he' |
| simp only [complexFourier, map_sum, map_smul, map_one, ha] at he' |
| intro x |
| have ht := congrArg (fun A : Module.End ℂ M => A x) he' |
| simpa only [LinearMap.sum_apply, LinearMap.smul_apply, Algebra.lsmul_apply, |
| Module.End.one_apply] using ht |
|
|
| end |
| end PartialPermutation |
|
|
| end |
|
|
| end OAI |
|
|
