| import Mathlib |
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| 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 |
|
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| namespace PartialPermutation |
|
|
| section Burnside |
| variable {R S : Type*} [Ring R] [Algebra ℂ R] [AddCommGroup S] |
| [Module ℂ S] [Module R S] [IsScalarTower ℂ R S] |
| [IsSimpleModule R S] [FiniteDimensional ℂ S] |
|
|
| lemma simple_action_surjective : Function.Surjective (Algebra.lsmul ℂ ℂ S (A := R)) := by |
| intro f |
| let E := Module.End R S |
| let f' : Module.End E S := |
| { toFun := f |
| map_add' := f.map_add |
| map_smul' := fun c x => by |
| obtain ⟨t, rfl⟩ := (IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed ℂ |
| (A := R) (V := S)).surjective c |
| change f (t • x) = t • f x |
| exact f.map_smul t x } |
| have : Module.Finite E S := Module.Finite.of_restrictScalars_finite ℂ E S |
| obtain ⟨r, hr⟩ := (Module.Finite.toModuleEnd_moduleEnd_surjective (R := R) (M := S)) f' |
| refine ⟨r, ?_⟩ |
| ext x |
| exact congrArg (fun A : Module.End E S => A x) hr |
|
|
| end Burnside |
|
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| def diagonalEnd {S : Type*} [AddCommGroup S] [Module ℂ S] (m : ℕ) : |
| Module.End ℂ S →ₐ[ℂ] Module.End ℂ (Fin m → S) where |
| toFun A := LinearMap.pi fun i => A.comp (LinearMap.proj i) |
| map_zero' := by ext; rfl |
| map_one' := by ext; rfl |
| map_add' := by intros; ext; rfl |
| map_mul' := by intros; ext; rfl |
| commutes' := by intro; ext; rfl |
|
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| section Isotypic |
| variable {R M S : Type*} [Ring R] [Algebra ℂ R] |
| [AddCommGroup M] [Module ℂ M] [Module R M] [IsScalarTower ℂ R M] |
| [AddCommGroup S] [Module ℂ S] [Module R S] [IsScalarTower ℂ R S] |
|
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| noncomputable def isotypicLift {m : ℕ} (e : M ≃ₗ[R] Fin m → S) : |
| Module.End ℂ S →ₐ[ℂ] Module.End ℂ M := |
| (e.restrictScalars ℂ).symm.conjAlgEquiv ℂ |>.toAlgHom.comp (diagonalEnd m) |
|
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| lemma isotypicLift_action {m : ℕ} (e : M ≃ₗ[R] Fin m → S) (r : R) : |
| isotypicLift e (Algebra.lsmul ℂ ℂ S r) = Algebra.lsmul ℂ ℂ M r := by |
| ext x |
| apply (e.restrictScalars ℂ).injective |
| simp [isotypicLift, diagonalEnd, LinearEquiv.conjAlgEquiv_apply, map_smul] |
| ext i |
| rfl |
|
|
| lemma isotypicLift_range [IsSimpleModule R S] [FiniteDimensional ℂ S] |
| {m : ℕ} (e : M ≃ₗ[R] Fin m → S) (A : Module.End ℂ S) : |
| ∃ r : R, isotypicLift e A = Algebra.lsmul ℂ ℂ M r := by |
| obtain ⟨r, rfl⟩ := simple_action_surjective (R := R) A |
| exact ⟨r, isotypicLift_action e r⟩ |
|
|
| noncomputable def isotypicOrbitMap {m : ℕ} (e : M ≃ₗ[R] Fin m → S) (v : M) : |
| Module.End ℂ S →ₗ[ℂ] M where |
| toFun A := isotypicLift e A v |
| map_add' A B := by simp |
| map_smul' c A := by simp |
|
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| lemma mem_isotypicOrbitMap_range {m : ℕ} (e : M ≃ₗ[R] Fin m → S) (v : M) : |
| v ∈ LinearMap.range (isotypicOrbitMap e v) := by |
| refine ⟨1, ?_⟩ |
| simp [isotypicOrbitMap] |
|
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| lemma isotypicOrbitMap_range_invariant {m : ℕ} (e : M ≃ₗ[R] Fin m → S) |
| (v : M) (r : R) (x : M) (hx : x ∈ LinearMap.range (isotypicOrbitMap e v)) : |
| r • x ∈ LinearMap.range (isotypicOrbitMap e v) := by |
| obtain ⟨A, rfl⟩ := hx |
| refine ⟨Algebra.lsmul ℂ ℂ S r * A, ?_⟩ |
| change isotypicLift e (_ * A) v = r • (isotypicLift e A v) |
| rw [map_mul, Module.End.mul_apply, isotypicLift_action] |
| rfl |
|
|
| lemma finrank_isotypicOrbitMap_range [FiniteDimensional ℂ S] {m : ℕ} |
| (e : M ≃ₗ[R] Fin m → S) (v : M) : |
| Module.finrank ℂ (LinearMap.range (isotypicOrbitMap e v)) ≤ |
| (Module.finrank ℂ S) ^ 2 := by |
| calc |
| _ ≤ Module.finrank ℂ (Module.End ℂ S) := LinearMap.finrank_range_le _ |
| _ = _ := by rw [Module.finrank_linearMap]; ring |
|
|
| end Isotypic |
|
|
| structure IsotypicMatrixModel (R M : Type*) [Ring R] [Algebra ℂ R] |
| [AddCommGroup M] [Module ℂ M] [Module R M] [IsScalarTower ℂ R M] where |
| degree : ℕ |
| degree_pos : 0 < degree |
| lift : Matrix (Fin degree) (Fin degree) ℂ →ₐ[ℂ] Module.End ℂ M |
| lift_image : ∀ A, ∃ r : R, lift A = Algebra.lsmul ℂ ℂ M r |
| orbit : ∀ v : M, ∃ W : Submodule ℂ M, v ∈ W ∧ |
| (∀ r : R, ∀ x ∈ W, r • x ∈ W) ∧ Module.finrank ℂ W ≤ degree ^ 2 |
|
|
| lemma exists_isotypicMatrixModel {R M : Type*} [Ring R] [Algebra ℂ R] |
| [AddCommGroup M] [Module ℂ M] [Module R M] [IsScalarTower ℂ R M] |
| [FiniteDimensional ℂ M] [Nontrivial M] [IsSemisimpleModule R M] |
| (h : IsIsotypic R M) : Nonempty (IsotypicMatrixModel R M) := by |
| have : Module.Finite R M := Module.Finite.of_restrictScalars_finite ℂ R M |
| obtain ⟨m, hm, S, hS, ⟨e⟩⟩ := h.linearEquiv_fun |
| let : IsSimpleModule R S := hS |
| have : Nontrivial S := IsSimpleModule.nontrivial R S |
| have : FiniteDimensional ℂ S := |
| Module.Finite.of_injective (S.subtype.restrictScalars ℂ) S.subtype_injective |
| 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 := ?_ |
| orbit := ?_ }⟩ |
| · intro A |
| exact isotypicLift_range e _ |
| · intro v |
| exact ⟨LinearMap.range (isotypicOrbitMap e v), mem_isotypicOrbitMap_range e v, |
| isotypicOrbitMap_range_invariant e v, finrank_isotypicOrbitMap_range e v⟩ |
|
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| end PartialPermutation |
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| end |
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| end OAI |
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