openAI-math-dataset / lean /OAI /RepresentationTheory /UniversalSquare /Balance /PackingSupport.lean
| import Mathlib | |
| import OAI.RepresentationTheory.Saxl.Main | |
| import OAI.RepresentationTheory.UniversalSquare.Support.ProductInduction | |
| import OAI.RepresentationTheory.UniversalSquare.Specht.CyclicRestriction | |
| import OAI.RepresentationTheory.UniversalSquare.Support.CoindProduct | |
| namespace OAI | |
| /-! Packing Support. -/ | |
| section | |
| noncomputable section | |
| open scoped TensorProduct MonoidAlgebra | |
| namespace Saxl | |
| theorem cyclic_supportLE_coind {G H X Y : Type*} | |
| [Group G] [Finite G] [Group H] [Finite H] | |
| [AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y] | |
| (ρ : Representation ℂ G X) (σ : Representation ℂ H Y) | |
| (φ : H →* G) (v : X) | |
| (T : Representation.IntertwiningMap σ (ρ.comp φ)) (hv : v ∈ T.range) : | |
| SupportLE (cyclic ρ v).toRepresentation (Representation.coind φ σ) := by | |
| apply intertwining_separates_of_simple_support | |
| intro S hS | |
| let := hS | |
| have hi : subrepInclusion S ≠ 0 := by | |
| intro hz | |
| let : Nontrivial S.toRepresentation.asModule := | |
| IsSimpleModule.nontrivial ℂ[G] S.toRepresentation.asModule | |
| let : Nontrivial S.toSubmodule := S.toRepresentation.asModuleEquiv.symm.toEquiv.nontrivial | |
| obtain ⟨x,hx⟩ := exists_ne (0 : S.toSubmodule) | |
| apply hx | |
| apply Subtype.ext | |
| exact congrArg (fun f : Representation.IntertwiningMap S.toRepresentation | |
| (cyclic ρ v).toRepresentation => f x) hz | |
| obtain ⟨F,hF⟩ := cyclic_restriction_support S.toRepresentation σ ρ φ v T hv | |
| (subrepInclusion S) hi | |
| exact ⟨coindLift φ F,coindLift_ne_zero φ F hF⟩ | |
| theorem SupportLE.cyclic_image {G X Y Z : Type*} [Group G] [Finite G] | |
| [AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y] | |
| [AddCommGroup Z] [Module ℂ Z] | |
| {ρ : Representation ℂ G X} {σ : Representation ℂ G Y} | |
| {τ : Representation ℂ G Z} (v : X) | |
| (T : Representation.IntertwiningMap ρ σ) (h : SupportLE (cyclic ρ v).toRepresentation τ) : | |
| SupportLE (cyclic σ (T v)).toRepresentation τ := by | |
| let Q := T.comp (subrepInclusion (cyclic ρ v)) | |
| have hm : T v ∈ Q.range := ⟨⟨v,mem_cyclic ρ v⟩,rfl⟩ | |
| obtain ⟨i,hi⟩ := subrepresentation_embeds_of_le_range Q (cyclic σ (T v)) | |
| ((cyclic_le _ _ _).mpr hm) | |
| exact (SupportLE.of_injective i hi).trans h | |
| def cyclicProductMap {n a b d : ℕ} (e : Fin n ≃ Fin a ⊕ Fin b) | |
| (v : WordSpace a d) (u : WordSpace b d) : | |
| Representation.IntertwiningMap | |
| (outer (cyclic (wordRep a d) v).toRepresentation (cyclic (wordRep b d) u).toRepresentation) | |
| ((wordRep n d).comp (sumPermHom e)) := | |
| externalWordMap e (subrepInclusion (cyclic (wordRep a d) v)) | |
| (subrepInclusion (cyclic (wordRep b d) u)) | |
| lemma cyclicProduct_mem_range {n a b d : ℕ} (e : Fin n ≃ Fin a ⊕ Fin b) | |
| (v : WordSpace a d) (u : WordSpace b d) : | |
| positionProduct e v u ∈ (cyclicProductMap e v u).range := by | |
| refine ⟨(⟨v,mem_cyclic _ _⟩ : (cyclic (wordRep a d) v).toSubmodule) ⊗ₜ[ℂ] | |
| (⟨u,mem_cyclic _ _⟩ : (cyclic (wordRep b d) u).toSubmodule), ?_⟩ | |
| exact positionTensor_tmul e v u | |
| theorem product_supportLE_coind {n a b d : ℕ} {G H X Y : Type*} | |
| [Group G] [Finite G] [Group H] [Finite H] | |
| [AddCommGroup X] [Module ℂ X] [FiniteDimensional ℂ X] | |
| [AddCommGroup Y] [Module ℂ Y] [FiniteDimensional ℂ Y] | |
| (φ : G →* Equiv.Perm (Fin a)) (ψ : H →* Equiv.Perm (Fin b)) | |
| (ρ : Representation ℂ G X) (σ : Representation ℂ H Y) | |
| (e : Fin n ≃ Fin a ⊕ Fin b) (v : WordSpace a d) (u : WordSpace b d) | |
| (hv : SupportLE (cyclic (wordRep a d) v).toRepresentation (Representation.coind φ ρ)) | |
| (hu : SupportLE (cyclic (wordRep b d) u).toRepresentation (Representation.coind ψ σ)) : | |
| SupportLE (cyclic (wordRep n d) (positionProduct e v u)).toRepresentation | |
| (Representation.coind ((sumPermHom e).comp (φ.prodMap ψ)) (outer ρ σ)) := by | |
| classical | |
| let : AddCommGroup ((cyclic (wordRep a d) v).toSubmodule ⊗[ℂ] | |
| (cyclic (wordRep b d) u).toSubmodule) := Module.addCommMonoidToAddCommGroup ℂ | |
| have h₀ := cyclic_supportLE_coind (wordRep n d) | |
| (outer (cyclic (wordRep a d) v).toRepresentation (cyclic (wordRep b d) u).toRepresentation) | |
| (sumPermHom e) (positionProduct e v u) (cyclicProductMap e v u) (cyclicProduct_mem_range e v u) | |
| have h₁ := (hv.outer hu).coind (sumPermHom e) | |
| have h₂ := (SupportLE.of_injective (coindOuter φ ψ ρ σ) | |
| (coindOuter_injective φ ψ ρ σ)).coind (sumPermHom e) | |
| have h₃ := SupportLE.of_injective (coindTrans (φ.prodMap ψ) (sumPermHom e) (outer ρ σ)) | |
| (coindTrans_injective (φ.prodMap ψ) (sumPermHom e) (outer ρ σ)) | |
| exact ((h₀.trans h₁).trans h₂).trans h₃ | |
| theorem young_cyclic_support {n : ℕ} (θ μ : YoungDiagram) (t : Tableau n μ) | |
| (hc : μ.card = θ.card) (hd : Dominates μ θ) | |
| (w : Fin n → Fin (θ.colLen 0)) (f : Equiv.Perm (Fin (θ.colLen 0))) | |
| (hw : ∀ j, (Finset.univ.filter (fun i => w i = f j)).card = θ.rowLen j) : | |
| ∃ F : Representation.IntertwiningMap (spechtRep t) | |
| (cyclic (wordRep n (θ.colLen 0)) (Pi.single w 1)).toRepresentation, F ≠ 0 := by | |
| obtain ⟨F,hF⟩ := young_orbit_map θ μ t hc hd w f hw | |
| let Q := F.comp (subrepInclusion (cyclic (wordRep n (θ.colLen 0)) (Pi.single w 1))) | |
| have hQ : Q ≠ 0 := by | |
| intro h0 | |
| exact hF (congrArg (fun T : Representation.IntertwiningMap | |
| (cyclic (wordRep n (θ.colLen 0)) (Pi.single w 1)).toRepresentation (spechtRep t) => | |
| T ⟨Pi.single w 1,mem_cyclic _ _⟩) h0) | |
| exact intertwining_reverse_nonzero Q hQ | |
| end Saxl | |
| end | |
| end | |
| end OAI | |