openAI-math-dataset / lean /OAI /RepresentationTheory /UniversalSquare /Support /SignedInduction.lean
| import Mathlib | |
| import OAI.RepresentationTheory.Saxl.Main | |
| import OAI.RepresentationTheory.UniversalSquare.Balance.BalancePacking | |
| namespace OAI | |
| /-! Signed Induction. -/ | |
| section | |
| noncomputable section | |
| open scoped TensorProduct | |
| namespace Saxl | |
| def externalTensorMap {a b : ℕ} {X X' Y Y' : Type*} | |
| [AddCommGroup X] [Module ℂ X] [AddCommGroup X'] [Module ℂ X'] | |
| [AddCommGroup Y] [Module ℂ Y] [AddCommGroup Y'] [Module ℂ Y'] | |
| {ρ : Representation ℂ (Equiv.Perm (Fin a)) X} | |
| {ρ' : Representation ℂ (Equiv.Perm (Fin a)) X'} | |
| {σ : Representation ℂ (Equiv.Perm (Fin b)) Y} | |
| {σ' : Representation ℂ (Equiv.Perm (Fin b)) Y'} | |
| (F : Representation.IntertwiningMap ρ ρ') (G : Representation.IntertwiningMap σ σ') : | |
| Representation.IntertwiningMap (externalTensor ρ σ) (externalTensor ρ' σ') where | |
| toLinearMap := TensorProduct.map F.toLinearMap G.toLinearMap | |
| isIntertwining' g := by | |
| ext x y | |
| change F (ρ g.1 x) ⊗ₜ[ℂ] G (σ g.2 y) = | |
| ρ' g.1 (F x) ⊗ₜ[ℂ] σ' g.2 (G y) | |
| rw [F.isIntertwining, G.isIntertwining] | |
| lemma externalTensorMap_injective {a b : ℕ} {X X' Y Y' : Type*} | |
| [AddCommGroup X] [Module ℂ X] [AddCommGroup X'] [Module ℂ X'] | |
| [AddCommGroup Y] [Module ℂ Y] [AddCommGroup Y'] [Module ℂ Y'] | |
| {ρ : Representation ℂ (Equiv.Perm (Fin a)) X} | |
| {ρ' : Representation ℂ (Equiv.Perm (Fin a)) X'} | |
| {σ : Representation ℂ (Equiv.Perm (Fin b)) Y} | |
| {σ' : Representation ℂ (Equiv.Perm (Fin b)) Y'} | |
| (F : Representation.IntertwiningMap ρ ρ') (G : Representation.IntertwiningMap σ σ') | |
| (hF : Function.Injective F) (hG : Function.Injective G) : | |
| Function.Injective (externalTensorMap F G) := | |
| TensorProduct.map_injective_of_flat_flat _ _ hF hG | |
| def signRestrictionMap {n a b : ℕ} {X Y Z : Type*} | |
| [AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y] | |
| [AddCommGroup Z] [Module ℂ Z] | |
| {ρ : Representation ℂ (Equiv.Perm (Fin n)) X} | |
| {σ : Representation ℂ (Equiv.Perm (Fin a)) Y} | |
| {τ : Representation ℂ (Equiv.Perm (Fin b)) Z} | |
| (e : Fin n ≃ Fin a ⊕ Fin b) | |
| (f : Representation.IntertwiningMap (ρ.comp (sumPermHom e)) (externalTensor σ τ)) : | |
| Representation.IntertwiningMap ((signTwist ρ).comp (sumPermHom e)) | |
| (externalTensor (signTwist σ) (signTwist τ)) where | |
| toLinearMap := f.toLinearMap | |
| isIntertwining' g := by | |
| ext x | |
| change f (signC (sumPerm e g.1 g.2) • ρ (sumPerm e g.1 g.2) x) = | |
| TensorProduct.map (signTwist σ g.1) (signTwist τ g.2) (f x) | |
| rw [map_smul] | |
| have hf := LinearMap.congr_fun (f.isIntertwining' (g.1,g.2)) x | |
| change f (ρ (sumPerm e g.1 g.2) x) = | |
| TensorProduct.map (σ g.1) (τ g.2) (f x) at hf | |
| rw [hf,signC_sumPerm] | |
| generalize f x = z | |
| induction z using TensorProduct.inductionOn with | |
| | add x y hx hy => simp only [map_add,smul_add,hx,hy] | |
| | tmul x y => | |
| change (signC g.1 * signC g.2) • (σ g.1 x ⊗ₜ[ℂ] τ g.2 y) = | |
| (signC g.1 • σ g.1 x) ⊗ₜ[ℂ] (signC g.2 • τ g.2 y) | |
| rw [TensorProduct.smul_tmul_smul] | |
| lemma signRestrictionMap_ne_zero {n a b : ℕ} {X Y Z : Type*} | |
| [AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y] | |
| [AddCommGroup Z] [Module ℂ Z] | |
| {ρ : Representation ℂ (Equiv.Perm (Fin n)) X} | |
| {σ : Representation ℂ (Equiv.Perm (Fin a)) Y} | |
| {τ : Representation ℂ (Equiv.Perm (Fin b)) Z} | |
| (e : Fin n ≃ Fin a ⊕ Fin b) | |
| (f : Representation.IntertwiningMap (ρ.comp (sumPermHom e)) (externalTensor σ τ)) | |
| (hf : f ≠ 0) : signRestrictionMap e f ≠ 0 := by | |
| intro h | |
| apply hf | |
| apply Representation.IntertwiningMap.ext | |
| apply LinearMap.ext | |
| intro x | |
| exact congrArg (fun H => H x) h | |
| def signDomainMap {n : ℕ} {X Y : Type*} [AddCommGroup X] [Module ℂ X] | |
| [AddCommGroup Y] [Module ℂ Y] | |
| {ρ : Representation ℂ (Equiv.Perm (Fin n)) X} | |
| {σ : Representation ℂ (Equiv.Perm (Fin n)) Y} | |
| (f : Representation.IntertwiningMap ρ (signTwist σ)) : | |
| Representation.IntertwiningMap (signTwist ρ) σ where | |
| toLinearMap := f.toLinearMap | |
| isIntertwining' g := by | |
| ext x | |
| change f (signC g • ρ g x) = σ g (f x) | |
| rw [map_smul] | |
| have h := LinearMap.congr_fun (f.isIntertwining' g) x | |
| change f (ρ g x) = signC g • σ g (f x) at h | |
| rw [h,smul_smul,signC_mul_self,one_smul] | |
| theorem signed_external_cyclic_support {n a b d : ℕ} {X Y Z : Type*} | |
| [AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y] | |
| [AddCommGroup Z] [Module ℂ Z] | |
| (ρ : Representation ℂ (Equiv.Perm (Fin n)) X) | |
| (σ : Representation ℂ (Equiv.Perm (Fin a)) Y) | |
| (τ : Representation ℂ (Equiv.Perm (Fin b)) Z) | |
| (e : Fin n ≃ Fin a ⊕ Fin b) (v : WordSpace a d) (u : WordSpace b d) | |
| (A : Fin d → Prop) (hv : v ∈ alphabetSub a d A) | |
| (hu : u ∈ alphabetSub b d (fun z => ¬ A z)) | |
| (F : Representation.IntertwiningMap σ (signTwist (cyclic (wordRep a d) v).toRepresentation)) | |
| (G : Representation.IntertwiningMap τ (signTwist (cyclic (wordRep b d) u).toRepresentation)) | |
| (hF : Function.Injective F) (hG : Function.Injective G) | |
| (f : Representation.IntertwiningMap (ρ.comp (sumPermHom e)) (externalTensor σ τ)) | |
| (hf : f ≠ 0) : | |
| ∃ H : Representation.IntertwiningMap (signTwist ρ) | |
| (cyclic (wordRep n d) (positionProduct e v u)).toRepresentation, H ≠ 0 := by | |
| exact external_cyclic_support (signTwist ρ) (signTwist σ) (signTwist τ) e v u A hv hu | |
| (signDomainMap F) (signDomainMap G) hF hG (signRestrictionMap e f) | |
| (signRestrictionMap_ne_zero e f hf) | |
| end Saxl | |
| end | |
| end | |
| end OAI | |