| import Mathlib |
| import OAI.RepresentationTheory.Saxl.Main |
| import OAI.RepresentationTheory.UniversalSquare.Band.OneExtra |
|
|
| namespace OAI |
|
|
| /-! Dual Transfer. -/ |
|
|
| section |
|
|
| noncomputable section |
| open scoped TensorProduct |
| namespace Saxl |
|
|
| def intertwiningDual {G X Y : Type*} [Group G] [AddCommGroup X] [Module ℂ X] |
| [AddCommGroup Y] [Module ℂ Y] {ρ : Representation ℂ G X} |
| {σ : Representation ℂ G Y} (f : Representation.IntertwiningMap ρ σ) : |
| Representation.IntertwiningMap σ.dual ρ.dual where |
| toLinearMap := f.toLinearMap.dualMap |
| isIntertwining' g := by |
| ext l x |
| change l (σ g⁻¹ (f x)) = l (f (ρ g⁻¹ x)) |
| rw [f.isIntertwining] |
|
|
| def intertwiningFlip {G X Y : Type*} [Group G] [AddCommGroup X] [Module ℂ X] |
| [AddCommGroup Y] [Module ℂ Y] {ρ : Representation ℂ G X} |
| {σ : Representation ℂ G Y} (f : Representation.IntertwiningMap ρ σ.dual) : |
| Representation.IntertwiningMap σ ρ.dual where |
| toLinearMap := f.toLinearMap.flip |
| isIntertwining' g := by |
| ext y x |
| change f x (σ g y) = f (ρ g⁻¹ x) y |
| have h := congrArg (fun l : Module.Dual ℂ Y => l y) (LinearMap.congr_fun (f.isIntertwining' (g⁻¹)) x) |
| change f (ρ g⁻¹ x) y = f x (σ (g⁻¹)⁻¹ y) at h |
| rw [inv_inv] at h |
| exact h.symm |
|
|
| lemma intertwiningFlip_ne_zero {G X Y : Type*} [Group G] |
| [AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y] |
| {ρ : Representation ℂ G X} {σ : Representation ℂ G Y} |
| (f : Representation.IntertwiningMap ρ σ.dual) (hf : f ≠ 0) : |
| intertwiningFlip f ≠ 0 := by |
| intro hz |
| apply hf |
| ext x y |
| exact congrArg (fun F => F y x) hz |
|
|
| theorem specht_support_of_dual {n : ℕ} {μ : YoungDiagram} (t : Tableau n μ) |
| {Y : Type*} [AddCommGroup Y] [Module ℂ Y] |
| (σ : Representation ℂ (Equiv.Perm (Fin n)) Y) |
| (f : Representation.IntertwiningMap (spechtRep t) σ.dual) (hf : f ≠ 0) : |
| ∃ h : Representation.IntertwiningMap (spechtRep t) σ, Function.Injective h := by |
| let F := (spechtSelfDual t).symm.toIntertwiningMap.comp (intertwiningFlip f) |
| have hF : F ≠ 0 := intertwining_comp_ne_zero |
| (spechtSelfDual t).symm.toIntertwiningMap (spechtSelfDual t).symm.injective |
| (intertwiningFlip f) (intertwiningFlip_ne_zero f hf) |
| let := specht_irreducible t |
| have hs : Function.Surjective F := |
| (Representation.IsIrreducible.surjective_or_eq_zero F).resolve_right hF |
| obtain ⟨h, hh⟩ := intertwining_lift_surjective F hs |
| (Representation.IntertwiningMap.id (spechtRep t)) |
| refine ⟨h, ?_⟩ |
| intro x y he |
| have hx := congrArg (fun H => H x) hh |
| have hy := congrArg (fun H => H y) hh |
| exact hx.symm.trans ((congrArg F he).trans hy) |
|
|
| def tensorDualEquiv {G X Y : Type*} [Group G] [AddCommGroup X] [Module ℂ X] |
| [AddCommGroup Y] [Module ℂ Y] [Module.Finite ℂ X] [Module.Finite ℂ Y] |
| (ρ : Representation ℂ G X) (σ : Representation ℂ G Y) : |
| (ρ.dual.tprod σ.dual).Equiv (ρ.tprod σ).dual where |
| toLinearEquiv := TensorProduct.dualDistribEquiv ℂ X Y |
| isIntertwining' g := by |
| ext l m x y |
| rfl |
|
|
| def tensorSignCancel {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) : |
| ((signTwist ρ).tprod (signTwist σ)).Equiv (ρ.tprod σ) where |
| toLinearEquiv := LinearEquiv.refl ℂ _ |
| isIntertwining' g := by |
| ext x y |
| change (signC g • ρ g x) ⊗ₜ[ℂ] (signC g • σ g y) = ρ g x ⊗ₜ[ℂ] σ g y |
| rw [TensorProduct.smul_tmul_smul, signC_mul_self, one_smul] |
|
|
| end Saxl |
| end |
| end |
|
|
| end OAI |
|
|
