| import Mathlib | |
| import OAI.Probability.BinarySweep.YoungTheory.YoungTranspose | |
| import OAI.Probability.BinarySweep.Conditional.ConditionalSpecht | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators Classical | |
| namespace BinaryCoordinateSweeps.Young | |
| open Irrep Representation | |
| variable (μ : YoungDiagram) | |
| lemma norm_signC (g : G μ) : ‖signC μ g‖=1 := by | |
| have h := congrArg Norm.norm (signC_sq μ g) | |
| rw [norm_mul,norm_one] at h | |
| nlinarith [norm_nonneg (signC μ g)] | |
| lemma hilbert_transpose_equiv : Nonempty ((hilbertSpecht μ).Equiv | |
| (signTwist μ ((hilbertSpecht μ.transpose).comp (transposeCells μ).permCongrHom.toMonoidHom))) := by | |
| obtain ⟨f⟩ := specht_transpose_sign μ | |
| let E := (spechtHilbertEquiv μ).symm.trans (f.toLinearEquiv.trans (spechtHilbertEquiv μ.transpose)) | |
| refine ⟨.mk E ?_⟩ | |
| intro g | |
| apply LinearMap.ext | |
| intro v | |
| obtain ⟨w,rfl⟩ := (spechtHilbertEquiv μ).surjective v | |
| change spechtHilbertEquiv μ.transpose | |
| (f.toLinearEquiv.toFun ((spechtHilbertEquiv μ).symm.toFun | |
| (spechtHilbertEquiv μ (spechtRep μ g ((spechtHilbertEquiv μ).symm.toFun | |
| (spechtHilbertEquiv μ w)))))) = | |
| signC μ g • spechtHilbertEquiv μ.transpose | |
| (spechtRep μ.transpose ((transposeCells μ).permCongr g) | |
| ((spechtHilbertEquiv μ.transpose).symm.toFun | |
| (spechtHilbertEquiv μ.transpose (f.toLinearEquiv.toFun | |
| ((spechtHilbertEquiv μ).symm.toFun (spechtHilbertEquiv μ w)))))) | |
| have hs (ν : YoungDiagram) (x : SpechtSpace ν) : | |
| (spechtHilbertEquiv ν).symm.toFun (spechtHilbertEquiv ν x) = x := | |
| (spechtHilbertEquiv ν).symm_apply_apply x | |
| simp only [hs] | |
| have h := Representation.IntertwiningMap.isIntertwining _ _ f.toIntertwiningMap g w | |
| change f.toLinearEquiv.toFun (spechtRep μ g w) = signC μ g • | |
| spechtRep μ.transpose ((transposeCells μ).permCongr g) (f.toLinearEquiv.toFun w) at h | |
| exact (congrArg (spechtHilbertEquiv μ.transpose).toFun h).trans (map_smul _ _ _) | |
| lemma hilbert_transpose_moment (p : G μ → ℂ) (q : ℕ) : | |
| evenMoment q (groupAverage (hilbertSpecht μ) p) = | |
| evenMoment q (groupAverage (hilbertSpecht μ.transpose) | |
| (fun g => p ((transposeCells μ).symm.permCongr g)*signC μ.transpose g)) := by | |
| obtain ⟨f⟩ := hilbert_transpose_equiv μ | |
| rw [equivalent_evenMoment _ _ f (hilbertSpecht_unitary μ) (fun g v => by | |
| change ‖signC μ g • hilbertSpecht μ.transpose ((transposeCells μ).permCongr g) v‖=‖v‖ | |
| rw [norm_smul,norm_signC,hilbertSpecht_unitary,one_mul])] | |
| congr 1 | |
| unfold groupAverage | |
| rw [←Equiv.sum_comp (transposeCells μ).permCongr] | |
| apply Finset.sum_congr rfl | |
| intro g _ | |
| change p g • (signC μ g • hilbertSpecht μ.transpose ((transposeCells μ).permCongr g)) = _ | |
| dsimp only | |
| rw [smul_smul,transpose_sign] | |
| rw [show (transposeCells μ).symm.permCongr ((transposeCells μ).permCongr g)=g from | |
| (transposeCells μ).permCongr.symm_apply_apply g] | |
| end BinaryCoordinateSweeps.Young | |
| namespace BinaryCoordinateSweeps | |
| open Young Irrep | |
| lemma realSign_permCongr_complex {X Y : Type*} [Fintype X] [Fintype Y] | |
| [DecidableEq X] [DecidableEq Y] (e : X≃Y) (g : Equiv.Perm X) : | |
| (realSign (e.permCongr g):ℂ)=(realSign g:ℂ) := by | |
| exact congrArg (fun sign : ℤˣ => (((sign : ℤ) : ℝ) : ℂ)) | |
| (Equiv.Perm.sign_permCongr e g) | |
| lemma conditional_transpose_moment {b h : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h) | |
| (μ : YoungDiagram) (e : Cell μ ≃ FreeSlot H 0) (z : ℝ) (q : ℕ) : | |
| evenMoment q (groupAverage ((hilbertSpecht μ).comp e.symm.permCongrHom.toMonoidHom) | |
| (fun g => (conditionalGroupLaw H z g:ℂ))) = | |
| evenMoment q (groupAverage ((hilbertSpecht μ.transpose).comp | |
| ((transposeCells μ).symm.trans e).symm.permCongrHom.toMonoidHom) | |
| (fun g => (conditionalGroupLaw H z g*realSign g:ℂ))) := by | |
| rw [groupAverage_comp_slots,groupAverage_comp_slots,hilbert_transpose_moment] | |
| congr 2 | |
| funext g | |
| congr 1 | |
| rw [realSign_permCongr_complex] | |
| change ((Equiv.Perm.sign g:ℤ):ℂ)=(((Equiv.Perm.sign g:ℤ):ℝ):ℂ) | |
| norm_cast | |
| end BinaryCoordinateSweeps | |
| end | |
| end OAI | |