| import Mathlib |
| import OAI.Probability.BinarySweep.Representations.IsotypicProjectionMoment |
| import OAI.Probability.BinarySweep.FiniteLaws.LinearEvenMoment |
| import OAI.Probability.BinarySweep.FiniteLaws.IndependentPush |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps |
| open Irrep Representation TraceHolder Density |
|
|
| lemma matrixMoment_linear {I : Type*} [Fintype I] [DecidableEq I] |
| (q : ℕ) (M : Matrix I I ℂ) : matrixMoment q M=evenMoment q M.toEuclideanLin := by |
| unfold matrixMoment evenMoment |
| rw [← trace_toEuclideanLinear] |
| simp only [euclidean_pow,Matrix.toLpLin_mul_same,←Module.End.mul_eq_comp, |
| Matrix.star_eq_conjTranspose,Matrix.toEuclideanLin_conjTranspose_eq_adjoint] |
|
|
| namespace Signed |
| variable {A X : Type*} [Fintype A] [DecidableEq A] [Fintype X] {N : ℕ} |
| {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] |
|
|
| def sampleMatrix (p : A → Bool) (w : X → ℂ) (f : X → Equiv.Perm (Fin N)) := |
| ∑x, w x • actionMatrix p (f x) |
|
|
| lemma sampleMatrix_linear (p : A → Bool) (w : X → ℂ) (f : X → Equiv.Perm (Fin N)) : |
| (sampleMatrix p w f).toEuclideanLin=∑x, w x • hilbertTensorRep p N (f x) := by |
| simp only [sampleMatrix,map_sum,map_smul,actionMatrix_linear] |
|
|
| lemma sampleMatrix_average (p : A → Bool) (w : X → ℂ) (f : X → Equiv.Perm (Fin N)) : |
| sampleMatrix p w f=signedAverage p (finitePush w f) := by |
| exact (finitePush_expectation w f (actionMatrix p)).symm |
|
|
| omit [FiniteDimensional ℂ V] in |
| lemma sampleMatrix_commute (p : A → Bool) |
| (ρ : Representation ℂ (Equiv.Perm (Fin N)) V) (w : X → ℂ) (f : X → Equiv.Perm (Fin N)) : |
| isotypicProjection p ρ*sampleMatrix p w f=sampleMatrix p w f*isotypicProjection p ρ := by |
| rw [sampleMatrix_average]; exact isotypicProjection_commutes_average p ρ _ |
|
|
| lemma sample_projected_moment (p : A → Bool) |
| (ρ : Representation ℂ (Equiv.Perm (Fin N)) V) [ρ.IsIrreducible] |
| (hρ : ∀g v, ‖ρ g v‖=‖v‖) (w : X → ℂ) (f : X → Equiv.Perm (Fin N)) {q : ℕ} (hq : 0<q) : |
| matrixMoment q (isotypicProjection p ρ*sampleMatrix p w f)= |
| (Module.finrank ℂ (IntertwiningMap ρ (hilbertTensorRep p N)):ℝ)* |
| evenMoment q (∑x, w x • ρ (f x)) := by |
| rw [sampleMatrix_average,local_projected_moment p ρ hρ _ hq] |
| rw [groupAverage,finitePush_expectation] |
|
|
| lemma sampleMoment_le_projected (p : A → Bool) |
| (ρ : Representation ℂ (Equiv.Perm (Fin N)) V) [ρ.IsIrreducible] |
| (hρ : ∀g v, ‖ρ g v‖=‖v‖) |
| (hocc : 0<Module.finrank ℂ (IntertwiningMap ρ (hilbertTensorRep p N))) |
| (w : X → ℂ) (f : X → Equiv.Perm (Fin N)) {q : ℕ} (hq : 0<q) : |
| evenMoment q (∑x, w x • ρ (f x))≤ |
| matrixMoment q (isotypicProjection p ρ*sampleMatrix p w f) := by |
| rw [sample_projected_moment p ρ hρ w f hq] |
| exact le_mul_of_one_le_left (evenMoment_nonneg _ _) (by exact_mod_cast hocc) |
|
|
| end Signed |
| end BinaryCoordinateSweeps |
|
|
| end |
|
|
| end OAI |
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|
