| import Mathlib |
| import OAI.Probability.BinarySweep.MatrixBounds.MatrixProjection |
| import OAI.Probability.BinarySweep.SparseBounds.CenteredMoment |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical ComplexConjugate |
|
|
| namespace BinaryCoordinateSweeps.Young |
| variable {I X : Type*} [Fintype I] [DecidableEq I] [Fintype X] [DecidableEq X] |
|
|
| def centeredMatrix (s : ℝ) (p : Equiv.Perm X → ℂ) : Matrix (I ↪ X) (I ↪ X) ℂ := |
| ∑A ∈ (Finset.univ : Finset I).powerset, |
| ((-1:ℂ)^(Fintype.card I-A.card)*(s:ℂ)^A.card/(s:ℂ)^Fintype.card I) • |
| ∑g, p g • partialKernel A g |
|
|
| omit [DecidableEq I] in |
| lemma centeredMatrix_toLin (s : ℝ) (p : Equiv.Perm X → ℂ) : |
| (centeredMatrix (I:=I) s p).toEuclideanLin=centeredHilbert s p := by |
| simp only [centeredMatrix,map_sum,map_smul,partialKernel_toLin,centeredHilbert_sum] |
|
|
| lemma evenMoment_one_matrix {J : Type*} [Fintype J] [DecidableEq J] |
| (M : Matrix J J ℂ) : Irrep.evenMoment 1 M.toEuclideanLin = ∑x,∑y, ‖M y x‖^2 := by |
| have hc (z : ℂ) : (star z*z).re=‖z‖^2 := by |
| rw [Complex.star_def,Complex.conj_mul',← Complex.ofReal_pow,Complex.ofReal_re] |
| unfold Irrep.evenMoment |
| rw [pow_one,← Matrix.toEuclideanLin_conjTranspose_eq_adjoint,Module.End.mul_eq_comp,← Matrix.toLpLin_mul_same, |
| Density.trace_toEuclideanLinear] |
| simp only [Matrix.trace,Matrix.diag,Matrix.mul_apply,Matrix.conjTranspose_apply, |
| Complex.re_sum] |
| apply Finset.sum_congr rfl |
| intro x _ |
| apply Finset.sum_congr rfl |
| intro y _ |
| exact hc (M y x) |
|
|
| theorem specht_moment_one_le_centered_matrix (μ : YoungDiagram) (s : ℝ) (hs : s≠0) |
| (p : G μ → ℂ) : |
| Irrep.evenMoment 1 (Irrep.groupAverage (hilbertSpecht μ) p) ≤ |
| ∑x : Tail μ ↪ Cell μ, ∑y, ‖centeredMatrix s p y x‖^2 := by |
| have he := centeredHilbert_moment μ s hs p 1 |
| rw [← centeredMatrix_toLin,evenMoment_one_matrix] at he |
| exact he |
|
|
| end BinaryCoordinateSweeps.Young |
|
|
| end |
|
|
| end OAI |
|
|
