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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | 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
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