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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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 | import Mathlib
import OAI.Probability.BinarySweep.FiniteLaws.LinearEvenMoment
import OAI.Probability.BinarySweep.YoungTheory.HookInducedMoment
import OAI.Probability.BinarySweep.MatrixBounds.BlockTrace
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps.Irrep
open TraceHolder
variable {I W : Type*} [Fintype I] [DecidableEq I]
[NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
local notation "HW" => CoindHilbertSpace (I:=I) (W:=W)
omit [DecidableEq I] [FiniteDimensional ℂ W] in
lemma fiber_sum (v : HW) : ∑i, fiberInclude i (v i)=v := by
classical
apply (WithLp.linearEquiv 2 ℂ (I → W)).injective
simp only [map_sum]
ext i
simp only [fiberInclude,LinearMap.comp_apply,LinearEquiv.coe_coe,
LinearEquiv.apply_symm_apply,LinearMap.single_apply,Finset.sum_apply,Pi.single_apply]
simp only [WithLp.coe_linearEquiv]
let : DecidableEq I := fun i j => Classical.propDecidable (i=j)
exact (Finset.sum_ite_eq Finset.univ i (fun x => v x)).trans (ite_eq_left (Finset.mem_univ i))
omit [DecidableEq I] [FiniteDimensional ℂ W] in
lemma fiber_identity : (∑i : I, (fiberInclude i).comp (fiberProject i))=(1 : HW →ₗ[ℂ] HW) := by
apply LinearMap.ext
intro v
simpa only [LinearMap.sum_apply,LinearMap.comp_apply, fiberProject, Module.End.one_apply,
LinearEquiv.coe_coe, LinearMap.proj_apply, WithLp.coe_linearEquiv] using fiber_sum v
omit [DecidableEq I] [FiniteDimensional ℂ W] in
lemma hilbertBlock_mul (A B : HW →ₗ[ℂ] HW) (i j : I) :
hilbertBlock (A*B) i j = ∑a, hilbertBlock A i a * hilbertBlock B a j := by
ext v
change A (B (fiberInclude j v)) i = _
conv_lhs => rw [← fiber_sum (B (fiberInclude j v))]
simp only [map_sum,LinearMap.sum_apply,Module.End.mul_apply]
change (WithLp.linearEquiv 2 ℂ (I → W)) (∑ x, A (fiberInclude x (B (fiberInclude j v) x))) i = _
rw [map_sum,Finset.sum_apply]
rfl
omit [Fintype I] [FiniteDimensional ℂ W] in
lemma hilbertBlock_one (i j : I) : hilbertBlock (1 : HW →ₗ[ℂ] HW) i j = if i=j then 1 else 0 := by
ext v
change fiberInclude j v i = _
rw [fiberInclude_apply]
split_ifs <;> rfl
omit [Fintype I] [DecidableEq I] [FiniteDimensional ℂ W] in
lemma hilbertBlock_sum {J : Type*} (s : Finset J) (A : J → HW →ₗ[ℂ] HW) (i j : I) :
hilbertBlock (∑a∈s,A a) i j = ∑a∈s,hilbertBlock (A a) i j := by
ext v
simp [hilbertBlock,LinearMap.sum_apply,LinearMap.comp_apply,map_sum]
omit [Fintype I] [DecidableEq I] [FiniteDimensional ℂ W] in
lemma hilbertBlock_smul (r : ℝ) (A : HW →ₗ[ℂ] HW) (i j : I) :
hilbertBlock (r • A) i j=r • hilbertBlock A i j := by
ext v
simp [hilbertBlock]
lemma fiber_adjoint (i : I) : (fiberInclude i : W →ₗ[ℂ] HW).adjoint=fiberProject i := by
symm
apply (LinearMap.eq_adjoint_iff _ _).mpr
intro x y
simp only [PiLp.inner_apply,fiberInclude_apply,apply_ite,inner_zero_right]
change _ = ∑j, if j=i then inner ℂ (x j) y else 0
simp only [fiberProject,LinearMap.comp_apply,LinearMap.proj_apply,LinearEquiv.coe_coe,
WithLp.coe_linearEquiv,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
lemma hilbertBlock_adjoint (A : HW →ₗ[ℂ] HW) (i j : I) :
hilbertBlock A.adjoint i j=(hilbertBlock A j i).adjoint := by
unfold hilbertBlock
rw [LinearMap.adjoint_comp,LinearMap.adjoint_comp,fiber_adjoint,
← fiber_adjoint (W:=W) j,LinearMap.adjoint_adjoint]
rfl
def hilbertBlocksHom : (HW →ₗ[ℂ] HW) →* Matrix I I (W →ₗ[ℂ] W) where
toFun A := hilbertBlock A
map_one' := by funext i j; exact hilbertBlock_one i j
map_mul' A B := by funext i j; exact hilbertBlock_mul A B i j
omit [DecidableEq I] in
lemma trace_hilbertBlocks (A : HW →ₗ[ℂ] HW) :
LinearMap.trace ℂ HW A = ∑i, LinearMap.trace ℂ W (hilbertBlock A i i) := by
conv_lhs => rw [← mul_one A,← fiber_identity]
rw [Finset.mul_sum,map_sum]
apply Finset.sum_congr rfl
intro i _
change LinearMap.trace ℂ HW ((A.comp (fiberInclude i)).comp (fiberProject i)) = _
rw [LinearMap.trace_comp_comm']
rfl
lemma trace_hilbertBlocks_prod {n : ℕ} (A : Fin (n+1) → HW →ₗ[ℂ] HW) :
LinearMap.trace ℂ HW (List.ofFn A).prod =
∑x : Fin (n+1) → I, LinearMap.trace ℂ W
(List.ofFn (fun j => hilbertBlock (A j) (x j) (x (finRotate (n+1) j)))).prod := by
rw [trace_hilbertBlocks]
rw [← map_sum]
change LinearMap.trace ℂ W (Matrix.trace (hilbertBlocksHom (I:=I) (W:=W) (List.ofFn A).prod)) = _
have hm := (hilbertBlocksHom (I:=I) (W:=W)).map_list_prod (List.ofFn A)
rw [List.map_ofFn] at hm
rw [hm,trace_prod_eq_sum_cycles,map_sum]
rfl
theorem hilbert_block_trace_holder {q : ℕ} (hq : 0<q) (A : Fin (2*q) → HW →ₗ[ℂ] HW) :
‖LinearMap.trace ℂ HW (List.ofFn A).prod‖ ≤
∑x : Fin (2*q) → I, ∏j,
evenMoment q (hilbertBlock (A j) (x j) (x (finRotate (2*q) j)))^(((2*q:ℕ):ℝ)⁻¹) := by
have ht : LinearMap.trace ℂ HW (List.ofFn A).prod =
∑x : Fin (2*q) → I, LinearMap.trace ℂ W
(List.ofFn (fun j => hilbertBlock (A j) (x j) (x (finRotate (2*q) j)))).prod := by
generalize hm : 2*q=m at A ⊢
cases m with
| zero => omega
| succ n => exact trace_hilbertBlocks_prod A
rw [ht]
exact (norm_sum_le _ _).trans (Finset.sum_le_sum (fun _ _ => linear_trace_holder hq _))
theorem evenMoment_blocks_le {q : ℕ} (hq : 0<q) (K : HW →ₗ[ℂ] HW) :
evenMoment q K ≤ ∑x : Fin (2*q) → I, ∏j,
(if j.val%2=0 then
evenMoment q (hilbertBlock K (x (finRotate (2*q) j)) (x j))
else evenMoment q (hilbertBlock K (x j) (x (finRotate (2*q) j))))^(((2*q:ℕ):ℝ)⁻¹) := by
have ht := hilbert_block_trace_holder hq
(fun j : Fin (2*q) => if j.val%2=0 then K.adjoint else K)
rw [alternating_prod] at ht
refine (Complex.re_le_norm _).trans (ht.trans_eq ?_)
apply Finset.sum_congr rfl
intro x _
apply Finset.prod_congr rfl
intro j _
split_ifs <;> simp only [hilbertBlock_adjoint,evenMoment_adjoint]
end BinaryCoordinateSweeps.Irrep
end
end OAI
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