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607ee45 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 | import Mathlib
import OAI.RepresentationTheory.Saxl.Main
import OAI.RepresentationTheory.UniversalSquare.Contraction.PairForms
import OAI.RepresentationTheory.UniversalSquare.Support.SingletonPairs
namespace OAI
/-! Singleton Forms. -/
section
noncomputable section
namespace Saxl
def singleWord {d : ℕ} (v : Fin d → ℂ) : WordSpace 1 d := fun w => v (w 0)
lemma wordMap_singleWord {a b : ℕ} (L : Fin a → Fin b → ℂ) (v : Fin a → ℂ) :
wordMap L (singleWord v) = singleWord (fun j => ∑ i, v i * L i j) := by
classical
funext w
change (∑ c : Fin 1 → Fin a, v (c 0) * ∏ i, L (c i) (w i)) = _
simpa only [Fintype.prod_unique, show (default : Fin 1) = 0 from rfl, singleWord] using
(Fintype.sum_equiv (Equiv.funUnique (Fin 1) (Fin a))
(fun c => v (c 0) * L (c 0) (w 0))
(fun i => v i * L i (w 0)) (fun _ => rfl))
def pairSingletonTensor {n a m d : ℕ} (e : Fin n ≃ Fin a ⊕ Fin 1)
(p : Fin a ≃ Fin m ⊕ Fin m) (B : Matrix (Fin d) (Fin d) ℂ) (v : Fin d → ℂ) :
WordSpace n d := positionProduct e (pairFormTensor p B) (singleWord v)
lemma wordMap_pairSingletonTensor {n a m d D : ℕ}
(e : Fin n ≃ Fin a ⊕ Fin 1) (p : Fin a ≃ Fin m ⊕ Fin m)
(L : Fin d → Fin D → ℂ) (B : Matrix (Fin d) (Fin d) ℂ) (v : Fin d → ℂ) :
wordMap L (pairSingletonTensor e p B v) = pairSingletonTensor e p
(fun j k => ∑ q : Fin d × Fin d, B q.1 q.2 * L q.1 j * L q.2 k)
(fun j => ∑ i, v i * L i j) := by
rw [pairSingletonTensor, wordMap_positionProduct, wordMap_pairFormTensor,
wordMap_singleWord]
rfl
def pairSingletonEquiv {n a m : ℕ} (e : Fin n ≃ Fin a ⊕ Fin 1)
(p : Fin a ≃ Fin m ⊕ Fin m) : Fin n ≃ Fin (m+1) ⊕ Fin m :=
e.trans ((Equiv.sumCongr p (Equiv.refl _)).trans
((Equiv.sumAssoc _ _ _).trans ((Equiv.sumCongr (Equiv.refl _)
(Equiv.sumComm _ _)).trans ((Equiv.sumAssoc _ _ _).symm.trans
(Equiv.sumCongr finSumFinEquiv (Equiv.refl _))))))
@[simp] lemma pairSingletonEquiv_left {n a m : ℕ}
(e : Fin n ≃ Fin a ⊕ Fin 1) (p : Fin a ≃ Fin m ⊕ Fin m) (k : Fin m) :
(pairSingletonEquiv e p).symm (Sum.inl k.castSucc) =
e.symm (Sum.inl (p.symm (Sum.inl k))) := by
change (pairSingletonEquiv e p).symm (Sum.inl (Fin.castAdd 1 k)) = _
simp [pairSingletonEquiv]
@[simp] lemma pairSingletonEquiv_right {n a m : ℕ}
(e : Fin n ≃ Fin a ⊕ Fin 1) (p : Fin a ≃ Fin m ⊕ Fin m) (k : Fin m) :
(pairSingletonEquiv e p).symm (Sum.inr k) =
e.symm (Sum.inl (p.symm (Sum.inr k))) := by
simp [pairSingletonEquiv]
@[simp] lemma pairSingletonEquiv_last {n a m : ℕ}
(e : Fin n ≃ Fin a ⊕ Fin 1) (p : Fin a ≃ Fin m ⊕ Fin m) :
(pairSingletonEquiv e p).symm (Sum.inl (Fin.last m)) = e.symm (Sum.inr 0) := by
have hh : Fin.last m = Fin.natAdd m (0 : Fin 1) := Fin.ext (by simp)
rw [hh]
simp [pairSingletonEquiv]
lemma pairSingletonTensor_one {n a m d : ℕ}
(e : Fin n ≃ Fin a ⊕ Fin 1) (p : Fin a ≃ Fin m ⊕ Fin m) (v : Fin d → ℂ) :
pairSingletonTensor e p (1 : Matrix (Fin d) (Fin d) ℂ) v =
symmetricPartialPairs (pairSingletonEquiv e p) Fin.castSuccEmb v := by
classical
funext w
have ht : leftWord (pairSingletonEquiv e p) w ∘ Fin.castSuccEmb =
leftWord p (leftWord e w) := by
funext k
simp [leftWord]
have hb : rightWord (pairSingletonEquiv e p) w = rightWord p (leftWord e w) := by
funext k
simp [rightWord,leftWord]
have hlast : Fin.last m ∉ Set.range (Fin.castSuccEmb : Fin m ↪ Fin (m+1)) := by
rintro ⟨k,hk⟩
have hh := congrArg Fin.val hk
have hh' := k.isLt
change k.val = m at hh
omega
have hcast (k : Fin m) : k.castSucc ∈ Set.range (Fin.castSuccEmb : Fin m ↪ Fin (m+1)) :=
⟨k,rfl⟩
have hprod : (∏ k ∈ Finset.univ.filter
(fun k => k ∉ Set.range (Fin.castSuccEmb : Fin m ↪ Fin (m+1))),
v (leftWord (pairSingletonEquiv e p) w k)) = v (rightWord e w 0) := by
rw [Finset.prod_filter, Fin.prod_univ_castSucc]
simp [leftWord,rightWord]
unfold symmetricPartialPairs
rw [ht,hb,hprod]
unfold pairSingletonTensor
rw [pairFormTensor_one]
change (if leftWord p (leftWord e w) = rightWord p (leftWord e w) then 1 else 0) *
v (rightWord e w 0) = _
split_ifs <;> simp
lemma pairSingletonTensor_smul {n a m d : ℕ}
(e : Fin n ≃ Fin a ⊕ Fin 1) (p : Fin a ≃ Fin m ⊕ Fin m)
(B : Matrix (Fin d) (Fin d) ℂ) (v : Fin d → ℂ) (c : ℂ) :
pairSingletonTensor e p (c • B) v = c^m • pairSingletonTensor e p B v := by
unfold pairSingletonTensor
rw [pairFormTensor_smul]
funext w
change c^m * pairFormTensor p B (leftWord e w) * singleWord v (rightWord e w) = _
exact mul_assoc _ _ _
theorem one_odd_row_pairSingleton_support {n a m d D : ℕ}
(e : Fin n ≃ Fin a ⊕ Fin 1) (p : Fin a ≃ Fin m ⊕ Fin m)
(B : Matrix (Fin d) (Fin d) ℂ) (v : Fin d → ℂ)
(L : Fin d → Fin D → ℂ) (c : ℂ) (hc : c ≠ 0)
(hL : (fun j k => ∑ q : Fin d × Fin d, B q.1 q.2 * L q.1 j * L q.2 k) =
c • (1 : Matrix (Fin D) (Fin D) ℂ))
(hv : (fun j => ∑ i, v i * L i j) ≠ 0)
(μ : YoungDiagram) (t : Tableau n μ)
(k : ℕ) (hk : Odd (μ.rowLen k)) (he : ∀ l, l ≠ k → Even (μ.rowLen l))
(hd : μ.colLen 0 ≤ D) :
∃ F : Representation.IntertwiningMap (spechtRep t)
(cyclic (wordRep n d) (pairSingletonTensor e p B v)).toRepresentation, F ≠ 0 := by
let v' := fun j => ∑ i, v i * L i j
have hmap : wordMap L (pairSingletonTensor e p B v) =
c^m • symmetricPartialPairs (pairSingletonEquiv e p) Fin.castSuccEmb v' := by
rw [wordMap_pairSingletonTensor, hL, pairSingletonTensor_smul, pairSingletonTensor_one]
have hcyc : cyclic (wordRep n D) (wordMap L (pairSingletonTensor e p B v)) =
cyclic (wordRep n D) (symmetricPartialPairs (pairSingletonEquiv e p) Fin.castSuccEmb v') := by
rw [hmap, cyclic_smul_eq _ _ _ (pow_ne_zero m hc)]
have hs := one_odd_row_symmetricPartialPairs_support (pairSingletonEquiv e p) Fin.castSuccEmb
v' hv μ t k hk he hd
rw [← hcyc] at hs
obtain ⟨F,hF⟩ := hs
exact cyclic_projection_support (wordMap L) (pairSingletonTensor e p B v) F hF
lemma shortPair_singleton_nonzero :
(fun j => ∑ i : Fin 4, (Pi.single 0 (1 : ℂ) : Fin 4 → ℂ) i * shortPairCongruence i j) ≠ 0 := by
intro h
have hh := congrFun h (0 : Fin 4)
norm_num [shortPairCongruence, Pi.single_apply, Fin.sum_univ_succ] at hh
theorem one_odd_row_shortPair_support {n a m : ℕ}
(e : Fin n ≃ Fin a ⊕ Fin 1) (p : Fin a ≃ Fin m ⊕ Fin m)
(μ : YoungDiagram) (t : Tableau n μ)
(k : ℕ) (hk : Odd (μ.rowLen k)) (he : ∀ l, l ≠ k → Even (μ.rowLen l))
(hd : μ.colLen 0 ≤ 4) :
∃ F : Representation.IntertwiningMap (spechtRep t)
(cyclic (wordRep n 4) (pairSingletonTensor e p shortPairForm (Pi.single 0 1))).toRepresentation,
F ≠ 0 :=
one_odd_row_pairSingleton_support e p shortPairForm (Pi.single 0 1)
shortPairCongruence 2 (by norm_num) shortPairCongruence_gram
shortPair_singleton_nonzero μ t k hk he hd
end Saxl
end
end
end OAI
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