File size: 5,821 Bytes
e3e0a0d | 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 | import Mathlib
import OAI.RepresentationTheory.Saxl.Main
import OAI.RepresentationTheory.UniversalSquare.Support.Candidate
namespace OAI
/-! Transpose Support. -/
section
noncomputable section
open scoped TensorProduct
namespace UniversalTensorSquare
def signTwistEquiv {n : ℕ} {X Y : Type*} [AddCommGroup X] [Module ℂ X]
[AddCommGroup Y] [Module ℂ Y]
{ρ : Representation ℂ (Equiv.Perm (Fin n)) X}
{σ : Representation ℂ (Equiv.Perm (Fin n)) Y} (e : ρ.Equiv σ) :
(Saxl.signTwist ρ).Equiv (Saxl.signTwist σ) where
toLinearEquiv := e.toLinearEquiv
isIntertwining' := (Saxl.signTwistMap e.toIntertwiningMap).isIntertwining'
lemma canonical_sign_transpose {n : ℕ} (μ : YoungDiagram) (hμ : μ.card = n) :
Nonempty ((Saxl.spechtRep (Saxl.canonicalTableau μ hμ)).Equiv
(Saxl.signTwist (Saxl.spechtRep
(Saxl.canonicalTableau μ.transpose ((Saxl.transpose_card μ).trans hμ))))) := by
obtain ⟨e⟩ := Saxl.specht_sign_transpose (Saxl.canonicalTableau μ hμ)
let e' := Saxl.equivOfSubrepEq
(Saxl.spechtSub (Saxl.transposeTableau (Saxl.canonicalTableau μ hμ)))
(Saxl.spechtSub
(Saxl.canonicalTableau μ.transpose ((Saxl.transpose_card μ).trans hμ)))
(Saxl.spechtSub_tableau_independent _ _)
exact ⟨e.trans (signTwistEquiv e')⟩
def spechtShapeEquiv {n : ℕ} {μ ν : YoungDiagram} (h : μ = ν)
(t : Saxl.Tableau n μ) (s : Saxl.Tableau n ν) :
(Saxl.spechtRep t).Equiv (Saxl.spechtRep s) := by
subst ν
exact Saxl.equivOfSubrepEq _ _ (Saxl.spechtSub_tableau_independent t s)
lemma canonical_self_sign {n : ℕ} (lam : YoungDiagram) (hlam : lam.card = n)
(hself : lam.transpose = lam) :
Nonempty ((Saxl.spechtRep (Saxl.canonicalTableau lam hlam)).Equiv
(Saxl.signTwist (Saxl.spechtRep (Saxl.canonicalTableau lam hlam)))) := by
obtain ⟨e⟩ := canonical_sign_transpose lam hlam
let e' := spechtShapeEquiv hself
(Saxl.canonicalTableau lam.transpose ((Saxl.transpose_card lam).trans hlam))
(Saxl.canonicalTableau lam hlam)
exact ⟨e.trans (signTwistEquiv e')⟩
lemma tensorAnti_injective {n : ℕ} {X Y Z : Type*}
[AddCommGroup X] [Module ℂ X] [AddCommGroup Y] [Module ℂ Y]
[AddCommGroup Z] [Module ℂ Z]
(ρ : Representation ℂ (Equiv.Perm (Fin n)) X)
{σ : Representation ℂ (Equiv.Perm (Fin n)) Y}
{τ : Representation ℂ (Equiv.Perm (Fin n)) Z}
(e : σ.Equiv (Saxl.signTwist τ)) :
Function.Injective (Saxl.tensorAnti ρ e.toIntertwiningMap) :=
(TensorProduct.congr (LinearEquiv.refl ℂ X) e.toLinearEquiv).injective
lemma kronecker_pos_transpose {n : ℕ} (lam : YoungDiagram) (hlam : lam.card = n)
(hself : lam.transpose = lam) (ν : YoungDiagram) (hν : ν.card = n)
(hpos : 0 < Saxl.kronecker (Saxl.canonicalTableau lam hlam)
(Saxl.canonicalTableau lam hlam) (Saxl.canonicalTableau ν hν)) :
0 < Saxl.kronecker (Saxl.canonicalTableau lam hlam)
(Saxl.canonicalTableau lam hlam)
(Saxl.canonicalTableau ν.transpose ((Saxl.transpose_card ν).trans hν)) := by
obtain ⟨eL⟩ := canonical_self_sign lam hlam hself
obtain ⟨eν⟩ := canonical_sign_transpose ν hν
obtain ⟨f, hf⟩ := (Saxl.kronecker_pos_iff _ _ _).mp hpos
let A := Saxl.tensorAnti (Saxl.spechtRep (Saxl.canonicalTableau lam hlam))
eL.toIntertwiningMap
let B := (Saxl.signTwistMap
(Y := Saxl.Specht (Saxl.canonicalTableau lam hlam) ⊗[ℂ]
Saxl.Specht (Saxl.canonicalTableau lam hlam)) f).comp (Saxl.signCancelMap eν.symm.toIntertwiningMap)
apply (Saxl.kronecker_pos_iff _ _ _).mpr
refine ⟨A.comp B, ?_⟩
intro hz
apply hf
ext x
have h := congrArg (fun F => F (eν x)) hz
have hi : Function.Injective A := tensorAnti_injective _ eL
apply hi
change A (f x) = A 0
change A (f (eν.symm (eν x))) = 0 at h
simpa only [eν.symm_apply_apply, map_zero] using h
theorem kronecker_pos_transpose_iff {n : ℕ} (lam : YoungDiagram) (hlam : lam.card = n)
(hself : lam.transpose = lam) (ν : YoungDiagram) (hν : ν.card = n) :
(0 < Saxl.kronecker (Saxl.canonicalTableau lam hlam)
(Saxl.canonicalTableau lam hlam) (Saxl.canonicalTableau ν hν)) ↔
(0 < Saxl.kronecker (Saxl.canonicalTableau lam hlam)
(Saxl.canonicalTableau lam hlam)
(Saxl.canonicalTableau ν.transpose ((Saxl.transpose_card ν).trans hν))) := by
constructor
· exact kronecker_pos_transpose lam hlam hself ν hν
· intro h
have ht := kronecker_pos_transpose lam hlam hself ν.transpose
((Saxl.transpose_card ν).trans hν) h
obtain ⟨f, hf⟩ := (Saxl.kronecker_pos_iff _ _ _).mp ht
let e := spechtShapeEquiv (YoungDiagram.transpose_transpose ν).symm
(Saxl.canonicalTableau ν hν)
(Saxl.canonicalTableau ν.transpose.transpose
((Saxl.transpose_card ν.transpose).trans ((Saxl.transpose_card ν).trans hν)))
apply (Saxl.kronecker_pos_iff _ _ _).mpr
refine ⟨f.comp e.toIntertwiningMap, ?_⟩
intro hz
apply hf
ext x
have hh := congrArg (fun F => F (e.symm x)) hz
change f (e (e.symm x)) = 0 at hh
simpa using hh
lemma fixedCandidate_support_transpose (n : ℕ) (hM : 3 ≤ staircaseIndex n)
(ν : YoungDiagram) (hν : ν.card = n) :
(0 < Saxl.kronecker (Saxl.canonicalTableau (fixedCandidate n) (fixedCandidate_card n hM))
(Saxl.canonicalTableau (fixedCandidate n) (fixedCandidate_card n hM))
(Saxl.canonicalTableau ν hν)) ↔
(0 < Saxl.kronecker (Saxl.canonicalTableau (fixedCandidate n) (fixedCandidate_card n hM))
(Saxl.canonicalTableau (fixedCandidate n) (fixedCandidate_card n hM))
(Saxl.canonicalTableau ν.transpose ((Saxl.transpose_card ν).trans hν))) :=
kronecker_pos_transpose_iff _ _ (fixedCandidate_transpose n) ν hν
end UniversalTensorSquare
end
end
end OAI
|