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f205175 | 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 | import Mathlib
import OAI.Probability.CriticalPercolation.LinearGroups.FieldSpanTransfer
import OAI.Probability.CriticalPercolation.LinearGroups.FullSpanAlternative
namespace OAI
section
noncomputable section
open Set Function Module
open scoped Classical
namespace CriticalPercolation.LinearGrowth
variable {K L T G n : Type*} [Field K] [Field L] [Field T] [Group G]
variable [Fintype n] [DecidableEq n]
def matrixRepresentation (ρ : G →* GL n K) : G →* ((n → K) ≃ₗ[K] (n → K)) :=
(LinearMap.GeneralLinearGroup.generalLinearEquiv K (n → K)).toMonoidHom.comp
(Matrix.GeneralLinearGroup.toLin.toMonoidHom.comp ρ)
@[simp] lemma matrixRepresentation_apply (ρ : G →* GL n K) (g : G) (v : n → K) :
matrixRepresentation ρ g v=(ρ g).val.mulVec v := rfl
def matrixFlatten : Matrix n n K ≃ₗ[K] (n × n → K) where
toFun M p := M p.1 p.2
invFun f i j := f (i,j)
left_inv _ := rfl
right_inv _ := rfl
map_add' _ _ := rfl
map_smul' _ _ := rfl
def FullMatrixSpan (ρ : G →* GL n K) : Prop :=
Submodule.span K (Set.range (fun g (p : n×n) => ρ g p.1 p.2))=⊤
def StrongFullMatrixSpan (ρ : G →* GL n K) : Prop :=
∀ H : Subgroup G, H.FiniteIndex → FullMatrixSpan (ρ.comp H.subtype)
lemma span_equiv_top {K V W J : Type*} [Field K] [AddCommGroup V] [Module K V]
[AddCommGroup W] [Module K W] (e : V ≃ₗ[K] W) (v : J → V)
(hv : Submodule.span K (Set.range v)=⊤) :
Submodule.span K (Set.range (fun j => e (v j)))=⊤ := by
rw [show Set.range (fun j => e (v j))=e '' Set.range v from (Set.range_comp _ _)]
change Submodule.span K (e.toLinearMap '' Set.range v)=⊤
rw [←Submodule.map_span e.toLinearMap,hv,Submodule.map_top]
exact e.range
lemma FullMatrixSpan.fullSpan (ρ : G →* GL n K) (hρ : FullMatrixSpan ρ) :
FullSpan (matrixRepresentation ρ) := by
let e := (matrixFlatten (K:=K) (n:=n)).symm.trans Matrix.toLin'
exact span_equiv_top e (fun g (p:n×n) => ρ g p.1 p.2) hρ
lemma StrongFullMatrixSpan.strongFullSpan (ρ : G →* GL n K) (hρ : StrongFullMatrixSpan ρ) :
StrongFullSpan (matrixRepresentation ρ) := by
intro H hH
exact FullMatrixSpan.fullSpan (ρ.comp H.subtype) (hρ H hH)
lemma FullMatrixSpan.transfer (f : K →+* T) (e : L →+* T)
(ρ : G →* GL n K) (τ : G →* GL n L)
(hm : ∀ g i j, f (ρ g i j)=e (τ g i j)) (hρ : FullMatrixSpan ρ) : FullMatrixSpan τ :=
pi_span_transfer f e (fun g (p:n×n) => ρ g p.1 p.2)
(fun g (p:n×n) => τ g p.1 p.2) (fun g p => hm g p.1 p.2) hρ
lemma StrongFullMatrixSpan.transfer (f : K →+* T) (e : L →+* T)
(ρ : G →* GL n K) (τ : G →* GL n L)
(hm : ∀ g i j, f (ρ g i j)=e (τ g i j)) (hρ : StrongFullMatrixSpan ρ) :
StrongFullMatrixSpan τ := by
intro H hH
exact FullMatrixSpan.transfer f e (ρ.comp H.subtype) (τ.comp H.subtype)
(fun g i j => hm g i j) (hρ H hH)
end CriticalPercolation.LinearGrowth
end
end
end OAI
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