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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 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import Mathlib
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical Pointwise BigOperators
namespace CriticalPercolation.LinearGrowth
variable {G R : Type*} [Group G] [CommRing R]
variable {n : Type*} [Fintype n] [DecidableEq n]
lemma exists_finiteType_matrix_ring [Group.FG G] (ρ : G →* GL n R) :
∃ D : Subalgebra ℤ R, Algebra.FiniteType ℤ D ∧ ∀ g i j, ρ g i j∈D := by
obtain ⟨S,hS,hSf⟩ := Group.fg_iff.mp (inferInstance : Group.FG G)
let T := hSf.toFinset ∪ hSf.toFinset⁻¹
let E := T.biUnion (fun g => Finset.univ.biUnion (fun i => Finset.univ.image (fun j => ρ g i j)))
let D := Algebra.adjoin ℤ (E : Set R)
have hD : Algebra.FiniteType ℤ D := (Subalgebra.fg_iff_finiteType _).mp (Subalgebra.fg_adjoin_finset E)
have hgen (g : G) (hg : g∈T) (i j : n) : ρ g i j∈D := by
apply Algebra.subset_adjoin
exact Finset.mem_biUnion.mpr ⟨g,hg,Finset.mem_biUnion.mpr
⟨i,Finset.mem_univ _,Finset.mem_image.mpr ⟨j,Finset.mem_univ _,rfl⟩⟩⟩
refine ⟨D,hD,?_⟩
intro g
have hg : g∈Subgroup.closure S := by rw [hS]; trivial
induction hg using Subgroup.closure_induction'' with
| mem a ha => exact hgen a (Finset.mem_union_left _ (hSf.mem_toFinset.mpr ha))
| inv_mem a ha =>
apply hgen a⁻¹
apply Finset.mem_union_right
simpa using hSf.mem_toFinset.mpr ha
| one =>
intro i j
rw [map_one]
by_cases h : i=j <;> simp [h,D.zero_mem,D.one_mem]
| mul a b ha hb iha ihb =>
intro i j
rw [map_mul]
change (∑ l, ρ a i l*ρ b l j)∈D
exact D.sum_mem (fun l _ => D.mul_mem (iha i l) (ihb l j))
variable (ρ : G →* GL n R) (D : Subalgebra ℤ R) (hD : ∀ g i j, ρ g i j∈D)
def liftMatrix (g : G) : Matrix n n D := fun i j => ⟨ρ g i j,hD g i j⟩
@[simp] lemma liftMatrix_map (g : G) :
(D.val.toRingHom.mapMatrix) (liftMatrix ρ D hD g)=(ρ g).val := rfl
lemma liftMatrix_one : liftMatrix ρ D hD 1=1 := by
apply Matrix.map_injective (f := D.val) Subtype.val_injective
change D.val.toRingHom.mapMatrix (liftMatrix ρ D hD 1)=D.val.toRingHom.mapMatrix 1
rw [map_one]
change (ρ 1).val=1
rw [map_one]
rfl
lemma liftMatrix_mul (g h : G) : liftMatrix ρ D hD (g*h)=liftMatrix ρ D hD g*liftMatrix ρ D hD h := by
apply Matrix.map_injective (f := D.val) Subtype.val_injective
change D.val.toRingHom.mapMatrix (liftMatrix ρ D hD (g*h))=D.val.toRingHom.mapMatrix _
rw [map_mul]
change (ρ (g*h)).val=(ρ g).val*(ρ h).val
rw [map_mul]
rfl
def liftMatrixGroup : G →* GL n D where
toFun g := ⟨liftMatrix ρ D hD g,liftMatrix ρ D hD g⁻¹,
by rw [←liftMatrix_mul,mul_inv_cancel,liftMatrix_one],
by rw [←liftMatrix_mul,inv_mul_cancel,liftMatrix_one]⟩
map_one' := Units.ext (liftMatrix_one ρ D hD)
map_mul' g h := Units.ext (liftMatrix_mul ρ D hD g h)
@[simp] lemma liftMatrixGroup_map (g : G) :
Matrix.GeneralLinearGroup.map D.val.toRingHom (liftMatrixGroup ρ D hD g)=ρ g := by
apply Units.ext
rfl
lemma liftMatrixGroup_injective (hρ : Injective ρ) : Injective (liftMatrixGroup ρ D hD) := by
intro x y he
apply hρ
apply Units.ext
ext i j
exact congrArg (fun M : GL n D => ((M i j : D) : R)) he
end CriticalPercolation.LinearGrowth
end
end
end OAI
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