openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /ConjugationMatrixGrowth.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Criticality.ExpandingCharacterGrowth | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical Matrix | |
| namespace CriticalPercolation.SparseGrowth | |
| variable {A : Type*} [Group A] [DecidableEq A] | |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| def latticeCharacter (w : ι → ℂ) : Multiplicative (ι → ℤ) →* Multiplicative ℂ where | |
| toFun x := Multiplicative.ofAdd (w ⬝ᵥ fun i => ((Multiplicative.toAdd x i : ℤ) : ℂ)) | |
| map_one' := by change Multiplicative.ofAdd _ = Multiplicative.ofAdd 0; congr 1; simp | |
| map_mul' x y := by | |
| change Multiplicative.ofAdd (w ⬝ᵥ fun i => (((Multiplicative.toAdd x i + | |
| Multiplicative.toAdd y i) : ℤ) : ℂ)) = Multiplicative.ofAdd (_+_) | |
| congr 1 | |
| simp [dotProduct,Finset.sum_add_distrib,mul_add] | |
| omit [DecidableEq ι] in | |
| lemma latticeCharacter_mulVec (w : ι → ℂ) (M : Matrix ι ι ℤ) (v : ι → ℤ) : | |
| Multiplicative.toAdd (latticeCharacter w (Multiplicative.ofAdd (M.mulVec v))) = | |
| (Matrix.vecMul w (M.map (Int.castRingHom ℂ))) ⬝ᵥ (fun i => (v i : ℂ)) := by | |
| change (w ⬝ᵥ fun i => ((M.mulVec v i : ℤ) : ℂ)) = _ | |
| have he : (fun i => ((M.mulVec v i : ℤ) : ℂ)) = | |
| (M.map (Int.castRingHom ℂ)).mulVec (fun i => (v i : ℂ)) := by | |
| funext i | |
| exact (Int.castRingHom ℂ).map_mulVec M v i | |
| rw [he,Matrix.dotProduct_mulVec] | |
| lemma latticeCharacter_single (w : ι → ℂ) (i : ι) : | |
| Multiplicative.toAdd (latticeCharacter w (Multiplicative.ofAdd (Pi.single i 1)))=w i := by | |
| change (w ⬝ᵥ fun j => (((Pi.single i (1:ℤ) : ι → ℤ) j : ℤ) : ℂ))=w i | |
| have he : (fun j => (((Pi.single i (1:ℤ) : ι → ℤ) j : ℤ) : ℂ)) = Pi.single i (1:ℂ) := by | |
| funext j | |
| by_cases h : i=j | |
| · subst j; simp | |
| · simp [Ne.symm h] | |
| rw [he,dotProduct_single,mul_one] | |
| theorem HasPolynomialDegree.conjugationMatrix_roots_norm_le_one {d : ℕ} | |
| (hg : HasPolynomialDegree A d) (H : Subgroup A) [H.Normal] (a : A) | |
| (p : H →* Multiplicative (ι → ℤ)) (hp : Surjective p) | |
| (M : Matrix ι ι ℤ) | |
| (hM : ∀ x : H, Multiplicative.toAdd (p ((MulAut.conjNormal a) x)) = | |
| M.mulVec (Multiplicative.toAdd (p x))) : | |
| ∀ z∈(M.map (Int.castRingHom ℂ)).charpoly.roots, ‖z‖≤1 := by | |
| intro z hz | |
| let C := M.map (Int.castRingHom ℂ) | |
| have hroot : (C.transpose.mulVecLin.charpoly).IsRoot z := by | |
| rw [Matrix.charpoly_mulVecLin,Matrix.charpoly_transpose] | |
| exact (Polynomial.mem_roots (Matrix.charpoly_monic C).ne_zero).mp hz | |
| have heig := (Module.End.hasEigenvalue_iff_isRoot_charpoly C.transpose.mulVecLin z).mpr hroot | |
| obtain ⟨w,hw⟩ := heig.exists_hasUnifEigenvector | |
| have hweq : Matrix.vecMul w C = z • w := by | |
| simpa only [Matrix.mulVecLin_apply,Matrix.mulVec_transpose] using hw.apply_eq_smul | |
| have hwn := hw.2 | |
| have hex : ∃ i, w i ≠ 0 := by | |
| by_contra! h | |
| exact hwn (funext h) | |
| obtain ⟨i,hi⟩ := hex | |
| obtain ⟨b,hb⟩ := hp (Multiplicative.ofAdd (Pi.single i 1)) | |
| let φ := (latticeCharacter w).comp p | |
| have hφ : ∀ x : H, Multiplicative.toAdd (φ ((MulAut.conjNormal a) x)) = | |
| z * Multiplicative.toAdd (φ x) := by | |
| intro x | |
| change Multiplicative.toAdd (latticeCharacter w (p ((MulAut.conjNormal a) x))) = | |
| z * Multiplicative.toAdd (latticeCharacter w (p x)) | |
| have hpx : p ((MulAut.conjNormal a) x) = | |
| Multiplicative.ofAdd (M.mulVec (Multiplicative.toAdd (p x))) := by | |
| exact congrArg Multiplicative.ofAdd (hM x) | |
| rw [hpx,latticeCharacter_mulVec] | |
| change (Matrix.vecMul w C) ⬝ᵥ (fun i => ((Multiplicative.toAdd (p x) i : ℤ) : ℂ)) = _ | |
| rw [hweq,smul_dotProduct] | |
| rfl | |
| have hφb : Multiplicative.toAdd (φ b) ≠ 0 := by | |
| change Multiplicative.toAdd (latticeCharacter w (p b)) ≠ 0 | |
| rw [hb,latticeCharacter_single] | |
| exact hi | |
| exact hg.norm_conj_eigencharacter_le_one H a φ z hφ b hφb | |
| end CriticalPercolation.SparseGrowth | |
| end | |
| end | |
| end OAI | |