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67aee1d | 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 | import OAI.Analysis.GroupDeterminants.ModularLifting
namespace OAI
noncomputable section
open scoped BigOperators
namespace GroupRingDeterminant
section RegularAction
open scoped ENNReal
variable {G : Type} [Group G] {n : ℕ}
def reindex (e : (Fin n × G) ≃ (Fin n × G)) : Hilbert G n →ₗᵢ[ℂ] Hilbert G n where
toFun ξ := ⟨fun a => ξ (e a), memℓp_gen
(((lp.memℓp ξ).summable (by norm_num)).comp_injective e.injective)⟩
map_add' ξ η := by ext a; rfl
map_smul' z ξ := by ext a; rfl
norm_map' ξ := by
rw [lp.norm_eq_tsum_rpow (by norm_num), lp.norm_eq_tsum_rpow (by norm_num)]
congr 1
exact e.tsum_eq (fun a => ‖ξ a‖ ^ (2 : ℝ≥0∞).toReal)
def cutComponent (i : Fin n) : Hilbert G n →L[ℂ] Hilbert G n := by
classical
let f : Hilbert G n →ₗ[ℂ] Hilbert G n := {
toFun ξ := ⟨fun a => if a.1 = i then ξ a else 0,
(lp.memℓp ξ).mono' (fun a => by split_ifs <;> simp)⟩
map_add' ξ η := by
ext a
change (if a.1 = i then ξ a + η a else 0) =
(if a.1 = i then ξ a else 0) + (if a.1 = i then η a else 0)
split_ifs <;> simp
map_smul' z ξ := by
ext a
change (if a.1 = i then z • ξ a else 0) = z • (if a.1 = i then ξ a else 0)
split_ifs <;> simp }
exact f.mkContinuous 1 (fun ξ => by
simpa only [one_mul] using lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)
(x := f ξ) (y := ξ) (fun a => by
change ‖ite (a.1 = i) (ξ a) 0‖ ≤ ‖ξ a‖
split_ifs <;> simp))
omit [Group G] in
@[simp] theorem cutComponent_apply (i : Fin n) (ξ : Hilbert G n) (a : Fin n × G) :
cutComponent i ξ a = if a.1 = i then ξ a else 0 := rfl
def elementaryOperator (i j : Fin n) (g : G) : Hilbert G n →L[ℂ] Hilbert G n :=
(cutComponent i).comp (reindex
(Equiv.prodCongr (Equiv.swap i j) (Equiv.mulLeft g⁻¹))).toContinuousLinearMap
theorem elementaryOperator_apply (i j : Fin n) (g : G) (ξ : Hilbert G n)
(a : Fin n) (h : G) :
elementaryOperator i j g ξ (a, h) = if a = i then ξ (j, g⁻¹ * h) else 0 := by
classical
by_cases hi : a = i
· subst a
simp [elementaryOperator, reindex, cutComponent_apply]
· simp [elementaryOperator, cutComponent_apply, hi]
def regularOperator (A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℤ G)) :
Hilbert G n →L[ℂ] Hilbert G n :=
∑ i : Fin n, ∑ j : Fin n, (A i j).coeff.sum
(fun g z => (z : ℂ) • elementaryOperator i j g)
theorem regularOperator_spec (A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℤ G)) :
IsRegularOperator A (regularOperator A) := by
classical
intro ξ i h
simp only [regularOperator, Finsupp.sum, sum_apply,
smul_apply, lp.coeFn_sum, Finset.sum_apply, lp.coeFn_smul,
Pi.smul_apply, smul_eq_mul, elementaryOperator_apply]
rw [Finset.sum_eq_single i]
· simp
· intro b _ hbi
simp [Ne.symm hbi]
· simp
theorem exists_regularOperator (A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℤ G)) :
∃ T, IsRegularOperator A T := ⟨regularOperator A, regularOperator_spec A⟩
end RegularAction
end GroupRingDeterminant
end
end OAI
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