| import OAI.RingTheory.DirectFiniteness.PackingFormulas |
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| namespace OAI |
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| universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 |
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| noncomputable section |
| open scoped Classical BigOperators |
| namespace KaplanskyCounterexample.CharacterTwist |
| open CharacterIdempotents AlgebraicPacking |
| variable {A : Type u_1} {N : Type u_2} {C : Type u_3} {V : Type u_4} {m : Type u_5} {K : Type u_6} [CommGroup A] [Group N] [CommGroup C] |
| variable [Field K] [Fintype A] [Fintype C] [Fintype V] [Fintype m] |
| variable (ι : V → A →* N) (hi : ∀ v, Function.Injective (ι v)) (v₀ : V) |
| variable (α : V → A →* C) (ξ : C →* Kˣ) |
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| def aMatrix (X : Matrix m V K) (slot : V → m) : Matrix m m (MonoidAlgebra K (L ι hi v₀ α)) := |
| aMat (Rectangular.U X (P ι hi v₀ α)) |
| (Jpack slot (uInv ι hi v₀ α) (fun v => e ι hi v₀ α ξ * P ι hi v₀ α v)) |
| (Matrix.diagonal (fun _ => e ι hi v₀ α ξ)) |
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| def bMatrix (X : Matrix m V K) (slot : V → m) : Matrix m m (MonoidAlgebra K (L ι hi v₀ α)) := |
| bMat (Cpack slot (u ι hi v₀ α) (fun v => e ι hi v₀ α ξ * P ι hi v₀ α v)) |
| (Rectangular.T X (P ι hi v₀ α)) (Matrix.diagonal (fun _ => e ι hi v₀ α ξ)) |
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| theorem matrix_products {F : Type u_7} [Fintype F] |
| (X : Matrix m V K) (Inc : F → Finset V) |
| (lineSum : F → MonoidAlgebra K (L ι hi v₀ α)) |
| (hdec : ∀ v, P (K := K) ι hi v₀ α v = |
| 1 + ∑ f : {f : F // v ∈ Inc f}, (lineSum f.val - 1)) |
| (hglobal : X * X.transpose = 1) |
| (hlocal : ∀ f i j, ∑ v ∈ Inc f, X i v * X j v = 0) |
| (hlt : Fintype.card m < Fintype.card V) |
| (hA : (Fintype.card A : K) ≠ 0) (hC : (Fintype.card C : K) ≠ 0) |
| (slot : V → m) (hξ : Function.Injective ξ) |
| (hα : ∀ v w, v ≠ w → slot v = slot w → α v ≠ α w) : |
| aMatrix ι hi v₀ α ξ X slot * bMatrix ι hi v₀ α ξ X slot=1 ∧ |
| bMatrix ι hi v₀ α ξ X slot * aMatrix ι hi v₀ α ξ X slot≠1 := by |
| have hp (v : V) : P (K := K) ι hi v₀ α v * P ι hi v₀ α v = P ι hi v₀ α v := |
| pushed_idempotent _ _ hA |
| obtain ⟨hUP,hPT,hUT⟩ := Rectangular.identities X (P ι hi v₀ α) lineSum Inc hp hdec hglobal hlocal |
| apply prescribed_packing_products (Rectangular.U X (P ι hi v₀ α)) |
| (Rectangular.T X (P ι hi v₀ α)) (P ι hi v₀ α) (Q ι hi v₀ α ξ) |
| (u ι hi v₀ α) (uInv ι hi v₀ α) slot (e ι hi v₀ α ξ) |
| (pushed_idempotent _ _ hC) (e_central ι hi v₀ α ξ) hp hUP hPT hUT |
| (uInv_mul_u ι hi v₀ α) (conjugate_projection ι hi v₀ α ξ) |
| _ (defect_ne_zero ι hi v₀ α ξ X hlt hA hC) |
| intro v w hvw hslot |
| apply pushed_orthogonal |
| intro heq |
| apply hα v w hvw hslot |
| apply MonoidHom.ext |
| intro a |
| apply hξ |
| have heq' := congrArg (fun χ : A →* Kˣ => χ a) heq |
| exact inv_injective heq' |
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| end KaplanskyCounterexample.CharacterTwist |
| end |
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| end OAI |
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