| import OAI.RingTheory.DirectFiniteness.CharacterFormulas |
|
|
| namespace OAI |
|
|
| universe u_1 u_2 u_3 u_4 u_5 |
|
|
| noncomputable section |
| open scoped Classical BigOperators |
| namespace KaplanskyCounterexample.FiniteBlock |
| variable {m : Type u_1} {C : Type u_2} [CommGroup C] |
|
|
| / |
| def permAction : Equiv.Perm m →* MulAut (m → C) where |
| toFun σ := |
| { toFun := fun w i => w (σ.symm i) |
| invFun := fun w i => w (σ i) |
| left_inv := by intro w; funext i; simp |
| right_inv := by intro w; funext i; simp |
| map_mul' := by intro w z; rfl } |
| map_one' := by ext w i; rfl |
| map_mul' σ τ := by ext w i; rfl |
|
|
| abbrev Auxiliary (m : Type u_3) (C : Type u_4) [CommGroup C] := |
| SemidirectProduct (m → C) (Equiv.Perm m) permAction |
| |
| instance [Fintype m] [Fintype C] : Fintype (Auxiliary m C) := |
| Fintype.ofEquiv ((m → C) × Equiv.Perm m) SemidirectProduct.equivProd.symm |
| |
| variable {K : Type u_5} [Field K] [Fintype m] [Fintype C] |
| variable (ξ : C →* Kˣ) |
| |
| def coordinate (i : m) : (m → C) →* Kˣ := ξ.comp (Pi.evalMonoidHom _ i) |
| |
| omit [Fintype m] in |
| lemma coordinate_ne (hξ : ξ ≠ 1) {i j : m} (hij : i ≠ j) : |
| coordinate ξ i ≠ coordinate ξ j := by |
| obtain ⟨c,hc⟩ : ∃ c, ξ c ≠ 1 := by |
| by_contra! h |
| exact hξ (MonoidHom.ext h) |
| intro h |
| have hh := congrArg (fun χ : (m → C) →* Kˣ => χ (Function.update 1 i c)) h |
| simp [coordinate, Function.update_of_ne hij.symm] at hh |
| exact hc hh |
| |
| open CharacterIdempotents |
| |
| def p (i : m) : MonoidAlgebra K (Auxiliary m C) := |
| pushed SemidirectProduct.inl (coordinate ξ i) |
| |
| def permElt (σ : Equiv.Perm m) : MonoidAlgebra K (Auxiliary m C) := |
| MonoidAlgebra.of K _ (SemidirectProduct.inr σ) |
| |
| omit [Fintype m] [Fintype C] in |
| lemma permElt_inv_mul (σ : Equiv.Perm m) : |
| permElt (K := K) (C := C) σ⁻¹ * permElt σ = 1 := by |
| simp [permElt, ← MonoidAlgebra.one_def] |
| omit [Fintype m] [Fintype C] in |
| lemma permElt_mul_inv (σ : Equiv.Perm m) : |
| permElt (K := K) (C := C) σ * permElt σ⁻¹ = 1 := by |
| simp [permElt, ← MonoidAlgebra.one_def] |
| |
| lemma p_ne_zero (hc : (Fintype.card (m → C) : K) ≠ 0) (i : m) : p ξ i ≠ 0 := |
| pushed_ne_zero _ SemidirectProduct.inl_injective _ hc |
| lemma p_idem (hc : (Fintype.card (m → C) : K) ≠ 0) (i : m) : p ξ i * p ξ i = p ξ i := |
| pushed_idempotent _ _ hc |
| lemma p_orth (hξ : ξ ≠ 1) {i j : m} (hij : i ≠ j) : p ξ i * p ξ j = 0 := |
| pushed_orthogonal _ _ _ (coordinate_ne ξ hξ hij) |
| |
| lemma translated (σ : Equiv.Perm m) (i : m) : |
| permElt σ * p ξ i * permElt σ⁻¹ = p ξ (σ i) := by |
| unfold p |
| rw [pushed_eq_sum, pushed_eq_sum] |
| simp only [mul_smul_comm, smul_mul_assoc, Finset.mul_sum, Finset.sum_mul] |
| congr 1 |
| calc |
| _ = ∑ w : m → C, MonoidAlgebra.single (SemidirectProduct.inl (permAction σ w)) |
| ((coordinate ξ (σ i) (permAction σ w) : K)⁻¹) := by |
| apply Finset.sum_congr rfl |
| intro w _ |
| simp only [permElt, MonoidAlgebra.of_apply, MonoidAlgebra.single_mul_single, |
| one_mul, mul_one] |
| rw [← SemidirectProduct.inl_aut] |
| congr 1 |
| simp [coordinate, permAction] |
| _ = _ := Equiv.sum_comp (permAction σ).toEquiv (fun w : m → C => |
| (MonoidAlgebra.single (SemidirectProduct.inl w : Auxiliary m C) |
| ((coordinate ξ (σ i) w : K)⁻¹))) |
| |
| end KaplanskyCounterexample.FiniteBlock |
| end |
| |
| end OAI |
| |
