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e3e0a0d | 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 92 93 | 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]
/-- The coordinate permutation action in the exact finite auxiliary group. -/
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
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