File size: 2,914 Bytes
ae00908 | 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 | import OAI.Algebra.Finitistic.Selection.CharacterModule
import OAI.Algebra.Finitistic.Selection.SelectionCharacter
import OAI.Algebra.Finitistic.Selection.SelectionAlgebra
namespace OAI
namespace LittleFinitistic
open MonoidAlgebra
noncomputable abbrev TestModule (m : ℕ) [NeZero m] :=
CharacterModule.eigenspace (quotientCentralEmbedding m) (selectionCharacter m)
(quotientCentralEmbedding_central m)
noncomputable instance testModule_nontrivial (m : ℕ) [NeZero m] : Nontrivial (TestModule m) :=
CharacterModule.eigenspace_nontrivial _ _ _ (quotientCentralEmbedding_injective m)
noncomputable instance testModule_finiteDimensional (m : ℕ) [NeZero m] :
FiniteDimensional ℂ (TestModule m) :=
CharacterModule.eigenspace_finiteDimensional _ _ _
noncomputable def testRingMap (m : ℕ) [NeZero m] :
SelectionRing →ₐ[ℂ] MonoidAlgebra ℂ (FiniteQuotientGroup m) :=
MonoidAlgebra.mapDomainAlgHom ℂ ℂ (finiteQuotientMap m)
noncomputable instance testModule_module (m : ℕ) [NeZero m] :
Module SelectionRing (TestModule m) :=
Module.compHom (TestModule m) (testRingMap m).toRingHom
@[simp] theorem testModule_smul_val (m : ℕ) [NeZero m]
(r : SelectionRing) (x : TestModule m) :
((r • x : TestModule m) : MonoidAlgebra ℂ (FiniteQuotientGroup m)) = testRingMap m r * x := rfl
instance testModule_scalarTower (m : ℕ) [NeZero m] : IsScalarTower ℂ SelectionRing (TestModule m) where
smul_assoc c r x := by
apply Subtype.ext
change testRingMap m (c • r) * x.val = c • (testRingMap m r * x.val)
rw [map_smul,smul_mul_assoc]
@[simp] theorem testRingMap_algebraZ (m : ℕ) [NeZero m] (q : ℤ) :
testRingMap m (algebraZ q) = single (finiteQuotientMap m (Presentation.z q)) 1 := by
simp [testRingMap,algebraZ]
theorem testModule_basis_action (m : ℕ) [NeZero m] (i : Fin m) (x : TestModule m) :
algebraZ i.val • x = (if i = lastIndex m then (-1 : ℂ) else 1) • x := by
apply Subtype.ext
change testRingMap m (algebraZ i.val) * x.val = _
rw [testRingMap_algebraZ,← quotientCentralEmbedding_basis]
have hx := x.property (Multiplicative.ofAdd (Pi.single i 1))
rw [selectionCharacter_basis] at hx
by_cases h : i = lastIndex m
· simp only [h,ite_true,Units.val_neg,Units.val_one] at hx ⊢
exact hx
· simp only [h,ite_false,Units.val_one] at hx ⊢
exact hx
/-- The first m−1 selecting idempotents act as identity; the next acts as zero. -/
theorem testModule_idempotent_action (m : ℕ) [NeZero m] (i : Fin m) (x : TestModule m) :
selectionIdempotent i.val • x = if i = lastIndex m then 0 else x := by
classical
unfold selectionIdempotent
rw [smul_assoc,add_smul,one_smul,testModule_basis_action]
split_ifs
· rw [show (-1 : ℂ) • x = -x from neg_one_smul ℂ x,add_neg_cancel,smul_zero]
· rw [one_smul,← two_smul ℂ x,smul_smul]
norm_num
end LittleFinitistic
end OAI
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