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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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Basic.Complex.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import OAI.Algebra.Finitistic.Selection.MatrixModel
namespace OAI
open scoped BigOperators
namespace LittleFinitistic
namespace CharacterModule
open MonoidAlgebra
variable {H F : Type*} [Group H] [Group F] [Fintype H]
variable (ψ : H →* F) (χ : H →* ℂˣ)
/-- The unnormalized character projector. Its identity coefficient is one,
so normalization is unnecessary for constructing the nonzero test module. -/
noncomputable def projector : MonoidAlgebra ℂ F :=
∑ h : H, single (ψ h) ((χ h : ℂ)⁻¹)
theorem projector_coeff_one (hψ : Function.Injective ψ) :
(projector ψ χ).coeff 1 = 1 := by
classical
have hp (h : H) : ψ h = 1 ↔ h = 1 := by
rw [← map_one ψ]
exact hψ.eq_iff
simp [projector, coeff_sum, coeff_single, Finsupp.single_apply, hp]
theorem projector_ne_zero (hψ : Function.Injective ψ) : projector ψ χ ≠ 0 := by
intro h
have := projector_coeff_one ψ χ hψ
rw [h] at this
simp at this
theorem projector_eigen (h : H) :
single (ψ h) (1 : ℂ) * projector ψ χ = (χ h : ℂ) • projector ψ χ := by
classical
unfold projector
rw [Finset.mul_sum, Finset.smul_sum]
have he := Equiv.sum_comp (Equiv.mulLeft h) (fun k : H =>
(χ h : ℂ) • single (ψ k) ((χ k : ℂ)⁻¹))
rw [← he]
apply Finset.sum_congr rfl
intro k _
change single (ψ h) (1 : ℂ) * single (ψ k) ((χ k : ℂ)⁻¹) =
(χ h : ℂ) • single (ψ (h*k)) ((χ (h*k) : ℂ)⁻¹)
simp only [map_mul,Units.val_mul,single_mul_single,one_mul,
smul_single,smul_eq_mul]
congr 1
field_simp
variable (hc : ∀ h g, Commute (ψ h) g)
/-- All vectors on which the central subgroup acts by the selected character. -/
noncomputable def eigenspace : Submodule (MonoidAlgebra ℂ F) (MonoidAlgebra ℂ F) where
carrier := {x | ∀ h, single (ψ h) (1 : ℂ) * x = (χ h : ℂ) • x}
zero_mem' := by simp
add_mem' := by
intro a b ha hb h
simp [mul_add,ha h,hb h,smul_add]
smul_mem' := by
intro a x hx h
change single (ψ h) (1 : ℂ) * (a*x) = (χ h : ℂ) • (a*x)
rw [← mul_assoc, (MonoidAlgebra.single_commute (hc h) Commute.one_left a).eq,
mul_assoc,hx h,mul_smul_comm]
theorem projector_mem : projector ψ χ ∈ eigenspace ψ χ hc :=
projector_eigen ψ χ
theorem eigenspace_nontrivial (hψ : Function.Injective ψ) :
Nontrivial (eigenspace ψ χ hc) := by
refine ⟨⟨⟨projector ψ χ,projector_mem ψ χ hc⟩,0,?_⟩⟩
intro h
exact projector_ne_zero ψ χ hψ (congrArg Subtype.val h)
omit [Fintype H] in
theorem eigenspace_finiteDimensional [Finite F] :
FiniteDimensional ℂ (eigenspace ψ χ hc) := by
have : FiniteDimensional ℂ (MonoidAlgebra ℂ F) :=
Module.Finite.equiv (MonoidAlgebra.coeffLinearEquiv ℂ : MonoidAlgebra ℂ F ≃ₗ[ℂ] (F →₀ ℂ)).symm
exact FiniteDimensional.of_injective ((eigenspace ψ χ hc).restrictScalars ℂ).subtype
Subtype.val_injective
end CharacterModule
end LittleFinitistic
end OAI
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