File size: 2,517 Bytes
9b6e014 | 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 | import Mathlib
namespace OAI
noncomputable section
open scoped BigOperators MonoidAlgebra
namespace BinaryCoordinateSweeps
namespace Irrep
variable {G V : Type*} [Group G] [Fintype G]
[AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V]
(ρ : Representation ℂ G V) [ρ.IsIrreducible]
omit [Fintype G] [FiniteDimensional ℂ V] in
lemma invariant_eq_zero_of_finrank_gt_one (hD : 1 < Module.finrank ℂ V)
(v : V) (hv : ∀ g, ρ g v = v) : v = 0 := by
let f : Representation.IntertwiningMap (1 : Representation ℂ G ℂ) ρ :=
(LinearMap.toSpanSingleton ℂ V v).intertwiningMap_of_isIntertwiningMap _ _ (by
intro g c
simp only [MonoidHom.one_apply, Module.End.one_apply, LinearMap.toSpanSingleton_apply,
map_smul, hv])
rcases Representation.IsIrreducible.surjective_or_eq_zero f with hf | hf
· have hd := LinearMap.finrank_le_finrank_of_surjective (f := f.toLinearMap) hf
simp only [Module.finrank_self] at hd
omega
· have hh := congrArg (fun a : Representation.IntertwiningMap
(1 : Representation ℂ G ℂ) ρ => a (1 : ℂ)) hf
simpa [f] using hh
omit [FiniteDimensional ℂ V] in
lemma sum_apply_eq_zero_of_finrank_gt_one (hD : 1 < Module.finrank ℂ V) (x : V) :
∑ g : G, ρ g x = 0 := by
apply invariant_eq_zero_of_finrank_gt_one ρ hD
intro g
simp only [map_sum, ← Module.End.mul_apply, ← map_mul]
exact Function.Bijective.sum_comp (Group.mulLeft_bijective g) (fun h : G => ρ h x)
omit [FiniteDimensional ℂ V] in
lemma sum_eq_zero_of_finrank_gt_one (hD : 1 < Module.finrank ℂ V) :
∑ g : G, ρ g = 0 := by
ext x
simpa using sum_apply_eq_zero_of_finrank_gt_one ρ hD x
include ρ in
omit [FiniteDimensional ℂ V] in
lemma finrank_le_card : Module.finrank ℂ V ≤ Fintype.card G := by
let : IsSimpleModule ℂ[G] ρ.asModule := inferInstance
let : Nontrivial ρ.asModule := IsSimpleModule.nontrivial ℂ[G] ρ.asModule
obtain ⟨v, hv⟩ := exists_ne (0 : ρ.asModule)
let f : ℂ[G] →ₗ[ℂ] ρ.asModule :=
(LinearMap.toSpanSingleton ℂ[G] ρ.asModule v).restrictScalars ℂ
have hf : Function.Surjective f := IsSimpleModule.toSpanSingleton_surjective ℂ[G] hv
have hd := LinearMap.finrank_le_finrank_of_surjective (f := f) hf
have he : Module.finrank ℂ ρ.asModule = Module.finrank ℂ V :=
ρ.asModuleEquiv.finrank_eq
rw [he] at hd
rw [Module.finrank_eq_card_basis (MonoidAlgebra.basis G ℂ)] at hd
exact hd
end Irrep
end BinaryCoordinateSweeps
end
end OAI
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