| import Mathlib |
| import OAI.RepresentationTheory.PartialPermutation.AlternatingTabloids |
|
|
| namespace OAI |
|
|
| section |
| namespace PartialPermutation |
| namespace YoungTabloid |
| noncomputable section |
| open Finset |
| open scoped Classical |
|
|
| def tabloidRep (μ : YoungDiagram) : |
| Representation ℂ (Equiv.Perm μ.cells) (EuclideanSpace ℂ (Tabloid μ)) where |
| toFun g := |
| { toFun := fun f => WithLp.toLp 2 (fun r => f (relabel μ g⁻¹ r)) |
| map_add' := by intros; rfl |
| map_smul' := by intros; rfl } |
| map_one' := by ext f r; simp [relabel_one] |
| map_mul' := by |
| intro g h |
| ext f r |
| change f (relabel μ (g*h)⁻¹ r)=f (relabel μ h⁻¹ (relabel μ g⁻¹ r)) |
| rw [mul_inv_rev,relabel_mul] |
|
|
| @[simp] lemma tabloidRep_apply (μ : YoungDiagram) (g : Equiv.Perm μ.cells) |
| (f : EuclideanSpace ℂ (Tabloid μ)) (r : Tabloid μ) : |
| tabloidRep μ g f r=f (relabel μ g⁻¹ r) := rfl |
|
|
| lemma tabloidRep_unitary (μ : YoungDiagram) (g : Equiv.Perm μ.cells) |
| (f h : EuclideanSpace ℂ (Tabloid μ)) : |
| inner ℂ (tabloidRep μ g f) (tabloidRep μ g h)=inner ℂ f h := by |
| simp only [PiLp.inner_apply,tabloidRep_apply] |
| exact Fintype.sum_equiv (relabelEquiv μ g⁻¹) _ _ (fun _ => rfl) |
|
|
| def antisym (μ : YoungDiagram) : Module.End ℂ (EuclideanSpace ℂ (Tabloid μ)) := |
| ∑ c : columnGroup μ, signC c.1 • tabloidRep μ c.1 |
|
|
| lemma antisym_apply (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ)) (r : Tabloid μ) : |
| antisym μ f r = ∑ c : columnGroup μ, signC c.1 * f (relabel μ c.1⁻¹ r) := by |
| simp [antisym,LinearMap.sum_apply,LinearMap.smul_apply] |
|
|
| lemma antisym_alternating (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ)) : |
| Alternating μ (antisym μ f) := by |
| intro d r |
| rw [antisym_apply,antisym_apply,Finset.mul_sum] |
| calc |
| _ = ∑ c : columnGroup μ, signC (d*c).1 * |
| f (relabel μ (d*c).1⁻¹ (relabel μ d.1 r)) := |
| (Fintype.sum_equiv (Equiv.mulLeft d) _ _ (fun _ => rfl)).symm |
| _ = _ := by |
| apply Finset.sum_congr rfl |
| intro c _ |
| simp only [Subgroup.coe_mul,signC_mul,mul_inv_rev,← relabel_mul] |
| simp only [mul_assoc,inv_mul_cancel,mul_one] |
|
|
| def delta (μ : YoungDiagram) (r : Tabloid μ) : EuclideanSpace ℂ (Tabloid μ) := |
| WithLp.toLp 2 (fun s => if s=r then 1 else 0) |
|
|
| def polytabloid (μ : YoungDiagram) : EuclideanSpace ℂ (Tabloid μ) := |
| antisym μ (delta μ (canonical μ)) |
|
|
| lemma canonical_fix_iff (μ : YoungDiagram) (c : columnGroup μ) : |
| relabel μ c (canonical μ)=canonical μ ↔ c=1 := by |
| constructor |
| · intro h |
| exact column_canonical_injective μ (h.trans (relabel_one μ _).symm) |
| · rintro rfl |
| rfl |
|
|
| lemma polytabloid_canonical (μ : YoungDiagram) : polytabloid μ (canonical μ)=1 := by |
| rw [polytabloid,antisym_apply] |
| change (∑ c : columnGroup μ, signC c.1 * |
| (if relabel μ c.1⁻¹ (canonical μ)=canonical μ then 1 else 0))=1 |
| have hh (c : columnGroup μ) : relabel μ c.1⁻¹ (canonical μ)=canonical μ ↔ c=1 := by |
| change relabel μ (c⁻¹:columnGroup μ) (canonical μ)=canonical μ ↔ c=1 |
| rw [canonical_fix_iff,inv_eq_one] |
| simp [hh] |
|
|
| lemma polytabloid_ne_zero (μ : YoungDiagram) : polytabloid μ≠0 := by |
| intro h |
| have hc := polytabloid_canonical μ |
| rw [h] at hc |
| exact zero_ne_one hc |
|
|
| lemma polytabloid_alternating (μ : YoungDiagram) : Alternating μ (polytabloid μ) := |
| antisym_alternating μ _ |
|
|
| lemma antisym_rank_one (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ)) : |
| antisym μ f = (antisym μ f (canonical μ)) • polytabloid μ := by |
| apply PiLp.ext |
| intro r |
| have ha : Alternating μ (fun s => antisym μ f (canonical μ)*polytabloid μ s) := by |
| intro c s |
| dsimp only |
| rw [polytabloid_alternating μ c s] |
| ring |
| exact congrFun (alternating_eq_of_canonical_eq μ _ _ (antisym_alternating μ f) ha |
| (by rw [polytabloid_canonical,mul_one])) r |
|
|
| @[simp] lemma signC_conj {α : Type*} [Fintype α] [DecidableEq α] (g : Equiv.Perm α) : |
| starRingEnd ℂ (signC g)=signC g := by simp [signC] |
|
|
| lemma antisym_selfadjoint (μ : YoungDiagram) (f h : EuclideanSpace ℂ (Tabloid μ)) : |
| inner ℂ (antisym μ f) h=inner ℂ f (antisym μ h) := by |
| have hu (c : columnGroup μ) : |
| inner ℂ (tabloidRep μ c.1 f) h=inner ℂ f (tabloidRep μ c.1⁻¹ h) := by |
| have hh := tabloidRep_unitary μ c.1 f (tabloidRep μ c.1⁻¹ h) |
| simpa using hh |
| simp only [antisym,LinearMap.sum_apply,LinearMap.smul_apply,inner_sum,sum_inner, |
| inner_smul_left,inner_smul_right,signC_conj,hu] |
| exact Fintype.sum_equiv (Equiv.inv (columnGroup μ)) _ _ (fun c => by |
| simp only [Equiv.inv_apply,Subgroup.coe_inv,signC_inv]) |
|
|
| lemma inner_delta (μ : YoungDiagram) (r : Tabloid μ) (f : EuclideanSpace ℂ (Tabloid μ)) : |
| inner ℂ (delta μ r) f=f r := by simp [PiLp.inner_apply,delta] |
|
|
| lemma antisym_eq_inner (μ : YoungDiagram) (f : EuclideanSpace ℂ (Tabloid μ)) : |
| antisym μ f=inner ℂ (polytabloid μ) f • polytabloid μ := by |
| rw [antisym_rank_one] |
| congr 1 |
| rw [polytabloid,antisym_selfadjoint,inner_delta] |
|
|
| end |
| end YoungTabloid |
| end PartialPermutation |
| end |
|
|
| end OAI |
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|
