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import OAI.RepresentationTheory.PartialPermutation.Tabloids
namespace OAI
section
namespace PartialPermutation
namespace YoungTabloid
noncomputable section
open Finset
open scoped Classical
lemma exists_sortingEquiv (α : Type*) [Fintype α] (r : α → ℕ) (hr : Function.Injective r)
(n : ℕ) (hn : Fintype.card α=n) :
∃ e : α ≃ Fin n, StrictMono (fun i => r (e.symm i)) := by
let : LinearOrder α := LinearOrder.lift' r hr
let e := Fintype.orderIsoFinOfCardEq α hn
exact ⟨e.toEquiv.symm, e.strictMono⟩
lemma strictMono_fin_nat_le {n : ℕ} (f : Fin n → ℕ) (hf : StrictMono f) (i : Fin n) :
i.val ≤ f i := by
obtain ⟨i,hi⟩ := i
induction i with
| zero => exact Nat.zero_le _
| succ i ih =>
have hi' : i<n := by omega
have h := hf (show (⟨i,hi'⟩:Fin n)<⟨i+1,hi⟩ by simp)
have ih' := ih hi'
change i ≤ f ⟨i,hi'⟩ at ih'
change i+1 ≤ f ⟨i+1,hi⟩
omega
def Transversal (μ : YoungDiagram) (r : Tabloid μ) : Prop :=
∀ x y : μ.cells, col μ x=col μ y → r.1 x=r.1 y → x=y
lemma transversal_col_injective (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r)
(j : ℕ) : Function.Injective (fun x : Col μ j => r.1 x.1) := by
intro x y h
apply Subtype.ext
exact hr x.1 y.1 (x.2.trans y.2.symm) h
def sortedColumn (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) (j : ℕ) :
Col μ j ≃ Fin (μ.colLen j) :=
(exists_sortingEquiv (Col μ j) (fun x => r.1 x.1)
(transversal_col_injective μ r hr j) _ (col_card μ j)).choose
lemma sortedColumn_mono (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) (j : ℕ) :
StrictMono (fun i => r.1 ((sortedColumn μ r hr j).symm i).1) :=
(exists_sortingEquiv (Col μ j) (fun x => r.1 x.1)
(transversal_col_injective μ r hr j) _ (col_card μ j)).choose_spec
def fiberPermutation {α ι : Type*} (r : α → ι)
(p : ∀ i, Equiv.Perm {a // r a=i}) : Equiv.Perm α :=
(Equiv.sigmaFiberEquiv r).symm.trans
((Equiv.sigmaCongrRight p).trans (Equiv.sigmaFiberEquiv r))
lemma fiberPermutation_apply {α ι : Type*} (r : α → ι)
(p : ∀ i, Equiv.Perm {a // r a=i}) (a : α) :
fiberPermutation r p a = (p (r a) ⟨a,rfl⟩).1 := rfl
lemma fiberPermutation_rank {α ι : Type*} (r : α → ι)
(p : ∀ i, Equiv.Perm {a // r a=i}) (a : α) :
r (fiberPermutation r p a)=r a := (p (r a) ⟨a,rfl⟩).2
def columnSorting (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
Equiv.Perm μ.cells :=
fiberPermutation (col μ) (fun j => (sortedColumn μ r hr j).trans (columnEquiv μ j).symm)
lemma columnSorting_mem (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
columnSorting μ r hr ∈ columnGroup μ :=
fiberPermutation_rank (col μ) _
lemma columnSorting_row (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r)
(x : μ.cells) : row μ (columnSorting μ r hr x) ≤ r.1 x := by
let e := sortedColumn μ r hr (col μ x)
have h := strictMono_fin_nat_le _ (sortedColumn_mono μ r hr (col μ x)) (e ⟨x,rfl⟩)
change (e ⟨x,rfl⟩).val ≤ r.1 (e.symm (e ⟨x,rfl⟩)).val at h
erw [Equiv.symm_apply_apply] at h
exact h
lemma transversal_eq_column (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
∃ c : columnGroup μ, r.1=fun x => row μ (c.1 x) := by
let c := columnSorting μ r hr
have hle := columnSorting_row μ r hr
have heq : (∑ x, row μ (c x))=∑ x, r.1 x := by
rw [rowSum]
exact Fintype.sum_equiv c _ _ (fun _ => rfl)
refine ⟨⟨c,columnSorting_mem μ r hr⟩,funext (fun x => ?_)⟩
exact (Finset.sum_eq_sum_iff_of_le (fun x _ => hle x)).mp heq x (Finset.mem_univ x) |>.symm
lemma column_row_injective (μ : YoungDiagram) :
Function.Injective (fun c : columnGroup μ => fun x => row μ (c.1 x)) := by
intro c d h
apply Subtype.ext
apply Equiv.ext
intro x
apply Subtype.ext
exact Prod.ext (congrFun h x) ((c.2 x).trans (d.2 x).symm)
lemma unique_column_of_transversal (μ : YoungDiagram) (r : Tabloid μ) (hr : Transversal μ r) :
∃! c : columnGroup μ, r.1=fun x => row μ (c.1 x) := by
obtain ⟨c,hc⟩ := transversal_eq_column μ r hr
exact ⟨c,hc,fun d hd => column_row_injective μ (hd.symm.trans hc)⟩
end
end YoungTabloid
end PartialPermutation
end
end OAI
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