| import Mathlib |
| import OAI.RepresentationTheory.PartialPermutation.DiagramCounts |
|
|
| namespace OAI |
|
|
| section |
| namespace PartialPermutation |
| namespace DiagramCounting |
| noncomputable section |
| open Finset |
|
|
| lemma shift_injective : Function.Injective (fun x : ℕ × ℕ => (x.1+1,x.2)) := by |
| rintro ⟨i,j⟩ ⟨k,l⟩ h |
| simp only [Prod.mk.injEq] at h |
| exact Prod.ext (Nat.add_right_cancel h.1) h.2 |
|
|
| def dropTop (μ : YoungDiagram) : YoungDiagram where |
| cells := μ.cells.preimage (fun x : ℕ × ℕ => (x.1+1,x.2)) shift_injective.injOn |
| isLowerSet := by |
| intro y x hxy hy |
| simp only [Finset.mem_coe,Finset.mem_preimage] at hy ⊢ |
| have hh : (x.1+1,x.2) ≤ (y.1+1,y.2) := ⟨Nat.add_le_add_right hxy.1 1,hxy.2⟩ |
| exact μ.isLowerSet hh hy |
|
|
| lemma mem_dropTop (μ : YoungDiagram) (i j : ℕ) : |
| (i,j) ∈ dropTop μ ↔ (i+1,j) ∈ μ := by |
| change (i,j) ∈ μ.cells.preimage (fun x : ℕ × ℕ => (x.1+1,x.2)) shift_injective.injOn ↔ _ |
| exact Finset.mem_preimage |
|
|
| def lowerEquiv (μ : YoungDiagram) : (dropTop μ).cells ≃ {x : μ.cells // x.1.1≠0} where |
| toFun x := ⟨⟨(x.1.1+1,x.1.2), (mem_dropTop μ _ _).mp x.2⟩,Nat.succ_ne_zero _⟩ |
| invFun x := ⟨(x.1.1.1-1,x.1.1.2), by |
| apply (mem_dropTop μ _ _).mpr |
| have he : x.1.1.1-1+1=x.1.1.1 := by have h:=x.2; omega |
| rw [he]; exact x.1.2⟩ |
| left_inv x := by apply Subtype.ext; simp |
| right_inv x := by |
| apply Subtype.ext; apply Subtype.ext; apply Prod.ext |
| · have h:=x.2; dsimp; omega |
| · rfl |
|
|
| def rowEquiv (μ : YoungDiagram) : {x : μ.cells // x.1.1=0} ≃ Fin (μ.rowLen 0) where |
| toFun x := ⟨x.1.1.2, by have h:=μ.mem_iff_lt_rowLen.mp x.1.2; simpa [x.2] using h⟩ |
| invFun i := ⟨⟨(0,i), μ.mem_iff_lt_rowLen.mpr i.2⟩,rfl⟩ |
| left_inv x := by apply Subtype.ext; apply Subtype.ext; exact Prod.ext x.2.symm rfl |
| right_inv _ := rfl |
|
|
| lemma dropTop_card (μ : YoungDiagram) : |
| (dropTop μ).cells.card = μ.cells.card-μ.rowLen 0 := by |
| have h := Fintype.card_congr (lowerEquiv μ) |
| rw [Fintype.card_subtype_compl] at h |
| have he := Fintype.card_congr (rowEquiv μ) |
| simpa only [Fintype.card_coe,Fintype.card_fin,he] using h |
|
|
| lemma eq_of_dropTop_row (μ ν : YoungDiagram) (hd : dropTop μ=dropTop ν) |
| (hr : μ.rowLen 0=ν.rowLen 0) : μ=ν := by |
| apply YoungDiagram.ext |
| apply Finset.ext |
| rintro ⟨i,j⟩ |
| cases i with |
| | zero => exact μ.mem_iff_lt_rowLen.trans (hr ▸ ν.mem_iff_lt_rowLen.symm) |
| | succ i => |
| change (i+1,j) ∈ μ ↔ (i+1,j) ∈ ν |
| rw [← mem_dropTop μ, ← mem_dropTop ν,hd] |
|
|
| lemma transpose_card (μ : YoungDiagram) : μ.transpose.cells.card=μ.cells.card := by |
| have e : μ.cells ≃ μ.transpose.cells := { |
| toFun := fun x => ⟨x.1.swap, by simpa only [YoungDiagram.mem_cells, |
| YoungDiagram.mem_transpose,Prod.swap_swap] using x.2⟩ |
| invFun := fun x => ⟨x.1.swap,YoungDiagram.mem_transpose.mp x.2⟩ |
| left_inv := fun _ => rfl |
| right_inv := fun _ => rfl } |
| simpa only [Fintype.card_coe] using (Fintype.card_congr e).symm |
|
|
| def ShapeDeficit (n k : ℕ) := |
| {μ : Shape n // μ.1.cells.card-max (μ.1.rowLen 0) (μ.1.colLen 0)=k} |
|
|
| instance (n k : ℕ) : Fintype (ShapeDeficit n k) := by |
| classical |
| unfold ShapeDeficit |
| infer_instance |
|
|
| def deficitCode (n k : ℕ) (μ : ShapeDeficit n k) : Bool × Shape k := |
| if h : μ.1.1.colLen 0 ≤ μ.1.1.rowLen 0 then |
| (true,⟨dropTop μ.1.1, by rw [dropTop_card]; simpa only [max_eq_left h] using μ.2⟩) |
| else |
| (false,⟨dropTop μ.1.1.transpose, by |
| rw [dropTop_card,transpose_card,YoungDiagram.rowLen_transpose] |
| simpa only [max_eq_right (le_of_not_ge h)] using μ.2⟩) |
|
|
| lemma deficitCode_injective (n k : ℕ) : Function.Injective (deficitCode n k) := by |
| intro μ ν he |
| unfold deficitCode at he |
| split_ifs at he with hμ hν hν |
| · have hd : dropTop μ.1.1=dropTop ν.1.1 := congrArg (fun t : Bool × Shape k => t.2.1) he |
| have hcμ := μ.2 |
| have hcν := ν.2 |
| rw [μ.1.2,max_eq_left hμ] at hcμ |
| rw [ν.1.2,max_eq_left hν] at hcν |
| have hleμ := rowLen_le_card μ.1.1 0 |
| have hleν := rowLen_le_card ν.1.1 0 |
| rw [μ.1.2] at hleμ |
| rw [ν.1.2] at hleν |
| apply Subtype.ext; apply Subtype.ext |
| exact eq_of_dropTop_row _ _ hd (by omega) |
| · have hh := congrArg Prod.fst he; cases hh |
| · have hh := congrArg Prod.fst he; cases hh |
| · have hd : dropTop μ.1.1.transpose=dropTop ν.1.1.transpose := |
| congrArg (fun t : Bool × Shape k => t.2.1) he |
| have hcμ := μ.2 |
| have hcν := ν.2 |
| rw [μ.1.2,max_eq_right (le_of_not_ge hμ)] at hcμ |
| rw [ν.1.2,max_eq_right (le_of_not_ge hν)] at hcν |
| have hleμ := colLen_le_card μ.1.1 0 |
| have hleν := colLen_le_card ν.1.1 0 |
| rw [μ.1.2] at hleμ |
| rw [ν.1.2] at hleν |
| have htr := eq_of_dropTop_row _ _ hd (by |
| simp only [YoungDiagram.rowLen_transpose]; omega) |
| apply Subtype.ext; apply Subtype.ext |
| have hh := congrArg YoungDiagram.transpose htr |
| simpa only [YoungDiagram.transpose_transpose] using hh |
|
|
| lemma deficit_count_le (n k : ℕ) : Fintype.card (ShapeDeficit n k) ≤ 2*partitionCount k := by |
| simpa only [Fintype.card_prod,Fintype.card_bool,partitionCount] using |
| Fintype.card_le_of_injective (deficitCode n k) (deficitCode_injective n k) |
|
|
| end |
| end DiagramCounting |
| end PartialPermutation |
| end |
|
|
| end OAI |
|
|
