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607ee45 | 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
import OAI.RepresentationTheory.Saxl.Main
import OAI.RepresentationTheory.UniversalSquare.Balance.ConstantOutputs
import OAI.RepresentationTheory.UniversalSquare.Balance.WordPackingLeaves
namespace OAI
/-! Word Packing Geometry. -/
section
noncomputable section
namespace Saxl.Balance
open FlagColumns Columns
theorem WordPacking.of_bounded_shape {rs ps : List ℕ} {μ : YoungDiagram} {d E : ℕ}
(hp : rs.Perm μ.transpose.rowLens) (hd : μ.colLen 0 ≤ d)
(hsum : ps.sum = rs.sum) (hlen : ps.length ≤ E)
(k : ℕ) (I : Set ℕ) (A : Fin (d*d) → Prop) (label : Fin (d*d) → ℕ)
(he : ∀ a, (A a ∧ label a = k) ↔ output a ∈ I)
(hs : ∀ t : Tableau rs.sum μ, SupportLE (wordRep rs.sum E)
(projectedSpechtTensor t t (inOutputs I) (inOutputs_invariant I)).toRepresentation) :
Nonempty (WordPacking rs ps d A label {k}) := by
obtain ⟨e,hr,hc⟩ := place_columns μ hp
exact ⟨WordPacking.bounded e hr hc hd hsum hlen k I A label he (hs _)⟩
theorem WordPacking.neighbor {r d : ℕ} (hr : 0 < r) (hd : r+1 ≤ d)
{ps : List ℕ} (hsum : ps.sum = 2*r+1) (hlen : ps.length ≤ 4)
(k : ℕ) (A : Fin (d*d) → Prop) (label : Fin (d*d) → ℕ)
(he : ∀ a, (A a ∧ label a = k) ↔ output a ∈ Set.Icc r (r+1)) :
Nonempty (WordPacking [r+1,r] ps d A label {k}) := by
have hrow : (ShortColumns.shape r 1).rowLens = [r+1,r] :=
YoungDiagram.rowLens_ofRowLens_eq_self (by simp; omega)
have hD : (ShortColumns.shape r 1).transpose.colLen 0 ≤ d := by
rw [YoungDiagram.colLen_transpose]
have hh := YoungDiagram.rowLen_ofRowLens (hw := by apply List.Pairwise.sortedGE; simp)
(show Fin [r+1,r].length from ⟨0,by simp⟩)
change (ShortColumns.shape r 1).rowLen 0 = r+1 at hh
rw [hh]
exact hd
apply WordPacking.of_bounded_shape (μ := (ShortColumns.shape r 1).transpose)
(by rw [YoungDiagram.transpose_transpose,hrow]) hD (by simp; omega) hlen k _ A label he
intro t
exact neighboring_word_support (by simp; omega) t
lemma replicate_sorted (m k : ℕ) : (List.replicate m k).SortedGE :=
List.Pairwise.sortedGE (List.pairwise_replicate.mpr (Or.inr le_rfl))
lemma repeated_column_length {m k n : ℕ} (hk : 0 < k)
(t : Tableau n (YoungDiagram.ofRowLens (List.replicate m k) (replicate_sorted m k)).transpose)
(i : Fin n) : (columnLengthWord t i).val+1 = k := by
have hm := YoungDiagram.mem_transpose.mp (t i).property
obtain ⟨hj,hi⟩ := YoungDiagram.mem_ofRowLens.mp hm
have he := YoungDiagram.rowLen_ofRowLens (hw := replicate_sorted m k)
(show Fin (List.replicate m k).length from ⟨(t i).val.2,hj⟩)
simp only [Fin.getElem_fin, List.getElem_replicate] at he
change (YoungDiagram.ofRowLens (List.replicate m k) (replicate_sorted m k)).transpose.colLen
(t i).val.2 - 1 + 1 = k
rw [YoungDiagram.colLen_transpose,he]
omega
theorem WordPacking.repeated {m k d : ℕ} (hm : 0 < m) (hk : 0 < k) (hd : k ≤ d)
(j : ℕ) (A : Fin (d*d) → Prop) (label : Fin (d*d) → ℕ)
(he : ∀ a, (A a ∧ label a = j) ↔ output a = k) :
Nonempty (WordPacking (List.replicate m k) [m*k] d A label {j}) := by
let μ := (YoungDiagram.ofRowLens (List.replicate m k) (replicate_sorted m k)).transpose
have hrow : μ.transpose.rowLens = List.replicate m k := by
rw [YoungDiagram.transpose_transpose]
apply YoungDiagram.rowLens_ofRowLens_eq_self
intro x hx
rw [(List.mem_replicate.mp hx).2]
exact hk
have hD : μ.colLen 0 ≤ d := by
rw [YoungDiagram.colLen_transpose]
have hh := YoungDiagram.rowLen_ofRowLens (hw := replicate_sorted m k)
(show Fin (List.replicate m k).length from ⟨0,by simpa⟩)
simp only [Fin.getElem_fin, List.getElem_replicate] at hh
rw [hh]
exact hd
apply WordPacking.of_bounded_shape (μ := μ) (E := 1) (by rw [hrow]) hD
(by simp) (by simp) j {k} A label he
intro t
exact constant_output_support t k (repeated_column_length hk t)
end Saxl.Balance
end
end
end OAI
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