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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 91 92 | import OAI.RepresentationTheory.YoungSymmetry.UnitaryModel
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace RowColumn
open CubeShuffle CubeShuffle.Specht CubeShuffle.UnitaryFinite
/-- Permutations preserving each fibre of a given line map. -/
def lineGroup {α β : Type*} (f : α → β) : Subgroup (Equiv.Perm α) where
carrier := {g | ∀ x, f (g x) = f x}
one_mem' := by intro x; rfl
mul_mem' := by
intro a b ha hb x
exact (ha (b x)).trans (hb x)
inv_mem' := by
intro a ha x
simpa using (ha (a⁻¹ x)).symm
abbrev Board (m n : ℕ) := Fin m × Fin n
abbrev rowGroup {m n : ℕ} (Ω : Finset (Board m n)) :=
lineGroup (fun x : Ω => x.1.1)
abbrev columnGroup {m n : ℕ} (Ω : Finset (Board m n)) :=
lineGroup (fun x : Ω => x.1.2)
/-- Zero-based form of the manuscript's (h,h)-hook. -/
def InHook (a : YoungDiagram) (h : ℕ) : Prop :=
∀ c ∈ a.cells, c.1 < h ∨ c.2 < h
section Copies
variable {G V W : Type*} [Group G]
[NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
[NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
/-- An actual equivariant inclusion, not an unspecified overlap matrix. -/
def Intertwines (ρ : Representation ℂ G V) (τ : Representation ℂ G W)
(I : V →ₗ[ℂ] W) : Prop :=
∀ g v, I (ρ g v) = τ g (I v)
/-- All orthogonal multiplicity copies of one irreducible type. The completeness
condition quantifies over genuine intertwiners, i.e. it is exactly the isotypic
space condition. -/
def CompleteCopies (ρ : Representation ℂ G V) (τ : Representation ℂ G W)
{u : ℕ} (I : Fin u → V →ₗᵢ[ℂ] W) : Prop :=
(∀ i, Intertwines ρ τ (I i).toLinearMap) ∧
(∀ i j, i ≠ j → ∀ v w, inner ℂ (I i v) (I j w) = 0) ∧
∀ J : V →ₗ[ℂ] W, Intertwines ρ τ J →
LinearMap.range J ≤ ⨆ i, LinearMap.range (I i).toLinearMap
end Copies
/-- The all-copy operator-overlap sum, with each squared operator norm. -/
def allCopyOverlap {V W X : Type*}
[NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
[NormedAddCommGroup W] [InnerProductSpace ℂ W] [FiniteDimensional ℂ W]
[NormedAddCommGroup X] [InnerProductSpace ℂ X] [FiniteDimensional ℂ X]
{u v : ℕ} (I : Fin u → V →ₗᵢ[ℂ] X) (J : Fin v → W →ₗᵢ[ℂ] X) : ℝ :=
∑ i, ∑ j, ‖((I i).toContinuousLinearMap.adjoint).comp
(J j).toContinuousLinearMap‖ ^ 2
/-- Exact occupied-board main. Carriers are in orthonormal coordinates; arbitrary
unitary irreducibles of the two actual line groups are quantified over (equivalently,
the tuples of line partitions in the manuscript). Labelling `e` chooses the action
of the Specht module on the occupied cells and forces |a| = |Ω|. -/
abbrev OccupiedOverlapEndpoint : Prop :=
∃ C : ℝ, 0 < C ∧ ∀ (m n : ℕ) (Ω : Finset (Board m n))
(a : YoungDiagram) (e : Ω ≃ Cell a) (h : ℕ),
1 ≤ h → h ≤ m * n → InHook a h →
∀ (r c u v : ℕ)
(ρ : Representation ℂ (rowGroup Ω) (EuclideanSpace ℂ (Fin r)))
(τ : Representation ℂ (columnGroup Ω) (EuclideanSpace ℂ (Fin c))),
Representation.IsIrreducible ρ → Representation.IsIrreducible τ →
IsUnitary ρ → IsUnitary τ →
∀ (I : Fin u → EuclideanSpace ℂ (Fin r) →ₗᵢ[ℂ] hilbertSpace a)
(J : Fin v → EuclideanSpace ℂ (Fin c) →ₗᵢ[ℂ] hilbertSpace a),
CompleteCopies (V := EuclideanSpace ℂ (Fin r)) (W := hilbertSpace a) ρ ((relabelledUnitary a e).comp (rowGroup Ω).subtype) I →
CompleteCopies (V := EuclideanSpace ℂ (Fin c)) (W := hilbertSpace a) τ ((relabelledUnitary a e).comp (columnGroup Ω).subtype) J →
allCopyOverlap (V := EuclideanSpace ℂ (Fin r)) (W := EuclideanSpace ℂ (Fin c))
(X := hilbertSpace a) I J ≤
Real.exp (C * (((m + n + 1 : ℕ) : ℝ) * (h : ℝ)^2 *
Real.log ((m * n : ℕ) + 2) + (m * n - Ω.card : ℕ))) *
min 1 ((r : ℝ) * c / Module.finrank ℂ (space a))
end RowColumn
end
end OAI
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