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67aee1d | 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 | import Mathlib
namespace OAI
noncomputable section
open scoped BigOperators
namespace GroupRingDeterminant
/-- The Hilbert space l²(G)^n, with the product coordinates flattened. -/
abbrev Hilbert (G : Type) (n : ℕ) := lp (fun _ : Fin n × G => ℂ) 2
/-- The identity vector in component i. -/
def identityVector {G : Type} [Group G] {n : ℕ} (i : Fin n) : Hilbert G n :=
by
classical
exact lp.single 2 (i, 1) 1
/-- The unnormalized matrix trace, linear in its operator argument. -/
def groupTrace {G : Type} [Group G] {n : ℕ}
(T : Hilbert G n →L[ℂ] Hilbert G n) : ℂ :=
∑ i : Fin n, inner ℂ (identityVector i) (T (identityVector i))
/-- The invertible-case Fuglede--Kadison determinant.
It is used below only for invertible regular operators. -/
def fkDet {G : Type} [Group G] {n : ℕ}
(T : Hilbert G n →L[ℂ] Hilbert G n) : ℝ :=
Real.exp ((groupTrace (CFC.log (star T * T))).re / 2)
/-- Exact coordinate specification of the bounded left-regular operator of A. -/
def IsRegularOperator {G : Type} [Group G] {n : ℕ}
(A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℤ G))
(T : Hilbert G n →L[ℂ] Hilbert G n) : Prop :=
∀ (ξ : Hilbert G n) (i : Fin n) (h : G),
(T ξ) (i, h) = ∑ j : Fin n,
(A i j).coeff.sum fun g z => (z : ℂ) * ξ (j, g⁻¹ * h)
/-- Coefficientwise extension from the integral to the rational group ring. -/
def rationalMatrix {G : Type} [Group G] {n : ℕ}
(A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℤ G)) :
Matrix (Fin n) (Fin n) (MonoidAlgebra ℚ G) :=
A.map (MonoidAlgebra.mapRingHom G (Int.castRingHom ℚ))
/-- An integral group-ring matrix with rational and bounded inverses and determinant in (0, 1). -/
def MainStatement : Prop :=
∃ (G : Type) (_ : Group G) (_ : Group.FG G) (n : ℕ),
1 ≤ n ∧ ∃ (A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℤ G))
(T : Hilbert G n →L[ℂ] Hilbert G n),
IsUnit (rationalMatrix A) ∧ IsRegularOperator A T ∧ IsUnit T ∧
0 < fkDet T ∧ fkDet T < 1
end GroupRingDeterminant
end
end OAI
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