Datasets:

roseking
/

openAI-math-dataset

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

	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