`BookProof.PermSector.signRep_mem_sectorCore` (n : ℕ) (σ : Equiv.Perm (Fin n)) (x : (Hs.pow n).carrier) (hx : x ∈ sectorCore Hs D₂ D n) : (signRep Hs n).act σ x ∈ sectorCore Hs D₂

Proved

BookProof.PermSector.signRep_mem_sectorCore

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterPermutationSectorEsa.

BookProof.PermSector.signRep_mem_sectorCore (n : ℕ) (σ : Equiv.Perm (Fin n)) (x : (Hs.pow n).carrier) (hx : x ∈ sectorCore Hs D₂ D n) : (signRep Hs n).act σ x ∈ sectorCore Hs D₂ D n

Formalization note: Lean 4 identifier BookProof.PermSector.signRep_mem_sectorCore.

Preamble

-- Generated from ChapterPermutationSectorEsa.lean — theorem BookProof.PermSector.signRep_mem_sectorCore
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterGraphCoreTransfer
import Definitions.Def_ChapterReducingSubspaceEsa
import Definitions.Def_ChapterGroupAverageEsa
import Definitions.Def_ChapterTensorPermutation
import Mathlib
import Definitions.Def_ChapterPermutationSectorEsa
import Definitions.Def_ChapterDirectSumEsa
import Definitions.Def_ChapterGaugeUnconstrainedSpectrum
import Definitions.Def_ChapterTensorGraphCore
import Definitions.Def_ChapterWignerLittleGroup
open BookProof.DirectSumEsa
open BookProof.ChapterGaugeUnconstrainedSpectrum
open BookProof.TensorCore
open BookProof.ChapterWignerLittleGroup
open BookProof.PermSector



open scoped TensorProduct
open BookProof.FarisLavine BookProof.GraphCore BookProof.ReducedEsa BookProof.TensorCore
open BookProof.GroupAverage BookProof.TensorPerm

noncomputable section

variable (Hs : IPSpace) (D₂ : Submodule ℂ Hs.carrier) (A : D₂ →ₗ[ℂ] Hs.carrier)
  (D : Submodule ℂ Hs.carrier)

Formal statement

theorem BookProof.PermSector.signRep_mem_sectorCore (n : ℕ) (σ : Equiv.Perm (Fin n)) (x : (Hs.pow n).carrier)
    (hx : x ∈ sectorCore Hs D₂ D n) : (signRep Hs n).act σ x ∈ sectorCore Hs D₂ D n := by sorry

View graph