Every automorphism acts on the commutator layer with the eigenvalue one (Theorem 11)

import Mathlib
import Definitions.Def_RInf_alg
import Definitions.Def_RInf_braid

namespace RInf

/-!  Milestone 5 (Theorem 11): every automorphism of the pure braid group acts
on the commutator layer with the eigenvalue one. The induced action of α
equals the representation value rep σ for some permutation σ by the
factorization of equation (8), and every value of the representation has a
nonzero fixed vector - the content of Theorems 8 and 11, where the
representation is identified with the irreducible representation of the
symmetric group of Young diagram shape with three single cells, and the
eigenvalue one is read off from the cyclic exponents via Stembridge's
theorem. -/
theorem induced_eigen_one {n : Nat} {P Q : Type*} [Group P] [AddCommGroup Q]
    (W : BraidWorld n P Q) (α : P ≃* P) : HasEigenOne (W.act α) := by sorry

end RInf