Complement Duality in the 0-Hecke Algebra: Eigenspaces, Fibonacci Support, and the Antipode Involution

Let Hₙ(0) be the 0-Hecke algebra of type Aₙ₋₁, acting on the regular representation ℂ[Sₙ] via the Norton–Krob–Thibon generators T₁,…,Tₙ₋₁. The transfer operator T = Σ Tᵢ has integer eigenvalues 0,1,…,n−1 with Eulerian multiplicities. Let Vₖ denote the generalized eigenspace for eigenvalue k, and let Mⱼ (J ⊆ S = {1,…,n−1}) be the simple Hₙ(0)-modules. We establish the complement duality: for all k ∈ {0,…,n−1} and J ⊆ S, [Vₖ : Mⱼ] = [Vₙ₋₁₋ₖ : Mₛ∖ⱼ] As consequences we show that (i) the support of V₁ consists precisely of the independent sets of the Dynkin diagram Aₙ₋₁, giving |Supp(V₁)| = Fₙ₊₁ − 1 (Fibonacci); (ii) the Ext¹-quiver restricted to Supp(V₁) is isomorphic to that of Supp(Vₙ₋₂) via J ↦ S ∖ J; and (iii) the duality is the shadow of the antipode of the Hopf algebra of quasisymmetric functions QSym. Complement duality is proved for n = 3 and n = 4 via an explicit involution argument; it is verified computationally for n = 3,…,7 (716 equalities, zero failures). A companion Lean 4 formal verification file machine-checks the multiplicity tables and duality identities for n = 3 and n = 4.