Unifying Möbius Inversion and Weyl Group Reflection Length via Poset Incidence Algebra — E8 Intelligence Research

FINDING: Möbius inversion on divisor lattices and Weyl group reflection length are unified by the combinatorial structure of posets — the Möbius function of the divisor lattice is μ(n) (the classical number-theoretic Möbius function), while the reflection length in a Coxeter group corresponds to the rank function of the root poset, with both governed by the same incidence algebra formalism. | MATH: Classical Möbius function: μ(n) = Σ_{d|n} μ(d) = δ_{n,1}; Möbius inversion: f(n) = Σ_{d|n} g(d) ⇔ g(n) = Σ_{d|n} μ(n/d) f(d). For a finite Coxeter group W with root system Φ, reflection length ℓ_R(w) = min{ k | w = s₁⋯s_k, sᵢ reflections } = rank of the smallest parabolic subgroup containing w. For Weyl groups, ℓ_R(w) = codimension of the fixed space of w in the reflection representation. Excess: e(w) = min{ℓ(x)+ℓ(y)−ℓ(w) | w=xy, x²=y²=1} — for finite W, e(w) = ℓ_R(w) − ℓ_T(w) where ℓ_T is the usual length. | CONNECTION: The divisor lattice of n is a product of chains (one per prime factor), Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com