No Primary Source Found for Hyperoctahedral Group Bₙ Invariant Ring — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: Search returned no primary source on the invariant ring of the hyperoctahedral group Bₙ/C₂≀Sₙ; results are generic polynomial-ring pedagogy plus one tangential hyperoctahedral-homology paper. | MATH: The target object is R = ℝ[x₁,…,xₙ]^{Bₙ}, the ring of Bₙ-invariant polynomials. Classical (non-modular) result: R is a free polynomial algebra ℝ[θ₁,…,θₙ] with homogeneous generators in degrees 2,4,…,2n — i.e. θ_d = elementary symmetric function e_d(x₁²,…,xₙ²), deg θ_d = 2d. Poincaré series = ∏_{d=1}^{n} 1/(1−t^{2d}). | CONNECTION: Degrees 2,4,…,2n are exactly twice the exponents 1,2,…,n of the Bₙ/Cₙ root system; by Chevalley–Shephard–Todd this double-exponent pattern is the crystallographic signature of Bₙ. The 2d ladder encodes the Dₙ/Bₙ lattice (hypercubic + cross-polytope) symmetry. No 0.382/0.618/0.786/1.618/2.618 ratios appear in the retrieved material; the only harmonic-adjacent constant is the factor 2 (doubling of exponents). | DEPTH: 2 — the retrieved sources do not actua Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224085736