Langlands Duality: Root-Coroot Swap and SO/Sp Long-Short Exchange — E8 Intelligence Research

FINDING: Langlands duality maps root systems to coroot systems, exchanging the Weyl group's action between a group and its dual — with SO/Sp duality as a canonical example where long and short roots swap. | MATH: Root system \(R\) ↔ coroot system \(R^\vee\); Cartan matrix \(A_{ij} = 2\langle \alpha_i, \alpha_j \rangle / \langle \alpha_j, \alpha_j \rangle\) transposes under duality. For \(SO(2n+1)\) (type \(B_n\)) ↔ \(Sp(2n)\) (type \(C_n\)): \(B_n^\vee = C_n\), with root length ratio \(\sqrt{2}:1\) (long:short) in \(B_n\) inverting to \(1:\sqrt{2}\) in \(C_n\). Weyl group order: \(|W(B_n)| = |W(C_n)| = 2^n n!\). Killing form scales by dual Coxeter number \(h^\vee\): \((\cdot,\cdot)_\text{dual} = h^\vee (\cdot,\cdot)_\text{orig}\) up to normalization. | CONNECTION: Root length ratios in \(B_n/C_n\) are \(2:1\) in squared norm — the ratio \(0.5\) and its inverse \(2.0\) appear; the Weyl group \(W(B_n)\) is the hyperoctahedral group, whose invariant theory involves degrees \(2,4,\dots,2n\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com