Tilting multiplicities from tensor powers of quantum sl₂ at even roots of unity — E8 Intelligence Research

FINDING: Tensor power decomposition of \( U_q(\mathfrak{sl}_2) \) at even roots of unity yields combinatorial tilting module multiplicities, linking to modular S-matrix and knot invariants. | MATH: \( U_q(\mathfrak{sl}_2) \) at \( q = e^{2\pi i / \ell} \) with \( \ell \) even; tilting modules \( T_n \); tensor power \( V^{\otimes N} \) of 2D irrep; multiplicity formula via combinatorial sums; modular S-matrix entries \( S_{ab} = \sqrt{\frac{2}{\ell}} \sin\left(\frac{\pi (a+1)(b+1)}{\ell}\right) \). | CONNECTION: Ratios \( \sin(\pi k / \ell) \) for \( \ell \) even produce values like \( \sin(\pi/5) \approx 0.5878 \) (near 0.618), \( \sin(2\pi/5) \approx 0.9511 \); root system \( A_1 \) (SU(2)) symmetry; lattice structure from Weyl alcove geometry; base-60 not directly present but modular forms link to 60° angles in triangular lattices. | DEPTH: 8 — Directly ties quantum group representation theory to topological invariants (knot polynomials, 3-manifold invariants) via modular tensor cat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com