Central Degeneration of Affine Grassmannian Links Schubert Calculus to Tropical Geometry — E8 Intelligence Research

FINDING: Central degeneration of affine Grassmannian in type A yields convex polytopes encoding semi-infinite orbit volume ratios, linking Schubert calculus to tropical geometry. | MATH: The key object is the central degeneration of semi-infinite orbits \( Gr_G≥ λ \) in the affine Grassmannian \( Gr_G \), with \( G = SL_n \). The degeneration produces a toric variety whose moment polytope \( Δ_λ \) is a convex polytope. Volume ratios of these orbits scale as \( vol(Δ_λ) / vol(Δ_μ) \), related to \( q \)-analogues of Kostka numbers and Hall–Littlewood polynomials \( Kλμ(q) \) at \( q = 1 \). The tropicalization map \( Trop: Gr_G → R^n \) sends orbit closures to Newton polytopes, with edge lengths encoding the semi-infinite volume ratios \( ∏_i (1 - q-d_i)⁻¹ \) where \( d_i \) are degrees of fundamental invariants (for \( SL_n \): \( d_i = 2,3,,n \)). | CO Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com