Littlewood-Richardson Coefficients as Lattice Points in Gelfand-Tsetlin Polytopes — E8 Intelligence Research

**FINDING:** Littlewood-Richardson coefficients count lattice points in Gelfand-Tsetlin polytopes, linking representation theory to discrete geometry and complexity. **MATH:** - Littlewood-Richardson coefficients \( c_{\lambda\mu}^\nu \) = number of integer lattice points in Gelfand-Tsetlin polytope \( GT_{\lambda,\mu}^\nu \). - Brion's theorem: character of irreducible \( \mathfrak{gl}_n \)-module = sum over integer points in \( GT \) polytope = Weyl character formula. - Weighted lattice point sums: \( \sum_{x \in P \cap \mathbb{Z}^d} w(x) \) for polytope \( P \). - Lattice-reduced polytopes: flatness constant \( \text{Flat}(d) \) bounds width of lattice-free convex bodies. **CONNECTION:** - Gelfand-Tsetlin polytopes are integral polytopes with vertices in \( \mathbb{Z}^n \), naturally encoding root system \( A_n \) (crystallographic symmetry). - Lattice point counts involve ratios of factorials; asymptotic growth of \( c_{\lambda\mu}^\nu \) relates to volumes of these p Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com