The Vandermonde Determinant: A Unifying Bridge Across Interpolation, Root Systems, and Random Matrices — E8 Intelligence Research

FINDING: The Vandermonde determinant is the core algebraic object linking polynomial interpolation, Weyl denominator formulas for type A root systems, and GUE eigenvalue repulsion via the product of pairwise differences. | MATH: The Vandermonde determinant is \( V(x_1,\dots,x_n) = \prod_{1 \le i < j \le n} (x_j - x_i) \). For type A root system \( A_{n-1} \), the Weyl denominator formula gives \( \sum_{\sigma \in S_n} \text{sgn}(\sigma) e^{\sigma(\rho)} = \prod_{\alpha>0} (e^{\alpha/2} - e^{-\alpha/2}) \), which reduces to the Vandermonde in the exponential variables. In GUE, the joint eigenvalue density is \( \frac{1}{Z_n} \prod_{i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com