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

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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. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22702008

September 11, 2026Open Access

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

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Key Points

To establish the unifying role of the Vandermonde determinant across polynomial interpolation, Weyl denominator formulas for type A root systems, and eigenvalue repulsion in random matrix theory.
Evaluated algebraic expressions for the Vandermonde determinant defined as the product of pairwise differences.
Analyzed the reduction of the Weyl denominator formula for Lie algebra type A root systems into the Vandermonde form.
Examined joint eigenvalue probability densities within the Gaussian Unitary Ensemble (GUE).
Demonstrated that the Weyl denominator formula for type A root systems directly reduces to the Vandermonde determinant when expressed in exponential variables.
Showed that pairwise difference products govern eigenvalue repulsion in the Gaussian Unitary Ensemble joint eigenvalue distribution.

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Implication

Mathematical analysis demonstrates how the Vandermonde determinant links root systems and random matrix ensembles, highlighting unified algebraic structures across diverse domains.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6aa7b3770926e14a848b2716 https://doi.org/10.5281/zenodo.22702008

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

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

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