Extension of the Differential Algebraic Closure Method to Hecke Algebras

This paper extends the differential algebraic closure framework for solving polynomial equations to the context of Hecke algebras. We construct a differential algebraic structure on Hecke algebras using Dunkl operators and develop an explicit analytical solution for the eigenvalue problems arising in representation theory. Weprovide complete constructive proofs using symmetric function theory and combinatorial analysis, derive explicit expressions for the correction coefficients γ(n)m (q), and present a detailed O(n2) algorithm. Extensive numerical validation with 256-bit precision demonstrates machine precision accuracy (residuals < 10−30) for degrees up to 25, including challenging cases from representation theory.This work establishes fundamental connections between differential algebra and Hecke algebra theory, providing new computational tools for representation-theoretic problems with applications to quantum groups and Kazhdan-Lusztig theory.