On the Jacobian Matrices of Generalized Chebyshev Polynomials

TL;DR

This paper presents a practical method for computing Jacobian matrices of generalized Chebyshev polynomials linked to semisimple Lie algebras, expressing entries as linear combinations of characters with integer coefficients.

Contribution

It introduces a novel computational approach to derive Jacobian matrices for generalized Chebyshev polynomials associated with any semisimple Lie algebra.

Findings

Entries of Jacobian matrices are linear combinations of irreducible characters.

Integer coefficients are computed via basic Weyl chamber calculations.

Method applies broadly to arbitrary semisimple Lie algebras.

Abstract

In this paper, we give a practical method to compute the Jacobian matrices of generalized Chebyshev polynomials associated to arbitrary semisimple Lie algebras. The entries of each Jacobian matrix can be expressed as a linear combination of characters of irreducible representations of the underlying Lie algebra with integer coefficients. These integer coefficients can be obtained by basic computations in the fundamental Weyl chamber.