The h-expansion of Macdonald operators and their expression by Dunkl operators

TL;DR

This paper explicitly computes the third-order term in the h-expansion of Macdonald operators, demonstrating they can be expressed as polynomials of Dunkl operators, thus deepening understanding of their algebraic structure.

Contribution

It introduces a method to calculate higher-order coefficients in the h-expansion of Macdonald operators and proves their expression as Dunkl operator polynomials.

Findings

Third-degree Dunkl operators appear in the h^3 coefficient.

Explicit formulas for the h^3 coefficients are derived.

Macdonald operators' h-expansion coefficients are polynomials of Dunkl operators.

Abstract

Macdonald operators are well known as the 'commutative family' acting on the symmetric functions over Q(q,t). If we suppose that q=exp(h) and t=exp(beta h) and observe the Taylor expansion around h=0, we can see the second-degree Dunkl operator appear especially as the coefficient of h^2. These Dunkl operators also consist of commutative family. Then, as to the coefficient of h^3, it is natural to expect that third-degree Dunkl operator appears. The object of this paper is to calculate the coefficients of h^3 in the h-expansion of Macdonald operators explicitly, to introduce the method of calculation, and to prove that they can be expressed as the polynomials of Dunkl operators.