Affine Hecke algebras and raising operators for Macdonald polynomials

TL;DR

This paper introduces raising and lowering operators for Macdonald polynomials using Dunkl operators, proving integrality of double Kostka coefficients and defining double multinomial coefficients.

Contribution

It presents a natural q-analogue of raising operators for Macdonald polynomials and establishes new properties like integrality of double Kostka coefficients.

Findings

Proved integrality of double Kostka coefficients

Introduced double analogs of multinomial coefficients

Developed raising and lowering operators for Macdonald polynomials

Abstract

We introduce certain raising and lowering operators for Macdonald polynomials (of type $A_{n-1}$) by means of Dunkl operators. The raising operators we discuss are a natural $q$-analogue of raising operators for Jack polynomials introduced by L.Lapointe and L.Vinet. As an application we prove the integrality of double Kostka coefficients. Double analog of the multinomial coefficients are introduced.