A generalized Macdonald operator

TL;DR

This paper introduces a generalized difference operator for Macdonald polynomials tied to arbitrary admissible root systems, extending classical operators and deriving new Pieri and Littlewood-Richardson formulas.

Contribution

It provides an explicit difference operator for Macdonald polynomials associated with arbitrary root systems, generalizing previous special cases and deriving new combinatorial formulas.

Findings

Explicit difference operator for Macdonald polynomials with arbitrary root systems

New Pieri formulas derived from duality symmetry

Explicit expansions and Littlewood-Richardson formulas for small weights

Abstract

We present an explicit difference operator diagonalized by the Macdonald polynomials associated with an (arbitrary) admissible pair of irreducible reduced crystallographic root systems. By the duality symmetry, this gives rise to an explicit Pieri formula for the Macdonald polynomials in question. The simplest examples of our construction recover Macdonald's celebrated difference operators and associated Pieri formulas pertaining to the minuscule and quasi-minuscule weights. As further by-products, explicit expansions and Littlewood-Richardson type formulas are obtained for the Macdonald polynomials associated with a special class of small weights.