Pieri-Type Formulas for the Nonsymmetric Macdonald Polynomials

TL;DR

This paper extends Pieri formulas to nonsymmetric Macdonald polynomials, deriving explicit decompositions and coefficients for specific cases, advancing the understanding of their algebraic structure.

Contribution

It provides the first nonsymmetric Pieri-type formulas for r=1 and r=n-1, including explicit decomposition formulas and binomial coefficient evaluations.

Findings

Derived nonsymmetric Pieri formulas for r=1 and r=n-1.

Expressed product decompositions in terms of nonsymmetric Macdonald polynomials.

Evaluated generalized binomial coefficients for specific cases.

Abstract

In symmetric Macdonald polynomial theory the Pieri formula gives the branching coefficients for the product of the rth elementary symmetric function and the Macdonald polynomial. In this paper we give the nonsymmetric analogues for the cases r=1 and r=n-1. We do this by first deducing the the decomposition for the product of any nonsymmetric Macdonald polynomial with a linear function in terms of nonsymmetric Macdonald polynomials. As a corollary of finding the branching coefficients of the product of the first elementary function with a nonsymmetric Macdonald polynomial we evaluate the generalised binomial coefficients associated with the nonsymmetric Macdonald polynomials for |u|=|v|+1.