Further Pieri-type formulas for the nonsymmetric Macdonald polynomials

TL;DR

This paper extends Pieri-type formulas for nonsymmetric Macdonald polynomials beyond the known r=1 case, providing a comprehensive generalization that aids in evaluating related binomial coefficients.

Contribution

It introduces the full generalization of Pieri-type formulas for nonsymmetric Macdonald polynomials, expanding the known explicit branching coefficient formulas.

Findings

Generalized Pieri formulas for all r values

Explicit formulas for binomial coefficients

Enhanced understanding of polynomial decompositions

Abstract

The branching coefficients in the expansion of the elementary symmetric function multiplied by a symmetric Macdonald polynomial $P_\kappa(z)$ are known explicitly. These formulas generalise the known $r=1$ case of the Pieri-type formulas for the nonsymmetric Macdonald polynomials $E_\eta(z)$. In this paper we extend beyond the case $r=1$ for the nonsymmetric Macdonald polynomials, giving the full generalisation of the Pieri-type formulas for symmetric Macdonald polynomials. The decomposition also allows the evaluation of the generalised binomial coefficients $\tbinom{\eta }{\nu }_{q,t}$ associated with the nonsymmetric Macdonald polynomials.