Combinatorial formula for Macdonald polynomials, Bethe Ansatz, and generic Macdonald polynomials

TL;DR

This paper introduces generic Macdonald polynomials with additional parameters, providing a direct combinatorial proof for interpolation Macdonald polynomials and connecting them to Bethe eigenfunctions and quantum immanants.

Contribution

It presents a new class of generic Macdonald polynomials that unify various Macdonald polynomials and offers a direct combinatorial proof for their formulas.

Findings

Generic Macdonald polynomials depend on extra parameters.

They specialize to all degree-d Macdonald polynomials.

The form resembles Bethe eigenfunctions and relates to quantum immanants.

Abstract

We give a direct proof of the combinatorial formula for interpolation Macdonald polynomials by introducing certain polynomials, which we call generic Macdonald polynomials, which depend on $d$ additional parameters and specialize to all Macdonald polynomials of degree $d$. The form of these generic polynomials is that of a Bethe eigenfunction and they imitate, on a more elementary level, the $R$-matrix construction of quantum immanants.