Clustering properties of rectangular Macdonald polynomials

TL;DR

This paper proves a conjectured factorization property of Macdonald polynomials, which are important in mathematical physics, by analyzing four related families of these polynomials.

Contribution

It provides the first proof of a conjectured clustering property of Macdonald polynomials involving four different polynomial families.

Findings

Proof of the clustering factorization formula for Macdonald polynomials

Application of four polynomial families to establish the result

Enhanced understanding of Macdonald polynomials' structure

Abstract

The clustering properties of Jack polynomials are relevant in the theoretical study of the fractional Hall states. In this context, some factorization properties have been conjectured for the $(q,t)$-deformed problem involving Macdonald polynomials. The present paper is devoted to the proof of this formula. To this aim we use four families of Jack/Macdonald polynomials: symmetric homogeneous, nonsymmetric homogeneous, shifted symmetric and shifted nonsymmetric.