Strong factorization property of Macdonald polynomials and higher-order Macdonald's positivity conjecture

TL;DR

This paper proves a strong factorization property of Macdonald polynomials as q approaches 1, introduces multivariate q,t-Kostka numbers, and conjectures their nonnegative integer coefficients, extending Macdonald's positivity conjecture.

Contribution

It establishes a new strong factorization property of Macdonald polynomials near q=1 and introduces multivariate q,t-Kostka numbers with conjectured positivity.

Findings

Proved strong factorization property of Macdonald polynomials as q approaches 1.

Showed multivariate q,t-Kostka numbers are polynomials in q,t with integer coefficients.

Conjectured nonnegativity of coefficients of multivariate q,t-Kostka numbers.

Abstract

We prove a strong factorization property of interpolation Macdonald polynomials when $q$ tends to $1$. As a consequence, we show that Macdonald polynomials have a strong factorization property when $q$ tends to $1$, which was posed as an open question in our previous paper with F\'eray. Furthermore, we introduce multivariate $q,t$-Kostka numbers and we show that they are polynomials in $q,t$ with integer coefficients by using the strong factorization property of Macdonald polynomials. We conjecture that multivariate $q,t$-Kostka numbers are in fact polynomials in $q,t$ with nonnegative integer coefficients, which generalizes the celebrated Macdonald's positivity conjecture.