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

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 Féray. 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.

26 pages, 2 figures

References in corpus (2)

• Gaussian fluctuations of characters of symmetric groups and of Young diagrams
• Gaussian fluctuations of Jack-deformed random Young diagrams

Cited by in corpus (3)

• Cumulants of Jack symmetric functions and $b$-conjecture
• Macdonald cumulants, $G$-inversion polynomials and $G$-parking functions
• Algebras with two multiplications and their cumulants