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.
Highlights
J Algebr Comb (2017) 46:135–163 parameters q, t. They were immediately hailed as a breakthrough in symmetric function theory as well as special functions, as they contained most of the previously studied families of symmetric functions such as Schur polynomials, Jack polynomials, Hall– Littlewood polynomials and Askey–Wilson polynomials as special cases. They satisfied many exciting properties, among which we just mention one, which led to a remarkable relation between Macdonald polynomials, representation theory and algebraic geometry
Garsia and Haiman [7] refined this conjecture, giving a representation theoretic interpretation for the coefficients in terms of Garsia-Haiman modules, an interpretation which was proved almost 10 years later by Haiman [12], who connected the problem to the study of the Hilbert scheme of N points in the plane from algebraic geometry
Instead of proving Theorem 1.3, we prove the stronger result that interpolation Macdonald polynomials have a small cumulant property when q → 1, from which
Summary
J Algebr Comb (2017) 46:135–163 parameters q, t They were immediately hailed as a breakthrough in symmetric function theory as well as special functions, as they contained most of the previously studied families of symmetric functions such as Schur polynomials, Jack polynomials, Hall– Littlewood polynomials and Askey–Wilson polynomials as special cases. They satisfied many exciting properties, among which we just mention one, which led to a remarkable relation between Macdonald polynomials, representation theory and algebraic geometry. Their fascinating and rich combinatorial structure is one of the most important object of interest in contemporary algebraic combinatorics
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