A raising operator formula for Macdonald polynomials

Abstract: We give an explicit raising operator formula for the modified Macdonald polynomials \(\tilde{H}_{\mu}(X;q,t)\), which follows from our recent formula for \(\nabla\) on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions \(\tilde{H}^{1,n}(X;q,t)\) that we call \(1,n\)-Macdonald polynomials, which reduce to a scalar multiple of \(\tilde{H}_{\mu}(X;q,t)\) when \(n=1\). We conjecture that the coefficients of \(1,n\)-Macdonald polynomials in terms of Schur functions belong to \(\mathbb{N}[q,t]\), generalizing Macdonald positivity.

Date published: Thursday, July 13, 2023