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.

Keywords

• 05E05
• 33D52