A raising operator formula for Macdonald polynomials

TL;DR

This paper derives an explicit raising operator formula for modified Macdonald polynomials, connecting recent advances in LLT polynomials and symmetric functions, and introduces a new family called 1,n-Macdonald polynomials.

Contribution

It provides a novel raising operator formula for modified Macdonald polynomials and introduces the 1,n-Macdonald polynomials, extending Macdonald positivity conjectures.

Findings

Derived explicit raising operator formula for modified Macdonald polynomials

Introduced the 1,n-Macdonald polynomials family

Conjectured positivity of coefficients in Schur function expansion

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.