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arXiv · 2307.06517

A raising operator formula for Macdonald polynomials

Jonah Blasiak·Mark Haiman·Jennifer Morse·Anna Pun·George Seelinger
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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.

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Jonah Blasiak, Mark Haiman, Jennifer Morse, Anna Pun, George Seelinger. 2023-07-13. A raising operator formula for Macdonald polynomials. https://arxiv.org/abs/2307.06517

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