A raising operator formula for Macdonald polynomials

2023/07/13 by Blasiak, Jonah, Haiman, Mark, Morse, Jennifer +2
#Combinatorics (math.CO) #FOS: Mathematics #Primary: 05E05 #Secondary: 33D52

paper · doi:10.48550/arxiv.2307.06517

Abstract

We give an explicit raising operator formula for the modified Macdonald polynomials H_μ(X;q,t), which follows from our recent formula for ∇ 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 H^1,n(X;q,t) that we call 1,n-Macdonald polynomials, which reduce to a scalar multiple of H_μ(X;q,t) when n=1. We conjecture that the coefficients of 1,n-Macdonald polynomials in terms of Schur functions belong to ℕ[q,t], generalizing Macdonald positivity.

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