Saturation for Non-Symmetric Macdonald Polynomials

TL;DR

This paper proves that supports of non-symmetric Macdonald polynomials are M-convex, confirming their saturated Newton polytope property, and explores implications for affine Demazure characters and Bruhat interval polytopes.

Contribution

It establishes M-convexity of supports for non-symmetric Macdonald polynomials and affine Demazure characters, resolving several open conjectures and connecting to generalized permutahedra.

Findings

Supports are M-convex for non-symmetric Macdonald polynomials

Affine Demazure characters have saturated Newton polytopes

Certain affine Bruhat interval polytopes are generalized permutahedra

Abstract

We prove that supports of non-symmetric Macdonald polynomials are $M$-convex. As a consequence, we resolve a 2019 conjecture of Monical, Tokcan, and Yong that they have the saturated Newton polytope property. As a corollary we show that affine Demazure characters of type $\mathrm{GL}$ have M-convex supports and therefore the saturated Newton polytope property answering a 2022 open question of Besson and Hong. By their results, we then find that certain affine analogs of Bruhat interval polytopes in type $\mathrm{GL}$ are generalized permutahedra. To prove these results, we find a novel interpretation of the Haglund--Haiman--Loehr formula for non-symmetric Macdonald polynomials in terms of colorings of Dyck graphs.