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Saturation for Non-Symmetric Macdonald Polynomials

Milo Bechtloff Weising, Alexander E. Black

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.

Saturation for Non-Symmetric Macdonald Polynomials

Abstract

We prove that supports of non-symmetric Macdonald polynomials are -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 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 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.

Paper Structure

This paper contains 14 sections, 22 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Funding
  3. Acknowledgments
  4. Non-symmetric Macdonald polynomials and their combinatorics
  5. Non-symmetric Macdonald polynomials
  6. Newton polytopes
  7. From Non-symmetric Macdonald Polynomials to Graph Coloring
  8. Symmetries from Dyck Graph Colorings
  9. M-convexity
  10. A Geometric Lemma
  11. Proof of main theorem
  12. Conjecture of Monical, Tokcan, and Yong
  13. Moment polytopes
  14. Recovering Non-symmetric Macdonald Polynomials from the Dyck Graph

Key Result

Theorem 1.1

The supports of non-symmetric Macdonald polynomials are M-convex. In particular, they have the saturated Newton polytope property.

Theorems & Definitions (50)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Proposition 2.7
  • ...and 40 more