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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.00336 |
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| _version_ | 1866908903812366336 |
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| author | Weising, Milo Bechtloff Black, Alexander E. |
| author_facet | Weising, Milo Bechtloff Black, Alexander E. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00336 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Saturation for Non-Symmetric Macdonald Polynomials Weising, Milo Bechtloff Black, Alexander E. Combinatorics 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. |
| title | Saturation for Non-Symmetric Macdonald Polynomials |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.00336 |