Majorization via positivity of Jack and Macdonald polynomial differences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915796758822912 |
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| author | Chen, Hong Khare, Apoorva Sahi, Siddhartha |
| author_facet | Chen, Hong Khare, Apoorva Sahi, Siddhartha |
| contents | Majorization inequalities have a long history, going back to Maclaurin and Newton. They were recently studied for several families of symmetric functions, including by Cuttler--Greene--Skandera (2011), Sra (2016), Khare--Tao (2021), McSwiggen--Novak (2022), and Chen--Sahi (2024+) among others. Here we extend the inequalities by these authors to Jack and Macdonald polynomials, and obtain conjectural characterizations of majorization and of weak majorization of the underlying partitions. We prove these characterizations for two variables. In fact, we upgrade -- and prove in the above cases -- the characterization of majorization, to containment of Jack and Macdonald differences lying in the Muirhead semiring. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_19649 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Majorization via positivity of Jack and Macdonald polynomial differences Chen, Hong Khare, Apoorva Sahi, Siddhartha Combinatorics Representation Theory Primary 05E05, 33D52, Secondary 06A07, 26D05 Majorization inequalities have a long history, going back to Maclaurin and Newton. They were recently studied for several families of symmetric functions, including by Cuttler--Greene--Skandera (2011), Sra (2016), Khare--Tao (2021), McSwiggen--Novak (2022), and Chen--Sahi (2024+) among others. Here we extend the inequalities by these authors to Jack and Macdonald polynomials, and obtain conjectural characterizations of majorization and of weak majorization of the underlying partitions. We prove these characterizations for two variables. In fact, we upgrade -- and prove in the above cases -- the characterization of majorization, to containment of Jack and Macdonald differences lying in the Muirhead semiring. |
| title | Majorization via positivity of Jack and Macdonald polynomial differences |
| topic | Combinatorics Representation Theory Primary 05E05, 33D52, Secondary 06A07, 26D05 |
| url | https://arxiv.org/abs/2509.19649 |