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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.09072 |
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| _version_ | 1866911248332881920 |
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| author | Novelli, Jean-Christophe Thibon, Jean-Yves |
| author_facet | Novelli, Jean-Christophe Thibon, Jean-Yves |
| contents | As shown in our paper [JCTA 177 (2021), Paper No. 105305], the chromatic quasi-symmetric function of Shareshian-Wachs can be lifted to ${\bf WQSym}$, the algebra of quasi-symmetric functions in noncommuting variables. We investigate here its behaviour with respect to classical transformations of alphabets and propose a noncommutative analogue of Macdonald polynomials compatible with a noncommutative version of the Haglund-Wilson formula. We also introduce a multi-$t$ version of these noncommutative analogues. For rectangular partitions, their commutative images at $q=0$ appear to coincide with the multi-$t$ Hall-Littlewood functions introduced in [Lett. Math. Phys. 35 (1995), 359]. This leads us to conjecture that for rectangular partitions, multi-$t$ Macdonald polynomials are obtained as equivariant traces of certain Yang-Baxter elements of Hecke algebras. We also conjecture that all (ordinary) Macdonald polynomials can be obtained in this way. We conclude with some remarks relating various aspects of quasi-symmetric chromatic functions to calculations in Hecke algebras. In particular, we show that all modular relations are given by the product formula of the Kazhdan-Lusztig basis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09072 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation Novelli, Jean-Christophe Thibon, Jean-Yves Combinatorics As shown in our paper [JCTA 177 (2021), Paper No. 105305], the chromatic quasi-symmetric function of Shareshian-Wachs can be lifted to ${\bf WQSym}$, the algebra of quasi-symmetric functions in noncommuting variables. We investigate here its behaviour with respect to classical transformations of alphabets and propose a noncommutative analogue of Macdonald polynomials compatible with a noncommutative version of the Haglund-Wilson formula. We also introduce a multi-$t$ version of these noncommutative analogues. For rectangular partitions, their commutative images at $q=0$ appear to coincide with the multi-$t$ Hall-Littlewood functions introduced in [Lett. Math. Phys. 35 (1995), 359]. This leads us to conjecture that for rectangular partitions, multi-$t$ Macdonald polynomials are obtained as equivariant traces of certain Yang-Baxter elements of Hecke algebras. We also conjecture that all (ordinary) Macdonald polynomials can be obtained in this way. We conclude with some remarks relating various aspects of quasi-symmetric chromatic functions to calculations in Hecke algebras. In particular, we show that all modular relations are given by the product formula of the Kazhdan-Lusztig basis. |
| title | Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.09072 |