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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.01041 |
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| _version_ | 1866913488462413824 |
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| author | Kim, Donghyun Oh, Jaeseong |
| author_facet | Kim, Donghyun Oh, Jaeseong |
| contents | We introduce the Macdonald piece polynomial $\operatorname{I}_{μ,λ,k}[X;q,t]$, which is a vast generalization of the Macdonald intersection polynomial in the science fiction conjecture by Bergeron and Garsia. We demonstrate a remarkable connection between $\operatorname{I}_{μ,λ,k}$, $\nabla s_λ$, and the Loehr--Warrington formula $\operatorname{LW}_λ$, thereby obtaining the Loehr--Warrington conjecture as a corollary. To connect $\operatorname{I}_{μ,λ,k}$ and $\nabla s_λ$, we employ the plethystic formula for the Macdonald polynomials of Garsia--Haiman--Tesler, and to connect $\operatorname{I}_{μ,λ,k}$ and $\operatorname{LW}_λ$, we use our new findings on the combinatorics of $P$-tableaux together with the column exchange rule. We also present an extension of the science fiction conjecture and the Macdonald positivity by exploiting $\operatorname{I}_{μ,λ,k}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01041 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extending the science fiction and the Loehr--Warrington formula Kim, Donghyun Oh, Jaeseong Combinatorics Representation Theory We introduce the Macdonald piece polynomial $\operatorname{I}_{μ,λ,k}[X;q,t]$, which is a vast generalization of the Macdonald intersection polynomial in the science fiction conjecture by Bergeron and Garsia. We demonstrate a remarkable connection between $\operatorname{I}_{μ,λ,k}$, $\nabla s_λ$, and the Loehr--Warrington formula $\operatorname{LW}_λ$, thereby obtaining the Loehr--Warrington conjecture as a corollary. To connect $\operatorname{I}_{μ,λ,k}$ and $\nabla s_λ$, we employ the plethystic formula for the Macdonald polynomials of Garsia--Haiman--Tesler, and to connect $\operatorname{I}_{μ,λ,k}$ and $\operatorname{LW}_λ$, we use our new findings on the combinatorics of $P$-tableaux together with the column exchange rule. We also present an extension of the science fiction conjecture and the Macdonald positivity by exploiting $\operatorname{I}_{μ,λ,k}$. |
| title | Extending the science fiction and the Loehr--Warrington formula |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2409.01041 |