A parking function interpretation for $(-1)^{k}\nabla m_{2^{k}1^{l}}$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908417031929856 |
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| author | Qu, Menghao Xin, Guoce |
| author_facet | Qu, Menghao Xin, Guoce |
| contents | Haglund, Morse, and Zabrocki introduced a family of creation operators of Hall-Littlewood polynomials, $\{C_{a}\}$ for any $a\in \mathbb{Z}$, in their compositional refinement of the shuffle (ex-)conjecture. For any $α\vDash n$, the combinatorial formula for $\nabla C_α$ is a weighted sum of parking functions. These summations can be converted to a weighted sum of certain LLT polynomials. Thus $\nabla C_α$ is Schur positive since Grojnowski and Haiman proved that all LLT polynomials are Schur positive. In this paper, we obtain a recursion that implies the $C$-positivity of $(-1)^{k} m_{2^{k}1^{l}}$, and hence prove the Schur positivity of $(-1)^{k}\nabla m_{2^{k}1^{l}}$. As a corollary, a parking function interpretation for $(-1)^{k}\nabla m_{2^{k}1^{l}}$ is obtained by using the compositional shuffle theorem of Carlsson and Mellit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16824 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A parking function interpretation for $(-1)^{k}\nabla m_{2^{k}1^{l}}$ Qu, Menghao Xin, Guoce Combinatorics Haglund, Morse, and Zabrocki introduced a family of creation operators of Hall-Littlewood polynomials, $\{C_{a}\}$ for any $a\in \mathbb{Z}$, in their compositional refinement of the shuffle (ex-)conjecture. For any $α\vDash n$, the combinatorial formula for $\nabla C_α$ is a weighted sum of parking functions. These summations can be converted to a weighted sum of certain LLT polynomials. Thus $\nabla C_α$ is Schur positive since Grojnowski and Haiman proved that all LLT polynomials are Schur positive. In this paper, we obtain a recursion that implies the $C$-positivity of $(-1)^{k} m_{2^{k}1^{l}}$, and hence prove the Schur positivity of $(-1)^{k}\nabla m_{2^{k}1^{l}}$. As a corollary, a parking function interpretation for $(-1)^{k}\nabla m_{2^{k}1^{l}}$ is obtained by using the compositional shuffle theorem of Carlsson and Mellit. |
| title | A parking function interpretation for $(-1)^{k}\nabla m_{2^{k}1^{l}}$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2312.16824 |